3.9.86 \(\int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx\) [886]

3.9.86.1 Optimal result

Integrand size = 31, antiderivative size = 1551 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \left (64 b^4 e^4 g^4+4 b^2 c e^3 g^3 (7 b e f-66 b d g-69 a e g)+c^4 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )+3 c^2 e^2 g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 e g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{3465 c^4 e g^4}+\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\frac {2 \left (48 b^3 e^3 g^3+b c e^2 g^2 (67 b e f-198 b d g-157 a e g)+c^3 \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )-c^2 e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e f g+99 d^2 g^2\right )\right )\right ) (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{3465 c^3 g^4}-\frac {2 e \left (8 b^2 e^2 g^2+c e g (19 b e f-33 b d g-18 a e g)+c^2 \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt {a+b x+c x^2}}{693 c^2 g^4}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {a+b x+c x^2}}{99 c g^4}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (128 b^5 e^3 g^5-8 b^3 c e^2 g^4 (7 b e f+66 b d g+87 a e g)+2 c^5 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+b c^2 e g^3 \left (771 a^2 e^2 g^2+6 a b e g (43 e f+396 d g)-b^2 \left (37 e^2 f^2-264 d e f g-792 d^2 g^2\right )\right )-c^4 g \left (b f \left (56 e^3 f^3-264 d e^2 f^2 g+495 d^2 e f g^2-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )-c^3 g^2 \left (6 a^2 e^2 g^2 (26 e f+231 d g)-9 a b e g \left (15 e^2 f^2-110 d e f g-319 d^2 g^2\right )+b^2 \left (37 e^3 f^3-198 d e^2 f^2 g+495 d^2 e f g^2+462 d^3 g^3\right )\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3465 c^5 g^5 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (64 b^4 e^3 g^4+4 b^2 c e^2 g^3 (7 b e f-66 b d g-69 a e g)-2 c^4 f \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )+3 c^2 e g^2 \left (50 a^2 e^2 g^2-a b e g (29 e f-297 d g)+3 b^2 \left (e^2 f^2-11 d e f g+44 d^2 g^2\right )\right )-c^3 g \left (6 a e g \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e f g^2+231 d^3 g^3\right )\right )\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{3465 c^5 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]

output

2/3465*(48*b^3*e^3*g^3+b*c*e^2*g^2*(-157*a*e*g-198*b*d*g+67*b*e*f)+c^3*(-5 
67*d^3*g^3+1107*d^2*e*f*g^2-843*d*e^2*f^2*g+233*e^3*f^3)-c^2*e*g*(2*a*e*g* 
(-231*d*g+74*e*f)-3*b*(99*d^2*g^2-88*d*e*f*g+24*e^2*f^2)))*(g*x+f)^(3/2)*( 
c*x^2+b*x+a)^(1/2)/c^3/g^4-2/693*e*(8*b^2*e^2*g^2+c*e*g*(-18*a*e*g-33*b*d* 
g+19*b*e*f)+c^2*(81*d^2*g^2-96*d*e*f*g+29*e^2*f^2))*(g*x+f)^(5/2)*(c*x^2+b 
*x+a)^(1/2)/c^2/g^4+2/99*e^2*(b*e*g-3*c*d*g+c*e*f)*(g*x+f)^(7/2)*(c*x^2+b* 
x+a)^(1/2)/c/g^4-2/3465*(64*b^4*e^4*g^4+4*b^2*c*e^3*g^3*(-69*a*e*g-66*b*d* 
g+7*b*e*f)+c^4*(315*d^4*g^4-798*d^3*e*f*g^3+1098*d^2*e^2*f^2*g^2-732*d*e^3 
*f^3*g+187*e^4*f^4)+3*c^2*e^2*g^2*(50*a^2*e^2*g^2-a*b*e*g*(-297*d*g+29*e*f 
)+3*b^2*(44*d^2*g^2-11*d*e*f*g+e^2*f^2))-c^3*e*g*(6*a*e*g*(165*d^2*g^2-33* 
d*e*f*g+2*e^2*f^2)+b*(231*d^3*g^3-99*d^2*e*f*g^2+8*e^3*f^3)))*(g*x+f)^(1/2 
)*(c*x^2+b*x+a)^(1/2)/c^4/e/g^4+2/11*(e*x+d)^4*(g*x+f)^(1/2)*(c*x^2+b*x+a) 
^(1/2)/e+1/3465*(128*b^5*e^3*g^5-8*b^3*c*e^2*g^4*(87*a*e*g+66*b*d*g+7*b*e* 
f)+2*c^5*f^2*(-231*d^3*g^3+396*d^2*e*f*g^2-264*d*e^2*f^2*g+64*e^3*f^3)+b*c 
^2*e*g^3*(771*a^2*e^2*g^2+6*a*b*e*g*(396*d*g+43*e*f)-b^2*(-792*d^2*g^2-264 
*d*e*f*g+37*e^2*f^2))-c^4*g*(b*f*(-462*d^3*g^3+495*d^2*e*f*g^2-264*d*e^2*f 
^2*g+56*e^3*f^3)-18*a*g*(77*d^3*g^3+88*d^2*e*f*g^2-33*d*e^2*f^2*g+6*e^3*f^ 
3))-c^3*g^2*(6*a^2*e^2*g^2*(231*d*g+26*e*f)-9*a*b*e*g*(-319*d^2*g^2-110*d* 
e*f*g+15*e^2*f^2)+b^2*(462*d^3*g^3+495*d^2*e*f*g^2-198*d*e^2*f^2*g+37*e^3* 
f^3)))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^...
 

3.9.86.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 37.07 (sec) , antiderivative size = 26600, normalized size of antiderivative = 17.15 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\text {Result too large to show} \]

input

Integrate[(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]
 

output

Result too large to show
 

3.9.86.3 Rubi [A] (verified)

Time = 5.94 (sec) , antiderivative size = 1597, normalized size of antiderivative = 1.03, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {1272, 2184, 27, 2184, 27, 2184, 27, 2184, 27, 1269, 1172, 321, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx\)

\(\Big \downarrow \) 1272

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}-\frac {\int \frac {(d+e x)^3 \left (-\left ((c e f-3 c d g+b e g) x^2\right )+2 (c d f-b e f+b d g-a e g) x+b d f-3 a e f+a d g\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{11 e}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}-\frac {\frac {2 \int \frac {e^2 g^4 \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) x^4+e g^3 \left (b g^2 (25 b f+7 a g) e^3-c g \left (2 a e g (10 e f+33 d g)-b \left (58 e^2 f^2-120 d e g f+27 d^2 g^2\right )\right ) e+3 c^2 \left (11 e^3 f^3-33 d e^2 g f^2+9 d^2 e g^2 f+27 d^3 g^3\right )\right ) x^3+3 g^2 \left (b f g^2 (9 b f+7 a g) e^4+c g \left (a e g \left (7 e^2 f^2-48 d e g f-9 d^2 g^2\right )+b \left (14 e^3 f^3-27 d e^2 g f^2-9 d^2 e g^2 f+15 d^3 g^3\right )\right ) e+c^2 \left (5 e^4 f^4-15 d e^3 g f^3+15 d^3 e g^3 f+9 d^4 g^4\right )\right ) x^2+g \left (b f^2 g^2 (11 b f+21 a g) e^4+2 c^2 \left (e^4 f^5-3 d e^3 g f^4+9 d^4 g^4 f\right )+c g \left (3 a e g \left (7 e^3 f^3-21 d e^2 g f^2-27 d^2 e g^2 f+3 d^3 g^3\right )+b \left (13 e^4 f^4-33 d e^3 g f^3+9 d^3 e g^3 f+18 d^4 g^4\right )\right )\right ) x+g \left (7 a b f^3 g^2 e^4+b^2 f^4 g e^4+a c g \left (7 e^4 f^4-21 d e^3 g f^3-27 d^3 e g^3 f+9 d^4 g^4\right )+b c \left (e^4 f^5-3 d e^3 g f^4+9 d^4 g^4 f\right )\right )}{2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{9 c g^5}-\frac {2 e^3 (f+g x)^{7/2} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}-\frac {\frac {\int \frac {e^2 g^4 \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) x^4+e g^3 \left (b g^2 (25 b f+7 a g) e^3-c g \left (2 a e g (10 e f+33 d g)-b \left (58 e^2 f^2-120 d e g f+27 d^2 g^2\right )\right ) e+3 c^2 \left (11 e^3 f^3-33 d e^2 g f^2+9 d^2 e g^2 f+27 d^3 g^3\right )\right ) x^3+3 g^2 \left (b f g^2 (9 b f+7 a g) e^4+c g \left (a e g \left (7 e^2 f^2-48 d e g f-9 d^2 g^2\right )+b \left (14 e^3 f^3-27 d e^2 g f^2-9 d^2 e g^2 f+15 d^3 g^3\right )\right ) e+c^2 \left (5 e^4 f^4-15 d e^3 g f^3+15 d^3 e g^3 f+9 d^4 g^4\right )\right ) x^2+g \left (b f^2 g^2 (11 b f+21 a g) e^4+2 c^2 \left (e^4 f^5-3 d e^3 g f^4+9 d^4 g^4 f\right )+c g \left (3 a e g \left (7 e^3 f^3-21 d e^2 g f^2-27 d^2 e g^2 f+3 d^3 g^3\right )+b \left (13 e^4 f^4-33 d e^3 g f^3+9 d^3 e g^3 f+18 d^4 g^4\right )\right )\right ) x+g \left (7 a b f^3 g^2 e^4+b^2 f^4 g e^4+a c g \left (7 e^4 f^4-21 d e^3 g f^3-27 d^3 e g^3 f+9 d^4 g^4\right )+b c \left (e^4 f^5-3 d e^3 g f^4+9 d^4 g^4 f\right )\right )}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{9 c g^5}-\frac {2 e^3 (f+g x)^{7/2} \sqrt {a+b x+c x^2} (b e g-3 c d g+c e f)}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) \sqrt {c x^2+b x+a} (f+g x)^{5/2}}{7 c}+\frac {2 \int -\frac {e \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) x^3 g^7+\left (8 b^2 g^3 (13 b f+5 a g) e^4-c g^2 \left (-f (146 e f-429 d g) b^2+11 a g (26 e f+15 d g) b+90 a^2 e g^2\right ) e^3-c^2 g \left (2 a e g \left (100 e^2 f^2-264 d e g f-297 d^2 g^2\right )-b \left (292 e^3 f^3-1044 d e^2 g f^2+1242 d^2 e g^2 f-315 d^3 g^3\right )\right ) e+c^3 \left (214 e^4 f^4-741 d e^3 g f^3+891 d^2 e^2 g^2 f^2-315 d^3 e g^3 f-189 d^4 g^4\right )\right ) x^2 g^6+\left (8 b^3 f^3 g^2 e^4+b^2 f^2 g \left (40 a e g^2+3 c f (4 e f-11 d g)\right ) e^3+b c f \left (a f g^2 (28 e f-165 d g) e^3+c \left (22 e^4 f^4-75 d e^3 g f^3+81 d^2 e^2 g^2 f^2-63 d^4 g^4\right )\right )-3 a c g \left (30 a e^4 f^2 g^2-c \left (32 e^4 f^4-111 d e^3 g f^3+135 d^2 e^2 g^2 f^2+63 d^3 e g^3 f-21 d^4 g^4\right )\right )\right ) g^5+\left (16 b^2 f g^3 (4 b f+5 a g) e^4-c f g^2 \left (-f (91 e f-264 d g) b^2+a g (101 e f+330 d g) b+180 a^2 e g^2\right ) e^3+2 c^3 \left (22 e^4 f^5-75 d e^3 g f^4+81 d^2 e^2 g^2 f^3-63 d^4 g^4 f\right )+c^2 g \left (a e g \left (107 e^3 f^3-519 d e^2 g f^2+1377 d^2 e g^2 f-63 d^3 g^3\right )+b \left (179 e^4 f^4-603 d e^3 g f^3+648 d^2 e^2 g^2 f^2-63 d^3 e g^3 f-126 d^4 g^4\right )\right )\right ) x g^5}{2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\int \frac {e \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) x^3 g^7+\left (8 b^2 g^3 (13 b f+5 a g) e^4-c g^2 \left (-f (146 e f-429 d g) b^2+11 a g (26 e f+15 d g) b+90 a^2 e g^2\right ) e^3-c^2 g \left (2 a e g \left (100 e^2 f^2-264 d e g f-297 d^2 g^2\right )-b \left (292 e^3 f^3-1044 d e^2 g f^2+1242 d^2 e g^2 f-315 d^3 g^3\right )\right ) e+c^3 \left (214 e^4 f^4-741 d e^3 g f^3+891 d^2 e^2 g^2 f^2-315 d^3 e g^3 f-189 d^4 g^4\right )\right ) x^2 g^6+\left (8 b^3 f^3 g^2 e^4+b^2 f^2 g \left (40 a e g^2+3 c f (4 e f-11 d g)\right ) e^3+b c f \left (a f g^2 (28 e f-165 d g) e^3+c \left (22 e^4 f^4-75 d e^3 g f^3+81 d^2 e^2 g^2 f^2-63 d^4 g^4\right )\right )-3 a c g \left (30 a e^4 f^2 g^2-c \left (32 e^4 f^4-111 d e^3 g f^3+135 d^2 e^2 g^2 f^2+63 d^3 e g^3 f-21 d^4 g^4\right )\right )\right ) g^5+\left (16 b^2 f g^3 (4 b f+5 a g) e^4-c f g^2 \left (-f (91 e f-264 d g) b^2+a g (101 e f+330 d g) b+180 a^2 e g^2\right ) e^3+2 c^3 \left (22 e^4 f^5-75 d e^3 g f^4+81 d^2 e^2 g^2 f^3-63 d^4 g^4 f\right )+c^2 g \left (a e g \left (107 e^3 f^3-519 d e^2 g f^2+1377 d^2 e g^2 f-63 d^3 g^3\right )+b \left (179 e^4 f^4-603 d e^3 g f^3+648 d^2 e^2 g^2 f^2-63 d^3 e g^3 f-126 d^4 g^4\right )\right )\right ) x g^5}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\frac {2 e \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a} g^5}{5 c}+\frac {2 \int -\frac {3 \left (\left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) x^2 g^9+\left (16 b^4 f^2 g^3 e^4+3 b^3 f g^2 \left (16 a e g^2+c f (3 e f-22 d g)\right ) e^3-b^2 c f g \left (2 a e g^2 (26 e f+99 d g)-c f \left (4 e^2 f^2-33 d e g f+99 d^2 g^2\right )\right ) e^2+a c^2 g \left (2 a f g^2 (e f+231 d g) e^3+c \left (73 e^4 f^4-288 d e^3 g f^3+432 d^2 e^2 g^2 f^2-882 d^3 e g^3 f+105 d^4 g^4\right )\right )-b c f \left (157 a^2 e^4 g^4+3 a c e^2 \left (8 e^2 f^2-55 d e g f-99 d^2 g^2\right ) g^2-c^2 \left (41 e^4 f^4-156 d e^3 g f^3+234 d^2 e^2 g^2 f^2-189 d^3 e g^3 f+105 d^4 g^4\right )\right )\right ) g^8+\left (2 f \left (41 e^4 f^4-156 d e^3 g f^3+234 d^2 e^2 g^2 f^2-189 d^3 e g^3 f+105 d^4 g^4\right ) c^4-g \left (22 a e g \left (2 e^3 f^3-15 d e^2 g f^2+54 d^2 e g^2 f+21 d^3 g^3\right )-3 b \left (46 e^4 f^4-192 d e^3 g f^3+321 d^2 e^2 g^2 f^2-280 d^3 e g^3 f+70 d^4 g^4\right )\right ) c^3+e^2 g^2 \left (f \left (13 e^2 f^2-132 d e g f+495 d^2 g^2\right ) b^2-3 a g \left (37 e^2 f^2-352 d e g f-99 d^2 g^2\right ) b+2 a^2 e g^2 (76 e f+231 d g)\right ) c^2-b e^3 g^3 \left (-f (37 e f-330 d g) b^2+2 a g (164 e f+99 d g) b+157 a^2 e g^2\right ) c+16 b^3 e^4 g^4 (5 b f+3 a g)\right ) x g^8\right )}{2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{5 c g^3}}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\frac {2 e g^5 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{5 c}-\frac {3 \int \frac {\left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) x^2 g^9+\left (16 b^4 f^2 g^3 e^4+3 b^3 f g^2 \left (16 a e g^2+c f (3 e f-22 d g)\right ) e^3-b^2 c f g \left (2 a e g^2 (26 e f+99 d g)-c f \left (4 e^2 f^2-33 d e g f+99 d^2 g^2\right )\right ) e^2+a c^2 g \left (2 a f g^2 (e f+231 d g) e^3+c \left (73 e^4 f^4-288 d e^3 g f^3+432 d^2 e^2 g^2 f^2-882 d^3 e g^3 f+105 d^4 g^4\right )\right )-b c f \left (157 a^2 e^4 g^4+3 a c e^2 \left (8 e^2 f^2-55 d e g f-99 d^2 g^2\right ) g^2-c^2 \left (41 e^4 f^4-156 d e^3 g f^3+234 d^2 e^2 g^2 f^2-189 d^3 e g^3 f+105 d^4 g^4\right )\right )\right ) g^8+\left (2 f \left (41 e^4 f^4-156 d e^3 g f^3+234 d^2 e^2 g^2 f^2-189 d^3 e g^3 f+105 d^4 g^4\right ) c^4-g \left (22 a e g \left (2 e^3 f^3-15 d e^2 g f^2+54 d^2 e g^2 f+21 d^3 g^3\right )-3 b \left (46 e^4 f^4-192 d e^3 g f^3+321 d^2 e^2 g^2 f^2-280 d^3 e g^3 f+70 d^4 g^4\right )\right ) c^3+e^2 g^2 \left (f \left (13 e^2 f^2-132 d e g f+495 d^2 g^2\right ) b^2-3 a g \left (37 e^2 f^2-352 d e g f-99 d^2 g^2\right ) b+2 a^2 e g^2 (76 e f+231 d g)\right ) c^2-b e^3 g^3 \left (-f (37 e f-330 d g) b^2+2 a g (164 e f+99 d g) b+157 a^2 e g^2\right ) c+16 b^3 e^4 g^4 (5 b f+3 a g)\right ) x g^8}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{5 c g^3}}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\frac {2 e g^5 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{5 c}-\frac {3 \left (\frac {2 \left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a} g^8}{3 c}+\frac {2 \int -\frac {e g^{10} \left (64 e^3 f g^4 b^5+4 e^2 g^3 \left (16 a e g^2-c f (5 e f+66 d g)\right ) b^4-c e g^2 \left (8 a e (49 e f+33 d g) g^2+9 c f \left (2 e^2 f^2-11 d e g f-44 d^2 g^2\right )\right ) b^3-c g \left (276 a^2 e^3 g^4-6 a c e \left (13 e^2 f^2+231 d e g f+66 d^2 g^2\right ) g^2+c^2 f \left (20 e^3 f^3-99 d e^2 g f^2+198 d^2 e g^2 f+231 d^3 g^3\right )\right ) b^2+c^2 \left (3 a^2 e^2 (178 e f+297 d g) g^4+a c \left (52 e^3 f^3-297 d e^2 g f^2-1782 d^2 e g^2 f-231 d^3 g^3\right ) g^2+c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) b+2 a c^2 g \left (75 a^2 e^3 g^4-9 a c e \left (e^2 f^2+66 d e g f+55 d^2 g^2\right ) g^2-c^2 f \left (16 e^3 f^3-66 d e^2 g f^2+99 d^2 e g^2 f-924 d^3 g^3\right )\right )+\left (2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^5-g \left (b f \left (56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right )\right ) c^4-g^2 \left (\left (37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right ) b^2-9 a e g \left (15 e^2 f^2-110 d e g f-319 d^2 g^2\right ) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right ) c^3+b e g^3 \left (-\left (\left (37 e^2 f^2-264 d e g f-792 d^2 g^2\right ) b^2\right )+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right ) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\right ) x\right )}{2 \sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{3 c g^2}\right )}{5 c g^3}}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\frac {2 e g^5 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{5 c}-\frac {3 \left (\frac {2 g^8 \left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{3 c}-\frac {e g^8 \int \frac {64 e^3 f g^4 b^5+4 e^2 g^3 \left (16 a e g^2-c f (5 e f+66 d g)\right ) b^4-c e g^2 \left (8 a e (49 e f+33 d g) g^2+9 c f \left (2 e^2 f^2-11 d e g f-44 d^2 g^2\right )\right ) b^3-c g \left (276 a^2 e^3 g^4-6 a c e \left (13 e^2 f^2+231 d e g f+66 d^2 g^2\right ) g^2+c^2 f \left (20 e^3 f^3-99 d e^2 g f^2+198 d^2 e g^2 f+231 d^3 g^3\right )\right ) b^2+c^2 \left (3 a^2 e^2 (178 e f+297 d g) g^4+a c \left (52 e^3 f^3-297 d e^2 g f^2-1782 d^2 e g^2 f-231 d^3 g^3\right ) g^2+c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) b+2 a c^2 g \left (75 a^2 e^3 g^4-9 a c e \left (e^2 f^2+66 d e g f+55 d^2 g^2\right ) g^2-c^2 f \left (16 e^3 f^3-66 d e^2 g f^2+99 d^2 e g^2 f-924 d^3 g^3\right )\right )+\left (2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^5-g \left (b f \left (56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right )\right ) c^4-g^2 \left (\left (37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right ) b^2-9 a e g \left (15 e^2 f^2-110 d e g f-319 d^2 g^2\right ) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right ) c^3+b e g^3 \left (-\left (\left (37 e^2 f^2-264 d e g f-792 d^2 g^2\right ) b^2\right )+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right ) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\right ) x}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{3 c}\right )}{5 c g^3}}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\frac {2 e g^5 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{5 c}-\frac {3 \left (\frac {2 g^8 \left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{3 c}-\frac {e g^8 \left (\frac {\left (c f^2-b g f+a g^2\right ) \left (-2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^4-g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^2 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^3 g^4\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {c x^2+b x+a}}dx}{g}+\frac {\left (2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^5-g \left (b f \left (56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right )\right ) c^4-g^2 \left (\left (37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right ) b^2-9 a e g \left (15 e^2 f^2-110 d e g f-319 d^2 g^2\right ) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right ) c^3+b e g^3 \left (-\left (\left (37 e^2 f^2-264 d e g f-792 d^2 g^2\right ) b^2\right )+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right ) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\right ) \int \frac {\sqrt {f+g x}}{\sqrt {c x^2+b x+a}}dx}{g}\right )}{3 c}\right )}{5 c g^3}}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\frac {2 e g^5 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{5 c}-\frac {3 \left (\frac {2 g^8 \left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{3 c}-\frac {e g^8 \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (-2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^4-g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^2 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^3 g^4\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^5-g \left (b f \left (56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right )\right ) c^4-g^2 \left (\left (37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right ) b^2-9 a e g \left (15 e^2 f^2-110 d e g f-319 d^2 g^2\right ) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right ) c^3+b e g^3 \left (-\left (\left (37 e^2 f^2-264 d e g f-792 d^2 g^2\right ) b^2\right )+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right ) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}\right )}{3 c}\right )}{5 c g^3}}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\frac {2 e g^5 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{5 c}-\frac {3 \left (\frac {2 g^8 \left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{3 c}-\frac {e g^8 \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (-2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^4-g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^2 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^3 g^4\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^5-g \left (b f \left (56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right )\right ) c^4-g^2 \left (\left (37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right ) b^2-9 a e g \left (15 e^2 f^2-110 d e g f-319 d^2 g^2\right ) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right ) c^3+b e g^3 \left (-\left (\left (37 e^2 f^2-264 d e g f-792 d^2 g^2\right ) b^2\right )+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right ) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {g \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}\right )}{3 c}\right )}{5 c g^3}}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {c x^2+b x+a}}{11 e}-\frac {\frac {\frac {2 e^2 g \left (\left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right ) c^2+e g (19 b e f-33 b d g-18 a e g) c+8 b^2 e^2 g^2\right ) (f+g x)^{5/2} \sqrt {c x^2+b x+a}}{7 c}-\frac {\frac {2 e g^5 \left (\left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right ) c^3-e g \left (2 a e g (74 e f-231 d g)-3 b \left (24 e^2 f^2-88 d e g f+99 d^2 g^2\right )\right ) c^2+b e^2 g^2 (67 b e f-198 b d g-157 a e g) c+48 b^3 e^3 g^3\right ) (f+g x)^{3/2} \sqrt {c x^2+b x+a}}{5 c}-\frac {3 \left (\frac {2 g^8 \left (\left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right ) c^4-e g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e^2 g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^3 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^4 g^4\right ) \sqrt {f+g x} \sqrt {c x^2+b x+a}}{3 c}-\frac {e g^8 \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^5-g \left (b f \left (56 e^3 f^3-264 d e^2 g f^2+495 d^2 e g^2 f-462 d^3 g^3\right )-18 a g \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right )\right ) c^4-g^2 \left (\left (37 e^3 f^3-198 d e^2 g f^2+495 d^2 e g^2 f+462 d^3 g^3\right ) b^2-9 a e g \left (15 e^2 f^2-110 d e g f-319 d^2 g^2\right ) b+6 a^2 e^2 g^2 (26 e f+231 d g)\right ) c^3+b e g^3 \left (-\left (\left (37 e^2 f^2-264 d e g f-792 d^2 g^2\right ) b^2\right )+6 a e g (43 e f+396 d g) b+771 a^2 e^2 g^2\right ) c^2-8 b^3 e^2 g^4 (7 b e f+66 b d g+87 a e g) c+128 b^5 e^3 g^5\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {c x^2+b x+a}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b g f+a g^2\right ) \left (-2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right ) c^4-g \left (6 a e g \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right )+b \left (8 e^3 f^3-99 d^2 e g^2 f+231 d^3 g^3\right )\right ) c^3+3 e g^2 \left (3 \left (e^2 f^2-11 d e g f+44 d^2 g^2\right ) b^2-a e g (29 e f-297 d g) b+50 a^2 e^2 g^2\right ) c^2+4 b^2 e^2 g^3 (7 b e f-66 b d g-69 a e g) c+64 b^4 e^3 g^4\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{c g \sqrt {f+g x} \sqrt {c x^2+b x+a}}\right )}{3 c}\right )}{5 c g^3}}{7 c g^4}}{9 c g^5}-\frac {2 e^3 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{9 c g^4}}{11 e}\)

input

Int[(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]
 

output

(2*(d + e*x)^4*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(11*e) - ((-2*e^3*(c*e 
*f - 3*c*d*g + b*e*g)*(f + g*x)^(7/2)*Sqrt[a + b*x + c*x^2])/(9*c*g^4) + ( 
(2*e^2*g*(8*b^2*e^2*g^2 + c*e*g*(19*b*e*f - 33*b*d*g - 18*a*e*g) + c^2*(29 
*e^2*f^2 - 96*d*e*f*g + 81*d^2*g^2))*(f + g*x)^(5/2)*Sqrt[a + b*x + c*x^2] 
)/(7*c) - ((2*e*g^5*(48*b^3*e^3*g^3 + b*c*e^2*g^2*(67*b*e*f - 198*b*d*g - 
157*a*e*g) + c^3*(233*e^3*f^3 - 843*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d 
^3*g^3) - c^2*e*g*(2*a*e*g*(74*e*f - 231*d*g) - 3*b*(24*e^2*f^2 - 88*d*e*f 
*g + 99*d^2*g^2)))*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5*c) - (3*((2*g 
^8*(64*b^4*e^4*g^4 + 4*b^2*c*e^3*g^3*(7*b*e*f - 66*b*d*g - 69*a*e*g) + c^4 
*(187*e^4*f^4 - 732*d*e^3*f^3*g + 1098*d^2*e^2*f^2*g^2 - 798*d^3*e*f*g^3 + 
 315*d^4*g^4) + 3*c^2*e^2*g^2*(50*a^2*e^2*g^2 - a*b*e*g*(29*e*f - 297*d*g) 
 + 3*b^2*(e^2*f^2 - 11*d*e*f*g + 44*d^2*g^2)) - c^3*e*g*(6*a*e*g*(2*e^2*f^ 
2 - 33*d*e*f*g + 165*d^2*g^2) + b*(8*e^3*f^3 - 99*d^2*e*f*g^2 + 231*d^3*g^ 
3)))*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(3*c) - (e*g^8*((Sqrt[2]*Sqrt[b^ 
2 - 4*a*c]*(128*b^5*e^3*g^5 - 8*b^3*c*e^2*g^4*(7*b*e*f + 66*b*d*g + 87*a*e 
*g) + 2*c^5*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3* 
g^3) + b*c^2*e*g^3*(771*a^2*e^2*g^2 + 6*a*b*e*g*(43*e*f + 396*d*g) - b^2*( 
37*e^2*f^2 - 264*d*e*f*g - 792*d^2*g^2)) - c^4*g*(b*f*(56*e^3*f^3 - 264*d* 
e^2*f^2*g + 495*d^2*e*f*g^2 - 462*d^3*g^3) - 18*a*g*(6*e^3*f^3 - 33*d*e^2* 
f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3)) - c^3*g^2*(6*a^2*e^2*g^2*(26*e*f ...
 

3.9.86.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1272

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*(d + e*x)^(m + 1)*Sqrt[f + g*x]*( 
Sqrt[a + b*x + c*x^2]/(e*(2*m + 5))), x] - Simp[1/(e*(2*m + 5))   Int[((d + 
 e*x)^m/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]))*Simp[b*d*f - 3*a*e*f + a*d*g 
 + 2*(c*d*f - b*e*f + b*d*g - a*e*g)*x - (c*e*f - 3*c*d*g + b*e*g)*x^2, x], 
 x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && IntegerQ[2*m] &&  !LtQ[m, 
-1]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

3.9.86.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3253\) vs. \(2(1469)=2938\).

Time = 3.12 (sec) , antiderivative size = 3254, normalized size of antiderivative = 2.10

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3254\)
risch	\(\text {Expression too large to display}\)	\(11966\)
default	\(\text {Expression too large to display}\)	\(32647\)

input

int((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

output

((g*x+f)*(c*x^2+b*x+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/11*e^3* 
x^4*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)+2/9*(b*e^3*g+3*c*d*e^2 
*g+f*c*e^3-2/11*e^3*(5*b*g+5*c*f))/c/g*x^3*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+ 
b*f*x+a*f)^(1/2)+2/7*(a*e^3*g+3*b*d*e^2*g+b*e^3*f+3*c*d^2*e*g+3*c*d*e^2*f- 
2/11*e^3*(9/2*a*g+9/2*b*f)-2/9*(b*e^3*g+3*c*d*e^2*g+f*c*e^3-2/11*e^3*(5*b* 
g+5*c*f))/c/g*(4*b*g+4*c*f))/c/g*x^2*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+ 
a*f)^(1/2)+2/5*(3*a*e^2*g*d+3/11*a*e^3*f+3*b*d^2*e*g+3*b*e^2*f*d+c*d^3*g+3 
*c*d^2*e*f-2/9*(b*e^3*g+3*c*d*e^2*g+f*c*e^3-2/11*e^3*(5*b*g+5*c*f))/c/g*(7 
/2*a*g+7/2*b*f)-2/7*(a*e^3*g+3*b*d*e^2*g+b*e^3*f+3*c*d^2*e*g+3*c*d*e^2*f-2 
/11*e^3*(9/2*a*g+9/2*b*f)-2/9*(b*e^3*g+3*c*d*e^2*g+f*c*e^3-2/11*e^3*(5*b*g 
+5*c*f))/c/g*(4*b*g+4*c*f))/c/g*(3*b*g+3*c*f))/c/g*x*(c*g*x^3+b*g*x^2+c*f* 
x^2+a*g*x+b*f*x+a*f)^(1/2)+2/3*(3*a*d^2*e*g+3*a*d*e^2*f+b*d^3*g+3*b*d^2*e* 
f+c*d^3*f-2/3*(b*e^3*g+3*c*d*e^2*g+f*c*e^3-2/11*e^3*(5*b*g+5*c*f))/c/g*f*a 
-2/7*(a*e^3*g+3*b*d*e^2*g+b*e^3*f+3*c*d^2*e*g+3*c*d*e^2*f-2/11*e^3*(9/2*a* 
g+9/2*b*f)-2/9*(b*e^3*g+3*c*d*e^2*g+f*c*e^3-2/11*e^3*(5*b*g+5*c*f))/c/g*(4 
*b*g+4*c*f))/c/g*(5/2*a*g+5/2*b*f)-2/5*(3*a*e^2*g*d+3/11*a*e^3*f+3*b*d^2*e 
*g+3*b*e^2*f*d+c*d^3*g+3*c*d^2*e*f-2/9*(b*e^3*g+3*c*d*e^2*g+f*c*e^3-2/11*e 
^3*(5*b*g+5*c*f))/c/g*(7/2*a*g+7/2*b*f)-2/7*(a*e^3*g+3*b*d*e^2*g+b*e^3*f+3 
*c*d^2*e*g+3*c*d*e^2*f-2/11*e^3*(9/2*a*g+9/2*b*f)-2/9*(b*e^3*g+3*c*d*e^2*g 
+f*c*e^3-2/11*e^3*(5*b*g+5*c*f))/c/g*(4*b*g+4*c*f))/c/g*(3*b*g+3*c*f))/...
 

3.9.86.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.15 (sec) , antiderivative size = 1741, normalized size of antiderivative = 1.12 \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\text {Too large to display} \]

input

integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas 
")
 

output

-2/10395*((128*c^6*e^3*f^6 - 24*(22*c^6*d*e^2 + 5*b*c^5*e^3)*f^5*g + 3*(26 
4*c^6*d^2*e + 176*b*c^5*d*e^2 - (11*b^2*c^4 - 68*a*c^5)*e^3)*f^4*g^2 - (46 
2*c^6*d^3 + 891*b*c^5*d^2*e - 165*(b^2*c^4 - 6*a*c^5)*d*e^2 + (20*b^3*c^3 
- 87*a*b*c^4)*e^3)*f^3*g^3 + 3*(231*b*c^5*d^3 - 66*(2*b^2*c^4 - 11*a*c^5)* 
d^2*e + 11*(5*b^3*c^3 - 21*a*b*c^4)*d*e^2 - (11*b^4*c^2 - 53*a*b^2*c^3 + 3 
4*a^2*c^4)*e^3)*f^2*g^4 + 3*(231*(b^2*c^4 - 6*a*c^5)*d^3 - 33*(9*b^3*c^3 - 
 41*a*b*c^4)*d^2*e + 22*(8*b^4*c^2 - 42*a*b^2*c^3 + 33*a^2*c^4)*d*e^2 - (4 
0*b^5*c - 246*a*b^3*c^2 + 329*a^2*b*c^3)*e^3)*f*g^5 - (231*(2*b^3*c^3 - 9* 
a*b*c^4)*d^3 - 99*(8*b^4*c^2 - 41*a*b^2*c^3 + 30*a^2*c^4)*d^2*e + 33*(16*b 
^5*c - 96*a*b^3*c^2 + 123*a^2*b*c^3)*d*e^2 - (128*b^6 - 888*a*b^4*c + 1599 
*a^2*b^2*c^2 - 450*a^3*c^3)*e^3)*g^6)*sqrt(c*g)*weierstrassPInverse(4/3*(c 
^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^ 
2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), 1/ 
3*(3*c*g*x + c*f + b*g)/(c*g)) + 3*(128*c^6*e^3*f^5*g - 8*(66*c^6*d*e^2 + 
7*b*c^5*e^3)*f^4*g^2 + (792*c^6*d^2*e + 264*b*c^5*d*e^2 - (37*b^2*c^4 - 10 
8*a*c^5)*e^3)*f^3*g^3 - (462*c^6*d^3 + 495*b*c^5*d^2*e - 198*(b^2*c^4 - 3* 
a*c^5)*d*e^2 + (37*b^3*c^3 - 135*a*b*c^4)*e^3)*f^2*g^4 + (462*b*c^5*d^3 - 
99*(5*b^2*c^4 - 16*a*c^5)*d^2*e + 66*(4*b^3*c^3 - 15*a*b*c^4)*d*e^2 - 2*(2 
8*b^4*c^2 - 129*a*b^2*c^3 + 78*a^2*c^4)*e^3)*f*g^5 - (462*(b^2*c^4 - 3*a*c 
^5)*d^3 - 99*(8*b^3*c^3 - 29*a*b*c^4)*d^2*e + 66*(8*b^4*c^2 - 36*a*b^2*...
 

3.9.86.6 Sympy [F]

\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\int \left (d + e x\right )^{3} \sqrt {f + g x} \sqrt {a + b x + c x^{2}}\, dx \]

input

integrate((e*x+d)**3*(g*x+f)**(1/2)*(c*x**2+b*x+a)**(1/2),x)
 

output

Integral((d + e*x)**3*sqrt(f + g*x)*sqrt(a + b*x + c*x**2), x)
 

3.9.86.7 Maxima [F]

\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{3} \sqrt {g x + f} \,d x } \]

input

integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima 
")
 

output

integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^3*sqrt(g*x + f), x)
 

3.9.86.8 Giac [F]

\[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\int { \sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{3} \sqrt {g x + f} \,d x } \]

input

integrate((e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

output

integrate(sqrt(c*x^2 + b*x + a)*(e*x + d)^3*sqrt(g*x + f), x)
 

3.9.86.9 Mupad [F(-1)]

Timed out. \[ \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\int \sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a} \,d x \]

input

int((f + g*x)^(1/2)*(d + e*x)^3*(a + b*x + c*x^2)^(1/2),x)
 

output

int((f + g*x)^(1/2)*(d + e*x)^3*(a + b*x + c*x^2)^(1/2), x)