3.9.54 \(\int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx\) [854]

3.9.54.1 Optimal result

Integrand size = 29, antiderivative size = 532 \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {\left (5 b^3 e^3 g^3+64 c^3 (e f-d g)^3-4 b c e^2 g^2 (6 b e f-2 b d g+a e g)+16 b c^2 e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )+2 c e g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right ) x\right ) \sqrt {a+b x+c x^2}}{64 c^3 e^4}+\frac {g^2 (24 c e f-14 c d g-5 b e g) \left (a+b x+c x^2\right )^{3/2}}{24 c^2 e^2}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}-\frac {\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (5 b^2 e^2 g^2-4 c e g (6 b e f-2 b d g+a e g)+16 c^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right )}{128 c^{7/2} e^5}+\frac {\sqrt {c d^2-b d e+a e^2} (e f-d g)^3 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{e^5} \]

output

1/24*g^2*(-5*b*e*g-14*c*d*g+24*c*e*f)*(c*x^2+b*x+a)^(3/2)/c^2/e^2+1/4*g^3* 
(e*x+d)*(c*x^2+b*x+a)^(3/2)/c/e^2-1/128*(4*c*e*(-b*e+2*c*d)*(16*c^2*e^2*f^ 
3+5*b^2*d*e*g^3-4*c*d*g^2*(a*e*g-2*b*d*g+6*b*e*f))-2*(4*c^2*d^2-1/2*b^2*e^ 
2-2*c*e*(-a*e+b*d))*g*(5*b^2*e^2*g^2-4*c*e*g*(a*e*g-2*b*d*g+6*b*e*f)+16*c^ 
2*(d^2*g^2-3*d*e*f*g+3*e^2*f^2)))*arctanh(1/2*(2*c*x+b)/c^(1/2)/(c*x^2+b*x 
+a)^(1/2))/c^(7/2)/e^5+(-d*g+e*f)^3*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x) 
/(a*e^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))*(a*e^2-b*d*e+c*d^2)^(1/2)/ 
e^5+1/64*(5*b^3*e^3*g^3+64*c^3*(-d*g+e*f)^3-4*b*c*e^2*g^2*(a*e*g-2*b*d*g+6 
*b*e*f)+16*b*c^2*e*g*(d^2*g^2-3*d*e*f*g+3*e^2*f^2)+2*c*e*g*(5*b^2*e^2*g^2- 
4*c*e*g*(a*e*g-2*b*d*g+6*b*e*f)+16*c^2*(d^2*g^2-3*d*e*f*g+3*e^2*f^2))*x)*( 
c*x^2+b*x+a)^(1/2)/c^3/e^4
 

3.9.54.2 Mathematica [A] (verified)

Time = 3.04 (sec) , antiderivative size = 513, normalized size of antiderivative = 0.96 \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\frac {\frac {2 e \sqrt {a+x (b+c x)} \left (15 b^3 e^3 g^3-2 b c e^2 g^2 (26 a e g+b (36 e f-12 d g+5 e g x))+16 c^3 \left (-12 d^3 g^3+6 d^2 e g^2 (6 f+g x)-2 d e^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )+3 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )\right )+8 c^2 e g \left (a e g (-8 d g+3 e (8 f+g x))+b \left (6 d^2 g^2-2 d e g (9 f+g x)+e^2 \left (18 f^2+6 f g x+g^2 x^2\right )\right )\right )\right )}{c^3}-768 \sqrt {-c d^2+b d e-a e^2} (-e f+d g)^3 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )-\frac {3 \left (-5 b^4 e^4 g^3+128 c^4 d (-e f+d g)^3+8 b^2 c e^3 g^2 (3 b e f-b d g+3 a e g)-16 c^2 e^2 g \left (a^2 e^2 g^2+2 a b e g (3 e f-d g)+b^2 \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )+64 c^3 e \left (b (e f-d g)^3+a e g \left (3 e^2 f^2-3 d e f g+d^2 g^2\right )\right )\right ) \log \left (b+2 c x-2 \sqrt {c} \sqrt {a+x (b+c x)}\right )}{c^{7/2}}}{384 e^5} \]

input

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

output

((2*e*Sqrt[a + x*(b + c*x)]*(15*b^3*e^3*g^3 - 2*b*c*e^2*g^2*(26*a*e*g + b* 
(36*e*f - 12*d*g + 5*e*g*x)) + 16*c^3*(-12*d^3*g^3 + 6*d^2*e*g^2*(6*f + g* 
x) - 2*d*e^2*g*(18*f^2 + 9*f*g*x + 2*g^2*x^2) + 3*e^3*(4*f^3 + 6*f^2*g*x + 
 4*f*g^2*x^2 + g^3*x^3)) + 8*c^2*e*g*(a*e*g*(-8*d*g + 3*e*(8*f + g*x)) + b 
*(6*d^2*g^2 - 2*d*e*g*(9*f + g*x) + e^2*(18*f^2 + 6*f*g*x + g^2*x^2)))))/c 
^3 - 768*Sqrt[-(c*d^2) + b*d*e - a*e^2]*(-(e*f) + d*g)^3*ArcTan[(Sqrt[c]*( 
d + e*x) - e*Sqrt[a + x*(b + c*x)])/Sqrt[-(c*d^2) + e*(b*d - a*e)]] - (3*( 
-5*b^4*e^4*g^3 + 128*c^4*d*(-(e*f) + d*g)^3 + 8*b^2*c*e^3*g^2*(3*b*e*f - b 
*d*g + 3*a*e*g) - 16*c^2*e^2*g*(a^2*e^2*g^2 + 2*a*b*e*g*(3*e*f - d*g) + b^ 
2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)) + 64*c^3*e*(b*(e*f - d*g)^3 + a*e*g*( 
3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)))*Log[b + 2*c*x - 2*Sqrt[c]*Sqrt[a + x*(b 
 + c*x)]])/c^(7/2))/(384*e^5)
 

3.9.54.3 Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 562, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {1267, 27, 2184, 27, 1231, 27, 1269, 1092, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1267

\(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a} \left (e^2 g^2 (24 c e f-14 c d g-5 b e g) x^2-2 e g \left (e (4 b d+a e) g^2-3 c \left (4 e^2 f^2-d^2 g^2\right )\right ) x+e \left (8 c e^2 f^3-d (3 b d+2 a e) g^3\right )\right )}{2 (d+e x)}dx}{4 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {c x^2+b x+a} \left (e^2 g^2 (24 c e f-14 c d g-5 b e g) x^2-2 e g \left (e (4 b d+a e) g^2-3 c \left (4 e^2 f^2-d^2 g^2\right )\right ) x+e \left (8 c e^2 f^3-d (3 b d+2 a e) g^3\right )\right )}{d+e x}dx}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {\frac {\int \frac {3 e^3 \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)+g \left (16 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (6 b e f-2 b d g+a e g) c+5 b^2 e^2 g^2\right ) x\right ) \sqrt {c x^2+b x+a}}{2 (d+e x)}dx}{3 c e^2}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e \int \frac {\left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)+g \left (16 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (6 b e f-2 b d g+a e g) c+5 b^2 e^2 g^2\right ) x\right ) \sqrt {c x^2+b x+a}}{d+e x}dx}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{4 c e^2}-\frac {\int \frac {4 c e (b d-2 a e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-d \left (-e b^2+4 c d b-4 a c e\right ) g \left (16 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (6 b e f-2 b d g+a e g) c+5 b^2 e^2 g^2\right )+\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (16 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (6 b e f-2 b d g+a e g) c+5 b^2 e^2 g^2\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x+a}}dx}{4 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{4 c e^2}-\frac {\int \frac {4 c e (b d-2 a e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-d \left (-e b^2+4 c d b-4 a c e\right ) g \left (16 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (6 b e f-2 b d g+a e g) c+5 b^2 e^2 g^2\right )+\left (4 c e (2 c d-b e) \left (16 c^2 e^2 f^3+5 b^2 d e g^3-4 c d g^2 (6 b e f-2 b d g+a e g)\right )-2 \left (4 c^2 d^2-\frac {b^2 e^2}{2}-2 c e (b d-a e)\right ) g \left (16 \left (3 e^2 f^2-3 d e g f+d^2 g^2\right ) c^2-4 e g (6 b e f-2 b d g+a e g) c+5 b^2 e^2 g^2\right )\right ) x}{(d+e x) \sqrt {c x^2+b x+a}}dx}{8 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{4 c e^2}-\frac {\frac {\left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac {b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {128 c^3 (e f-d g)^3 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 1092

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{4 c e^2}-\frac {\frac {2 \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac {b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right ) \int \frac {1}{4 c-\frac {(b+2 c x)^2}{c x^2+b x+a}}d\frac {b+2 c x}{\sqrt {c x^2+b x+a}}}{e}-\frac {128 c^3 (e f-d g)^3 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac {b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right )}{\sqrt {c} e}-\frac {128 c^3 (e f-d g)^3 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{e}}{8 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{4 c e^2}-\frac {\frac {256 c^3 (e f-d g)^3 \left (a e^2-b d e+c d^2\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{e}+\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac {b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right )}{\sqrt {c} e}}{8 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{4 c e^2}-\frac {\frac {\text {arctanh}\left (\frac {b+2 c x}{2 \sqrt {c} \sqrt {a+b x+c x^2}}\right ) \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac {b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right )}{\sqrt {c} e}-\frac {128 c^3 (e f-d g)^3 \sqrt {a e^2-b d e+c d^2} \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{e}}{8 c e^2}\right )}{2 c}+\frac {e g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{3 c}}{8 c e^3}+\frac {g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2}\)

input

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

output

(g^3*(d + e*x)*(a + b*x + c*x^2)^(3/2))/(4*c*e^2) + ((e*g^2*(24*c*e*f - 14 
*c*d*g - 5*b*e*g)*(a + b*x + c*x^2)^(3/2))/(3*c) + (e*(((5*b^3*e^3*g^3 + 6 
4*c^3*(e*f - d*g)^3 - 4*b*c*e^2*g^2*(6*b*e*f - 2*b*d*g + a*e*g) + 16*b*c^2 
*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) + 2*c*e*g*(5*b^2*e^2*g^2 - 4*c*e*g* 
(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*x) 
*Sqrt[a + b*x + c*x^2])/(4*c*e^2) - (((4*c*e*(2*c*d - b*e)*(16*c^2*e^2*f^3 
 + 5*b^2*d*e*g^3 - 4*c*d*g^2*(6*b*e*f - 2*b*d*g + a*e*g)) - 2*(4*c^2*d^2 - 
 (b^2*e^2)/2 - 2*c*e*(b*d - a*e))*g*(5*b^2*e^2*g^2 - 4*c*e*g*(6*b*e*f - 2* 
b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)))*ArcTanh[(b + 2 
*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(Sqrt[c]*e) - (128*c^3*Sqrt[c*d^ 
2 - b*d*e + a*e^2]*(e*f - d*g)^3*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/( 
2*Sqrt[c*d^2 - b*d*e + a*e^2]*Sqrt[a + b*x + c*x^2])])/e)/(8*c*e^2)))/(2*c 
))/(8*c*e^3)
 

3.9.54.3.1 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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

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

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1267

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

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 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]))
 

3.9.54.4 Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 879, normalized size of antiderivative = 1.65

method	result	size
default	\(\frac {\left (-d^{3} g^{3}+3 d^{2} e f \,g^{2}-3 d \,e^{2} f^{2} g +e^{3} f^{3}\right ) \left (\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (e^{2} a -b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{4}}+\frac {g \left (d^{2} g^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+e^{2} g^{2} \left (\frac {x \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{4 c}-\frac {5 b \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )}{8 c}-\frac {a \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{4 c}\right )+3 e^{2} f^{2} \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )-3 d e f g \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )+\left (-d e \,g^{2}+3 e^{2} f g \right ) \left (\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{3 c}-\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{2 c}\right )\right )}{e^{3}}\)	\(879\)
risch	\(-\frac {\left (-48 g^{3} c^{3} e^{3} x^{3}-8 b \,c^{2} e^{3} g^{3} x^{2}+64 c^{3} d \,e^{2} g^{3} x^{2}-192 c^{3} e^{3} f \,g^{2} x^{2}-24 a \,c^{2} e^{3} g^{3} x +10 b^{2} c \,e^{3} g^{3} x +16 b \,c^{2} d \,e^{2} g^{3} x -48 b \,c^{2} e^{3} f \,g^{2} x -96 c^{3} d^{2} e \,g^{3} x +288 c^{3} d \,e^{2} f \,g^{2} x -288 c^{3} e^{3} f^{2} g x +52 a b c \,e^{3} g^{3}+64 a \,c^{2} d \,e^{2} g^{3}-192 a \,c^{2} e^{3} f \,g^{2}-15 b^{3} e^{3} g^{3}-24 b^{2} c d \,e^{2} g^{3}+72 b^{2} c \,e^{3} f \,g^{2}-48 b \,c^{2} d^{2} e \,g^{3}+144 b \,c^{2} d \,e^{2} f \,g^{2}-144 b \,c^{2} e^{3} f^{2} g +192 c^{3} d^{3} g^{3}-576 c^{3} d^{2} e f \,g^{2}+576 c^{3} d \,e^{2} f^{2} g -192 c^{3} e^{3} f^{3}\right ) \sqrt {c \,x^{2}+b x +a}}{192 c^{3} e^{4}}-\frac {-\frac {128 \left (a \,d^{3} e^{2} g^{3}-3 a \,d^{2} e^{3} f \,g^{2}+3 a d \,e^{4} f^{2} g -f^{3} a \,e^{5}-b \,d^{4} e \,g^{3}+3 b \,d^{3} e^{2} f \,g^{2}-3 b \,d^{2} e^{3} f^{2} g +b d \,e^{4} f^{3}+d^{5} g^{3} c -3 c \,d^{4} e f \,g^{2}+3 c \,d^{3} e^{2} f^{2} g -c \,d^{2} e^{3} f^{3}\right ) c^{3} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}+\frac {\left (16 a^{2} c^{2} e^{4} g^{3}-24 a \,b^{2} c \,e^{4} g^{3}-32 a b \,c^{2} d \,e^{3} g^{3}+96 a b \,c^{2} e^{4} f \,g^{2}-64 a \,c^{3} d^{2} e^{2} g^{3}+192 a \,c^{3} d \,e^{3} f \,g^{2}-192 a \,c^{3} e^{4} f^{2} g +5 b^{4} e^{4} g^{3}+8 b^{3} c d \,e^{3} g^{3}-24 b^{3} c \,e^{4} f \,g^{2}+16 b^{2} c^{2} d^{2} e^{2} g^{3}-48 b^{2} c^{2} d \,e^{3} f \,g^{2}+48 b^{2} c^{2} e^{4} f^{2} g +64 b \,c^{3} d^{3} e \,g^{3}-192 b \,c^{3} d^{2} e^{2} f \,g^{2}+192 b \,c^{3} d \,e^{3} f^{2} g -64 b \,c^{3} e^{4} f^{3}-128 c^{4} d^{4} g^{3}+384 c^{4} d^{3} e f \,g^{2}-384 c^{4} d^{2} e^{2} f^{2} g +128 c^{4} d \,e^{3} f^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{e \sqrt {c}}}{128 c^{3} e^{4}}\)	\(961\)

input

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

output

(-d^3*g^3+3*d^2*e*f*g^2-3*d*e^2*f^2*g+e^3*f^3)/e^4*(((x+d/e)^2*c+(b*e-2*c* 
d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)/e*ln((1/2*(b*e 
-2*c*d)/e+c*(x+d/e))/c^(1/2)+((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d 
*e+c*d^2)/e^2)^(1/2))/c^(1/2)-(a*e^2-b*d*e+c*d^2)/e^2/((a*e^2-b*d*e+c*d^2) 
/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2- 
b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c* 
d^2)/e^2)^(1/2))/(x+d/e)))+g/e^3*(d^2*g^2*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^( 
1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))+ 
e^2*g^2*(1/4*x*(c*x^2+b*x+a)^(3/2)/c-5/8*b/c*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/ 
2*b/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2 
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))))-1/4*a/c*(1/4*(2*c*x+b)/c*(c*x^2+b*x 
+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/ 
2))))+3*e^2*f^2*(1/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/ 
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2)))-3*d*e*f*g*(1/4*(2*c*x+b)/c 
*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2 
+b*x+a)^(1/2)))+(-d*e*g^2+3*e^2*f*g)*(1/3*(c*x^2+b*x+a)^(3/2)/c-1/2*b/c*(1 
/4*(2*c*x+b)/c*(c*x^2+b*x+a)^(1/2)+1/8*(4*a*c-b^2)/c^(3/2)*ln((1/2*b+c*x)/ 
c^(1/2)+(c*x^2+b*x+a)^(1/2)))))
 

3.9.54.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.9.54.6 Sympy [F]

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

input

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

output

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

3.9.54.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]

input

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

output

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

3.9.54.8 Giac [F(-2)]

Exception generated. \[ \int \frac {(f+g x)^3 \sqrt {a+b x+c x^2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

3.9.54.9 Mupad [F(-1)]

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

input

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

output

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