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

Optimal. Leaf size=1551 \[ -\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 (\sin ^{-1}\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}} F\left (\sin ^{-1}\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}} \]

[Out]

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*(-567*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+3
96*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*(7
7*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))^(1/2)*2^(1/2),(-2*g*(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)
^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(g*x+f)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^5/g^5/(c*x^2
+b*x+a)^(1/2)/(c*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)+2/3465*(a*g^2-b*f*g+c*f^2)*(64*b^4*e^3*g^4+4*
b^2*c*e^2*g^3*(-69*a*e*g-66*b*d*g+7*b*e*f)-2*c^4*f*(-231*d^3*g^3+396*d^2*e*f*g^2-264*d*e^2*f^2*g+64*e^3*f^3)+3
*c^2*e*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*g*(6*a*e*g*(16
5*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)))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2
)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*g*(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2
^(1/2)*(-4*a*c+b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(
1/2)/c^5/g^5/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 5.34, antiderivative size = 1551, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {932, 1667, 857, 732, 435, 430} \begin {gather*} \frac {2 \sqrt {f+g x} \sqrt {c x^2+b x+a} (d+e x)^4}{11 e}+\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 (\text {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 )}{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 {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}} F\left (\text {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 )}{3465 c^5 g^5 \sqrt {f+g x} \sqrt {c x^2+b x+a}}+\frac {2 e^2 (c e f-3 c d g+b e g) (f+g x)^{7/2} \sqrt {c x^2+b x+a}}{99 c g^4}-\frac {2 e \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}}{693 c^2 g^4}+\frac {2 \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}}{3465 c^3 g^4}-\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}}{3465 c^4 e g^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(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 + 1
098*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])/(3465*c^4*e*g^4) + (2*(d +
e*x)^4*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(11*e) + (2*(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])/(3465*c^3*g^4)
- (2*e*(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])/(693*c^2*g^4) + (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) + (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 + 231*d*g) - 9*a*b*e*g*(15*e^2*f^2 - 110*d*e*f*g - 319*d^2*g^2)
+ b^2*(37*e^3*f^3 - 198*d*e^2*f^2*g + 495*d^2*e*f*g^2 + 462*d^3*g^3)))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x
^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sq
rt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(3465*c^5*g^5*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b
^2 - 4*a*c])*g)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*f^2 - b*f*g + a*g^2)*(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*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 -
231*d^3*g^3) + 3*c^2*e*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*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[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipt
icF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*g)/(2*c*f -
(b + Sqrt[b^2 - 4*a*c])*g)])/(3465*c^5*g^5*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m^2, 1/4]

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(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] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rule 932

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] - Dist[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] && NeQ[e*f - d*g, 0]
&& NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[2*m] &&  !LtQ[m, -1]

Rule 1667

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]]}, Simp[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] + Dist[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] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int (d+e x)^3 \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx &=\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 (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\right )}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{11 e}\\ &=\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}+\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 {2 \int \frac {\frac {1}{2} g \left (b^2 e^4 f^4 g+7 a b e^4 f^3 g^2+a c g \left (7 e^4 f^4-21 d e^3 f^3 g-27 d^3 e f g^3+9 d^4 g^4\right )+b c \left (e^4 f^5-3 d e^3 f^4 g+9 d^4 f g^4\right )\right )+\frac {1}{2} g \left (b e^4 f^2 g^2 (11 b f+21 a g)+2 c^2 \left (e^4 f^5-3 d e^3 f^4 g+9 d^4 f g^4\right )+c g \left (3 a e g \left (7 e^3 f^3-21 d e^2 f^2 g-27 d^2 e f g^2+3 d^3 g^3\right )+b \left (13 e^4 f^4-33 d e^3 f^3 g+9 d^3 e f g^3+18 d^4 g^4\right )\right )\right ) x+\frac {3}{2} g^2 \left (b e^4 f g^2 (9 b f+7 a g)+c^2 \left (5 e^4 f^4-15 d e^3 f^3 g+15 d^3 e f g^3+9 d^4 g^4\right )+c e g \left (a e g \left (7 e^2 f^2-48 d e f g-9 d^2 g^2\right )+b \left (14 e^3 f^3-27 d e^2 f^2 g-9 d^2 e f g^2+15 d^3 g^3\right )\right )\right ) x^2+\frac {1}{2} e g^3 \left (b e^3 g^2 (25 b f+7 a g)+3 c^2 \left (11 e^3 f^3-33 d e^2 f^2 g+9 d^2 e f g^2+27 d^3 g^3\right )-c e g \left (2 a e g (10 e f+33 d g)-b \left (58 e^2 f^2-120 d e f g+27 d^2 g^2\right )\right )\right ) x^3+\frac {1}{2} e^2 g^4 \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 ) x^4}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{99 c e g^5}\\ &=\frac {2 (d+e x)^4 \sqrt {f+g x} \sqrt {a+b x+c x^2}}{11 e}-\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 {4 \int \frac {-\frac {1}{4} g^5 \left (8 b^3 e^4 f^3 g^2+b^2 e^3 f^2 g \left (40 a e g^2+3 c f (4 e f-11 d g)\right )+b c f \left (a e^3 f g^2 (28 e f-165 d g)+c \left (22 e^4 f^4-75 d e^3 f^3 g+81 d^2 e^2 f^2 g^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 f^3 g+135 d^2 e^2 f^2 g^2+63 d^3 e f g^3-21 d^4 g^4\right )\right )\right )-\frac {1}{4} g^5 \left (16 b^2 e^4 f g^3 (4 b f+5 a g)+2 c^3 \left (22 e^4 f^5-75 d e^3 f^4 g+81 d^2 e^2 f^3 g^2-63 d^4 f g^4\right )-c e^3 f g^2 \left (180 a^2 e g^2-b^2 f (91 e f-264 d g)+a b g (101 e f+330 d g)\right )+c^2 g \left (a e g \left (107 e^3 f^3-519 d e^2 f^2 g+1377 d^2 e f g^2-63 d^3 g^3\right )+b \left (179 e^4 f^4-603 d e^3 f^3 g+648 d^2 e^2 f^2 g^2-63 d^3 e f g^3-126 d^4 g^4\right )\right )\right ) x-\frac {1}{4} g^6 \left (8 b^2 e^4 g^3 (13 b f+5 a g)+c^3 \left (214 e^4 f^4-741 d e^3 f^3 g+891 d^2 e^2 f^2 g^2-315 d^3 e f g^3-189 d^4 g^4\right )-c e^3 g^2 \left (90 a^2 e g^2-b^2 f (146 e f-429 d g)+11 a b g (26 e f+15 d g)\right )-c^2 e g \left (2 a e g \left (100 e^2 f^2-264 d e f g-297 d^2 g^2\right )-b \left (292 e^3 f^3-1044 d e^2 f^2 g+1242 d^2 e f g^2-315 d^3 g^3\right )\right )\right ) x^2-\frac {1}{4} e g^7 \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 ) x^3}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{693 c^2 e g^9}\\ &=\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 {8 \int \frac {\frac {3}{8} g^8 \left (16 b^4 e^4 f^2 g^3+3 b^3 e^3 f g^2 \left (16 a e g^2+c f (3 e f-22 d g)\right )-b^2 c e^2 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 f g+99 d^2 g^2\right )\right )+a c^2 g \left (2 a e^3 f g^2 (e f+231 d g)+c \left (73 e^4 f^4-288 d e^3 f^3 g+432 d^2 e^2 f^2 g^2-882 d^3 e f g^3+105 d^4 g^4\right )\right )-b c f \left (157 a^2 e^4 g^4+3 a c e^2 g^2 \left (8 e^2 f^2-55 d e f g-99 d^2 g^2\right )-c^2 \left (41 e^4 f^4-156 d e^3 f^3 g+234 d^2 e^2 f^2 g^2-189 d^3 e f g^3+105 d^4 g^4\right )\right )\right )+\frac {3}{8} g^8 \left (16 b^3 e^4 g^4 (5 b f+3 a g)+2 c^4 f \left (41 e^4 f^4-156 d e^3 f^3 g+234 d^2 e^2 f^2 g^2-189 d^3 e f g^3+105 d^4 g^4\right )-b c e^3 g^3 \left (157 a^2 e g^2-b^2 f (37 e f-330 d g)+2 a b g (164 e f+99 d g)\right )+c^2 e^2 g^2 \left (2 a^2 e g^2 (76 e f+231 d g)-3 a b g \left (37 e^2 f^2-352 d e f g-99 d^2 g^2\right )+b^2 f \left (13 e^2 f^2-132 d e f g+495 d^2 g^2\right )\right )-c^3 g \left (22 a e g \left (2 e^3 f^3-15 d e^2 f^2 g+54 d^2 e f g^2+21 d^3 g^3\right )-3 b \left (46 e^4 f^4-192 d e^3 f^3 g+321 d^2 e^2 f^2 g^2-280 d^3 e f g^3+70 d^4 g^4\right )\right )\right ) x+\frac {3}{8} g^9 \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 ) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3465 c^3 e g^{12}}\\ &=-\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 {16 \int \frac {-\frac {3}{16} e g^{10} \left (64 b^5 e^3 f g^4+4 b^4 e^2 g^3 \left (16 a e g^2-c f (5 e f+66 d g)\right )-b^3 c e g^2 \left (8 a e g^2 (49 e f+33 d g)+9 c f \left (2 e^2 f^2-11 d e f g-44 d^2 g^2\right )\right )+2 a c^2 g \left (75 a^2 e^3 g^4-9 a c e g^2 \left (e^2 f^2+66 d e f g+55 d^2 g^2\right )-c^2 f \left (16 e^3 f^3-66 d e^2 f^2 g+99 d^2 e f g^2-924 d^3 g^3\right )\right )+b c^2 \left (3 a^2 e^2 g^4 (178 e f+297 d g)+a c g^2 \left (52 e^3 f^3-297 d e^2 f^2 g-1782 d^2 e f g^2-231 d^3 g^3\right )+c^2 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 )\right )-b^2 c g \left (276 a^2 e^3 g^4-6 a c e g^2 \left (13 e^2 f^2+231 d e f g+66 d^2 g^2\right )+c^2 f \left (20 e^3 f^3-99 d e^2 f^2 g+198 d^2 e f g^2+231 d^3 g^3\right )\right )\right )-\frac {3}{16} e g^{10} \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 ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{10395 c^4 e g^{14}}\\ &=-\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 {\left (\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 )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{3465 c^4 g^5}+\frac {\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 ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{3465 c^4 g^5}\\ &=-\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 {\left (\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}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3465 c^5 g^5 \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}+\frac {\left (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-b g-\sqrt {b^2-4 a c} g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{3465 c^5 g^5 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=-\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 (\sin ^{-1}\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}} F\left (\sin ^{-1}\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}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 35.03, size = 26600, normalized size = 17.15 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(32646\) vs. \(2(1469)=2938\).
time = 0.28, size = 32647, normalized size = 21.05

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.70, size = 1782, normalized size = 1.15 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/10395*((462*c^6*d^3*f^3*g^3 - 693*b*c^5*d^3*f^2*g^4 - 693*(b^2*c^4 - 6*a*c^5)*d^3*f*g^5 + 231*(2*b^3*c^3 - 9
*a*b*c^4)*d^3*g^6 - (128*c^6*f^6 - 120*b*c^5*f^5*g - 3*(11*b^2*c^4 - 68*a*c^5)*f^4*g^2 - (20*b^3*c^3 - 87*a*b*
c^4)*f^3*g^3 - 3*(11*b^4*c^2 - 53*a*b^2*c^3 + 34*a^2*c^4)*f^2*g^4 - 3*(40*b^5*c - 246*a*b^3*c^2 + 329*a^2*b*c^
3)*f*g^5 + (128*b^6 - 888*a*b^4*c + 1599*a^2*b^2*c^2 - 450*a^3*c^3)*g^6)*e^3 + 33*(16*c^6*d*f^5*g - 16*b*c^5*d
*f^4*g^2 - 5*(b^2*c^4 - 6*a*c^5)*d*f^3*g^3 - (5*b^3*c^3 - 21*a*b*c^4)*d*f^2*g^4 - 2*(8*b^4*c^2 - 42*a*b^2*c^3
+ 33*a^2*c^4)*d*f*g^5 + (16*b^5*c - 96*a*b^3*c^2 + 123*a^2*b*c^3)*d*g^6)*e^2 - 99*(8*c^6*d^2*f^4*g^2 - 9*b*c^5
*d^2*f^3*g^3 - 2*(2*b^2*c^4 - 11*a*c^5)*d^2*f^2*g^4 - (9*b^3*c^3 - 41*a*b*c^4)*d^2*f*g^5 + (8*b^4*c^2 - 41*a*b
^2*c^3 + 30*a^2*c^4)*d^2*g^6)*e)*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*(462*c^6*d^3*f^2*g^4 - 462*b*c^5*d^3*f*g^5 + 462*(b^2*c^4 - 3*a*c^5)*d^3*g^6 -
(128*c^6*f^5*g - 56*b*c^5*f^4*g^2 - (37*b^2*c^4 - 108*a*c^5)*f^3*g^3 - (37*b^3*c^3 - 135*a*b*c^4)*f^2*g^4 - 2
*(28*b^4*c^2 - 129*a*b^2*c^3 + 78*a^2*c^4)*f*g^5 + (128*b^5*c - 696*a*b^3*c^2 + 771*a^2*b*c^3)*g^6)*e^3 + 66*(
8*c^6*d*f^4*g^2 - 4*b*c^5*d*f^3*g^3 - 3*(b^2*c^4 - 3*a*c^5)*d*f^2*g^4 - (4*b^3*c^3 - 15*a*b*c^4)*d*f*g^5 + (8*
b^4*c^2 - 36*a*b^2*c^3 + 21*a^2*c^4)*d*g^6)*e^2 - 99*(8*c^6*d^2*f^3*g^3 - 5*b*c^5*d^2*f^2*g^4 - (5*b^2*c^4 - 1
6*a*c^5)*d^2*f*g^5 + (8*b^3*c^3 - 29*a*b*c^4)*d^2*g^6)*e)*sqrt(c*g)*weierstrassZeta(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), 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*(693*c^6*d^3*g^6*x + 231*c^6*d^3*f*g^5 + 231*b*c^5*d^3*g^6 + (315*c^6*g^6*x^4 - 64*c^6*f^4*g^2 + 20*b*c^
5*f^3*g^3 + 2*(9*b^2*c^4 - 23*a*c^5)*f^2*g^4 + 10*(2*b^3*c^3 - 7*a*b*c^4)*f*g^5 - 2*(32*b^4*c^2 - 138*a*b^2*c^
3 + 75*a^2*c^4)*g^6 + 35*(c^6*f*g^5 + b*c^5*g^6)*x^3 - 10*(4*c^6*f^2*g^4 - b*c^5*f*g^5 + (4*b^2*c^4 - 9*a*c^5)
*g^6)*x^2 + (48*c^6*f^3*g^3 - 13*b*c^5*f^2*g^4 - (13*b^2*c^4 - 32*a*c^5)*f*g^5 + (48*b^3*c^3 - 157*a*b*c^4)*g^
6)*x)*e^3 + 33*(35*c^6*d*g^6*x^3 + 8*c^6*d*f^3*g^3 - 3*b*c^5*d*f^2*g^4 - (3*b^2*c^4 - 8*a*c^5)*d*f*g^5 + (8*b^
3*c^3 - 27*a*b*c^4)*d*g^6 + 5*(c^6*d*f*g^5 + b*c^5*d*g^6)*x^2 - 2*(3*c^6*d*f^2*g^4 - b*c^5*d*f*g^5 + (3*b^2*c^
4 - 7*a*c^5)*d*g^6)*x)*e^2 + 99*(15*c^6*d^2*g^6*x^2 - 4*c^6*d^2*f^2*g^4 + 2*b*c^5*d^2*f*g^5 - 2*(2*b^2*c^4 - 5
*a*c^5)*d^2*g^6 + 3*(c^6*d^2*f*g^5 + b*c^5*d^2*g^6)*x)*e)*sqrt(c*x^2 + b*x + a)*sqrt(g*x + f))/(c^6*g^6)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d + e x\right )^{3} \sqrt {f + g x} \sqrt {a + b x + c x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {f+g\,x}\,{\left (d+e\,x\right )}^3\,\sqrt {c\,x^2+b\,x+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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