3.878 \(\int \frac{(f+g x)^4}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=496 \[ -\frac{2 \left (-b^2 \left (a^2 e g^4+4 a c d f g^3+c^2 e f^4\right )+x \left (2 c^2 g^2 \left (a^2 (-g) (4 e f-d g)-3 a b f (e f-2 d g)+3 b^2 d f^2\right )-b c g^3 \left (-3 a^2 e g-4 a b (e f-d g)+4 b^2 d f\right )+b^3 g^4 (b d-a e)+c^3 f^2 (4 a g (2 e f-3 d g)-b f (4 d g+e f))+2 c^4 d f^4\right )+b c \left (a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (3 d g+2 e f)+c^2 d f^4\right )+2 a c \left (a^2 e g^4-2 a c f g^2 (3 e f-2 d g)+c^2 f^3 (e f-4 d g)\right )+a b^3 d g^4\right )}{c^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{g^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (-3 b e g-2 c d g+8 c e f)}{2 c^{5/2} e^2}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}+\frac{(e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 2.32804, antiderivative size = 496, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{2 \left (-b^2 \left (a^2 e g^4+4 a c d f g^3+c^2 e f^4\right )+x \left (2 c^2 g^2 \left (a^2 (-g) (4 e f-d g)-3 a b f (e f-2 d g)+3 b^2 d f^2\right )-b c g^3 \left (-3 a^2 e g-4 a b (e f-d g)+4 b^2 d f\right )+b^3 g^4 (b d-a e)+c^3 f^2 (4 a g (2 e f-3 d g)-b f (4 d g+e f))+2 c^4 d f^4\right )+b c \left (a^2 g^3 (4 e f-3 d g)+2 a c f^2 g (3 d g+2 e f)+c^2 d f^4\right )+2 a c \left (a^2 e g^4-2 a c f g^2 (3 e f-2 d g)+c^2 f^3 (e f-4 d g)\right )+a b^3 d g^4\right )}{c^2 \left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{g^3 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) (-3 b e g-2 c d g+8 c e f)}{2 c^{5/2} e^2}+\frac{g^4 \sqrt{a+b x+c x^2}}{c^2 e}+\frac{(e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**4/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

Mathematica [A]  time = 5.97108, size = 587, normalized size = 1.18 \[ \frac{-\frac{2 e \left (b^2 \left (3 a^2 e^2 g^4+a c g^3 \left (d^2 g+4 d e (2 f+3 g x)+e^2 x (g x-8 f)\right )+c^2 \left (d^2 g^4 x^2-12 d e f^2 g^2 x+2 e^2 f^4\right )\right )-2 b c \left (a^2 e g^3 (-5 d g+4 e f+5 e g x)+2 a c g \left (d^2 g^3 x+d e g \left (3 f^2+6 f g x-g^2 x^2\right )+e^2 f^2 (2 f-3 g x)\right )+c^2 e f^3 (d (f-4 g x)-e f x)\right )-4 c \left (2 a^3 e^2 g^4+a^2 c g^2 \left (d^2 g^2+d e g (4 f+g x)+e^2 \left (-6 f^2-4 f g x+g^2 x^2\right )\right )+a c^2 \left (d^2 g^4 x^2-2 d e f^2 g (2 f+3 g x)+e^2 f^3 (f+4 g x)\right )+c^3 d e f^4 x\right )+b^3 g^3 (3 a e g (e x-d)+c d x (d g+8 e f-e g x))-3 b^4 d e g^4 x\right )}{c^2 \left (b^2-4 a c\right ) \sqrt{a+x (b+c x)} \left (e (b d-a e)-c d^2\right )}+\frac{g^3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) (-3 b e g-2 c d g+8 c e f)}{c^{5/2}}+\frac{2 (e f-d g)^4 \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^{3/2}}-\frac{2 (e f-d g)^4 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}}}{2 e^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(f + g*x)^4/((d + e*x)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

((-2*e*(-3*b^4*d*e*g^4*x + b^3*g^3*(3*a*e*g*(-d + e*x) + c*d*x*(8*e*f + d*g - e*
g*x)) + b^2*(3*a^2*e^2*g^4 + c^2*(2*e^2*f^4 - 12*d*e*f^2*g^2*x + d^2*g^4*x^2) +
a*c*g^3*(d^2*g + e^2*x*(-8*f + g*x) + 4*d*e*(2*f + 3*g*x))) - 2*b*c*(a^2*e*g^3*(
4*e*f - 5*d*g + 5*e*g*x) + c^2*e*f^3*(-(e*f*x) + d*(f - 4*g*x)) + 2*a*c*g*(d^2*g
^3*x + e^2*f^2*(2*f - 3*g*x) + d*e*g*(3*f^2 + 6*f*g*x - g^2*x^2))) - 4*c*(2*a^3*
e^2*g^4 + c^3*d*e*f^4*x + a*c^2*(d^2*g^4*x^2 - 2*d*e*f^2*g*(2*f + 3*g*x) + e^2*f
^3*(f + 4*g*x)) + a^2*c*g^2*(d^2*g^2 + d*e*g*(4*f + g*x) + e^2*(-6*f^2 - 4*f*g*x
+ g^2*x^2)))))/(c^2*(b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*Sqrt[a + x*(b + c*
x)]) + (2*(e*f - d*g)^4*Log[d + e*x])/(c*d^2 + e*(-(b*d) + a*e))^(3/2) + (g^3*(8
*c*e*f - 2*c*d*g - 3*b*e*g)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^
(5/2) - (2*(e*f - d*g)^4*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e
*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(c*d^2 + e*(-(b*d) + a*e))^(3/2))/(2*e^
2)

_______________________________________________________________________________________

Maple [B]  time = 0.035, size = 4453, normalized size = 9. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^4/(e*x+d)/(c*x^2+b*x+a)^(3/2),x)

[Out]

e/(a*e^2-b*d*e+c*d^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)*f^4-e/(a*e^2-b*d*e+c*d^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))*f^4-12/e/(a*
e^2-b*d*e+c*d^2)/(4*a*c-b^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*b*c*d^2*f^2*g^2+8/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*b*c*d^3*f*g^3+8*g^3/e
^2*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d*f+4*g^3/e^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*
x+a)^(1/2)*d*f+4*g^3/e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*f+4*g^4/e/c*a*b/(
4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+16*g^3/e^3*d^2*f/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2
)*c*x-g^4/e^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*d+2/e^4/(a*e^2-b*d*e+c*d^2
)/(4*a*c-b^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)*
b*c*d^5*g^4+4/e^4/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*c^2*d^5*g^4+4/e^2/(a*e^2-b*d*e+c*d^2)/(4*a
*c-b^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)*b^2*d^
3*f*g^3-6/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b^2*d^2*f^2*g^2-4/(a*e^2-b*d*e+c*d^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)*d*f^3*g-6*g^2/e/c/(c*x^2+b
*x+a)^(1/2)*f^2+g^4/e*x^2/c/(c*x^2+b*x+a)^(1/2)-g^4/e^2/c^(3/2)*ln((1/2*b+c*x)/c
^(1/2)+(c*x^2+b*x+a)^(1/2))*d+4*g^3/e/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+
a)^(1/2))*f-3/4*g^4/e*b^2/c^3/(c*x^2+b*x+a)^(1/2)-3/2*g^4/e*b/c^(5/2)*ln((1/2*b+
c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2*g^4/e/c^2*a/(c*x^2+b*x+a)^(1/2)-g^4/e^3/c/(c
*x^2+b*x+a)^(1/2)*d^2-24*g^2/e^2*d*f^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x+1/e^3
/(a*e^2-b*d*e+c*d^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)*d^4*g^4-2*e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*b*c*f^4+8/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*b*c*d*f^3*g
-2/e^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*b*c*d^4*g^4-16/e^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*c^2*d^4*f*g^3+
24/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*c^2*d^3*f^2*g^2-16/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*c^2*d^2*f^3*
g-8/e^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*d^4*f*g^3+12/e^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*d^3*f^2*g^2
-8/e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*d^2*f^3*g+8*g^3/e^3*d^2*f/(4*a*c-b^2)/(c*x^2+b*x+a)^(
1/2)*b+16*g/e*f^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x-12*g^2/e^2*d*f^2/(4*a*c-b^
2)/(c*x^2+b*x+a)^(1/2)*b-4*g^4/e^4*d^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*c*x-1/2*g
^4/e^2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d+2*g^3/e*b^3/c^2/(4*a*c-b^2)/(c*
x^2+b*x+a)^(1/2)*f-3/2*g^4/e*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+2*g^4/e/c
^2*a*b^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)-2*g^4/e^3*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(
1/2)*x*d^2-12*g^2/e*b/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x*f^2-1/e^3/(a*e^2-b*d*e+c
*d^2)/(4*a*c-b^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)*b^2*d^4*g^4+2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b*c*d*f^4+4/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*b^2*d*f^3*g+4/
e^2/(a*e^2-b*d*e+c*d^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))*d^3*f*g^3-6/e/(a*e^2-b
*d*e+c*d^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))*d^2*f^2*g^2+4/(a*e^2-b*d*e+c*d^2)/
(4*a*c-b^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*
c^2*d*f^4-g^4/e^3*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*d^2-6*g^2/e*b^2/c/(4*a*c
-b^2)/(c*x^2+b*x+a)^(1/2)*f^2-3/4*g^4/e*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+
g^4/e^2*x/c/(c*x^2+b*x+a)^(1/2)*d-4*g^3/e*x/c/(c*x^2+b*x+a)^(1/2)*f-1/2*g^4/e^2*
b/c^2/(c*x^2+b*x+a)^(1/2)*d+2*g^3/e*b/c^2/(c*x^2+b*x+a)^(1/2)*f+4*g^3/e^2/c/(c*x
^2+b*x+a)^(1/2)*d*f+8*g/e*f^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*b-2*g^4/e^4*d^3/(4
*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*b+3/2*g^4/e*b/c^2*x/(c*x^2+b*x+a)^(1/2)-e/(a*e^2-b
*d*e+c*d^2)/(4*a*c-b^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)*b^2*f^4-1/e^3/(a*e^2-b*d*e+c*d^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))*d^
4*g^4-4/e^2/(a*e^2-b*d*e+c*d^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)*d^3*f*g^3+6/e/(a*e^2-b*d*e+c*d^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)*d^2*f^2*g^2+4/(a*e^2-b*d*e+c*d^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))*d*f^3*g

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^4/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^4/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**4/(e*x+d)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x + f)^4/((c*x^2 + b*x + a)^(3/2)*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: TypeError