3.2381 \(\int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=295 \[ -\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )^2}+\frac{4 (d+e x) \left (x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )+b^2 \left (-\left (5 c d^2 e-a e^3\right )\right )+4 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{e^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.832536, antiderivative size = 295, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )}{3 c^2 \left (b^2-4 a c\right )^2}+\frac{4 (d+e x) \left (x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )+b^2 \left (-\left (5 c d^2 e-a e^3\right )\right )+4 b c d \left (3 a e^2+c d^2\right )-4 a c e \left (3 a e^2+c d^2\right )\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2}}-\frac{2 (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac{e^4 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(a + b*x + c*x^2)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 112.107, size = 303, normalized size = 1.03 \[ \frac{2 \left (d + e x\right )^{3} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}} + \frac{4 \left (d + e x\right ) \left (- 12 a^{2} c e^{3} + a b^{2} e^{3} + 12 a b c d e^{2} - 4 a c^{2} d^{2} e - 5 b^{2} c d^{2} e + 4 b c^{2} d^{3} + x \left (b e - 2 c d\right ) \left (- 8 a c e^{2} + b^{2} e^{2} + 4 b c d e - 4 c^{2} d^{2}\right )\right )}{3 c \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}}} - \frac{2 e \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}} \left (- 20 a c e^{2} + 3 b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right )}{3 c^{2} \left (- 4 a c + b^{2}\right )^{2}} + \frac{e^{4} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

2*(d + e*x)**3*(2*a*e - b*d + x*(b*e - 2*c*d))/(3*(-4*a*c + b**2)*(a + b*x + c*x
**2)**(3/2)) + 4*(d + e*x)*(-12*a**2*c*e**3 + a*b**2*e**3 + 12*a*b*c*d*e**2 - 4*
a*c**2*d**2*e - 5*b**2*c*d**2*e + 4*b*c**2*d**3 + x*(b*e - 2*c*d)*(-8*a*c*e**2 +
b**2*e**2 + 4*b*c*d*e - 4*c**2*d**2))/(3*c*(-4*a*c + b**2)**2*sqrt(a + b*x + c*
x**2)) - 2*e*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)*(-20*a*c*e**2 + 3*b**2*e**2 +
8*b*c*d*e - 8*c**2*d**2)/(3*c**2*(-4*a*c + b**2)**2) + e**4*atanh((b + 2*c*x)/(2
*sqrt(c)*sqrt(a + b*x + c*x**2)))/c**(5/2)

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Mathematica [A]  time = 1.39792, size = 395, normalized size = 1.34 \[ \frac{e^4 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{5/2}}-\frac{2 \left (b^3 \left (3 a^2 e^4-18 a c e^4 x^2+c^2 d \left (d^3+12 d^2 e x-18 d e^2 x^2-4 e^3 x^3\right )\right )-2 b^2 c \left (21 a^2 e^4 x+2 a c e \left (-2 d^3+18 d^2 e x-6 d e^2 x^2+7 e^3 x^3\right )+3 c^2 d^2 x \left (d^2-8 d e x+2 e^2 x^2\right )\right )-4 b c \left (5 a^3 e^4+12 a^2 c d e^2 (d-2 e x)+3 a c^2 d \left (d^3-4 d^2 e x+6 d e^2 x^2-4 e^3 x^3\right )+2 c^3 d^3 x^2 (3 d-4 e x)\right )+8 c^2 \left (a^3 e^3 (8 d+3 e x)+4 a^2 c e \left (d^3+3 d e^2 x^2+e^3 x^3\right )-3 a c^2 d^2 x \left (d^2+2 e^2 x^2\right )-2 c^3 d^4 x^3\right )+2 b^4 e^4 x \left (3 a+2 c x^2\right )+3 b^5 e^4 x^2\right )}{3 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(a + b*x + c*x^2)^(5/2),x]

[Out]

(-2*(3*b^5*e^4*x^2 + 2*b^4*e^4*x*(3*a + 2*c*x^2) + b^3*(3*a^2*e^4 - 18*a*c*e^4*x
^2 + c^2*d*(d^3 + 12*d^2*e*x - 18*d*e^2*x^2 - 4*e^3*x^3)) - 4*b*c*(5*a^3*e^4 + 2
*c^3*d^3*x^2*(3*d - 4*e*x) + 12*a^2*c*d*e^2*(d - 2*e*x) + 3*a*c^2*d*(d^3 - 4*d^2
*e*x + 6*d*e^2*x^2 - 4*e^3*x^3)) + 8*c^2*(-2*c^3*d^4*x^3 + a^3*e^3*(8*d + 3*e*x)
- 3*a*c^2*d^2*x*(d^2 + 2*e^2*x^2) + 4*a^2*c*e*(d^3 + 3*d*e^2*x^2 + e^3*x^3)) -
2*b^2*c*(21*a^2*e^4*x + 3*c^2*d^2*x*(d^2 - 8*d*e*x + 2*e^2*x^2) + 2*a*c*e*(-2*d^
3 + 18*d^2*e*x - 6*d*e^2*x^2 + 7*e^3*x^3))))/(3*c^2*(b^2 - 4*a*c)^2*(a + x*(b +
c*x))^(3/2)) + (e^4*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2)

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Maple [B]  time = 0.014, size = 1550, normalized size = 5.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+b*x+a)^(5/2),x)

[Out]

1/6*d*e^3*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+4/3*d*e^3*b^4/c^2/(4*a*c-b^2)^
2/(c*x^2+b*x+a)^(1/2)-4/3*d^3*e*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+1/2*e^4*b^
2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-16*d*e^3*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+
b*x+a)^(1/2)+d^2*e^2*b^2/c/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-8/3*d*e^3*a/c^2/(c*
x^2+b*x+a)^(3/2)+1/6*d*e^3*b^2/c^3/(c*x^2+b*x+a)^(3/2)-3*d^2*e^2*x/c/(c*x^2+b*x+
a)^(3/2)+4*d^2*e^2*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+4*d^2*e^2*a/(4*a*c-b^
2)/(c*x^2+b*x+a)^(3/2)*x-1/24*e^4*b^4/c^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*x-1/3*
e^4*b^4/c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/4*e^4*b^3/c^3*a/(4*a*c-b^2)/(c
*x^2+b*x+a)^(3/2)+2*e^4*b^3/c^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+e^4/c^2*b^2/
(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)*x+16*d^2*e^2*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)
*b-d*e^3*b/c^2*x/(c*x^2+b*x+a)^(3/2)-4*d*e^3*b/c*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/
2)*x+2/3*d^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b-1/3*e^4*x^3/c/(c*x^2+b*x+a)^(3/2)
-1/48*e^4*b^3/c^4/(c*x^2+b*x+a)^(3/2)-e^4/c^2*x/(c*x^2+b*x+a)^(1/2)+1/2*e^4/c^3*
b/(c*x^2+b*x+a)^(1/2)-4/3*d^3*e/c/(c*x^2+b*x+a)^(3/2)+2*d^2*e^2*a/c/(4*a*c-b^2)/
(c*x^2+b*x+a)^(3/2)*b-2*d*e^3*b^2/c^2*a/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-32*d*e^3
*b*a/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+4*e^4*b^2/c*a/(4*a*c-b^2)^2/(c*x^2+b*x+
a)^(1/2)*x+32*d^2*e^2*a*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-64/3*d^3*e*b*c/(4*
a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/3*d*e^3*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/
2)*x+8/3*d*e^3*b^3/c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x-8/3*d^3*e*b/(4*a*c-b^2)
/(c*x^2+b*x+a)^(3/2)*x+1/2*d^2*e^2*b^3/c^2/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)+8*d^2
*e^2*b^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+e^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+
(c*x^2+b*x+a)^(1/2))+1/2*d^2*e^2*b/c^2/(c*x^2+b*x+a)^(3/2)-4*d*e^3*x^2/c/(c*x^2+
b*x+a)^(3/2)+1/2*e^4*b/c^2*x^2/(c*x^2+b*x+a)^(3/2)+1/8*e^4*b^2/c^3*x/(c*x^2+b*x+
a)^(3/2)-1/48*e^4*b^5/c^4/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)-32/3*d^3*e*b^2/(4*a*c-
b^2)^2/(c*x^2+b*x+a)^(1/2)+16/3*d^4*c/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*b-1/6*e^
4*b^5/c^3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)+1/3*e^4*b/c^3*a/(c*x^2+b*x+a)^(3/2)+
1/2*e^4/c^3*b^3/(4*a*c-b^2)/(c*x^2+b*x+a)^(1/2)+4/3*d^4/(4*a*c-b^2)/(c*x^2+b*x+a
)^(3/2)*c*x+32/3*d^4*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x

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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((e*x + d)^4/(c*x^2 + b*x + a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.475024, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + b*x + a)^(5/2),x, algorithm="fricas")

[Out]

[1/6*(4*(48*a^2*b*c^2*d^2*e^2 - 64*a^3*c^2*d*e^3 - (b^3*c^2 - 12*a*b*c^3)*d^4 -
8*(a*b^2*c^2 + 4*a^2*c^3)*d^3*e - (3*a^2*b^3 - 20*a^3*b*c)*e^4 + 4*(4*c^5*d^4 -
8*b*c^4*d^3*e + 3*(b^2*c^3 + 4*a*c^4)*d^2*e^2 + (b^3*c^2 - 12*a*b*c^3)*d*e^3 - (
b^4*c - 7*a*b^2*c^2 + 8*a^2*c^3)*e^4)*x^3 + 3*(8*b*c^4*d^4 - 16*b^2*c^3*d^3*e +
6*(b^3*c^2 + 4*a*b*c^3)*d^2*e^2 - 8*(a*b^2*c^2 + 4*a^2*c^3)*d*e^3 - (b^5 - 6*a*b
^3*c)*e^4)*x^2 + 6*(12*a*b^2*c^2*d^2*e^2 - 16*a^2*b*c^2*d*e^3 + (b^2*c^3 + 4*a*c
^4)*d^4 - 2*(b^3*c^2 + 4*a*b*c^3)*d^3*e - (a*b^4 - 7*a^2*b^2*c + 4*a^3*c^2)*e^4)
*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*
x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4*x^3 + (b^6 - 6*a*b^4*c + 32*a^3
*c^3)*e^4*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^4*x + (a^2*b^4 - 8*a^3*
b^2*c + 16*a^4*c^2)*e^4)*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x
^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4
+ (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*
b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3
*c^3 + 16*a^3*b*c^4)*x)*sqrt(c)), 1/3*(2*(48*a^2*b*c^2*d^2*e^2 - 64*a^3*c^2*d*e^
3 - (b^3*c^2 - 12*a*b*c^3)*d^4 - 8*(a*b^2*c^2 + 4*a^2*c^3)*d^3*e - (3*a^2*b^3 -
20*a^3*b*c)*e^4 + 4*(4*c^5*d^4 - 8*b*c^4*d^3*e + 3*(b^2*c^3 + 4*a*c^4)*d^2*e^2 +
(b^3*c^2 - 12*a*b*c^3)*d*e^3 - (b^4*c - 7*a*b^2*c^2 + 8*a^2*c^3)*e^4)*x^3 + 3*(
8*b*c^4*d^4 - 16*b^2*c^3*d^3*e + 6*(b^3*c^2 + 4*a*b*c^3)*d^2*e^2 - 8*(a*b^2*c^2
+ 4*a^2*c^3)*d*e^3 - (b^5 - 6*a*b^3*c)*e^4)*x^2 + 6*(12*a*b^2*c^2*d^2*e^2 - 16*a
^2*b*c^2*d*e^3 + (b^2*c^3 + 4*a*c^4)*d^4 - 2*(b^3*c^2 + 4*a*b*c^3)*d^3*e - (a*b^
4 - 7*a^2*b^2*c + 4*a^3*c^2)*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) + 3*((b^4*c^
2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e
^4*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*e^4*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^
3*b*c^2)*e^4*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^4)*arctan(1/2*(2*c*x + b
)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^
4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2
*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^
3*c^3 + 16*a^3*b*c^4)*x)*sqrt(-c))]

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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((e*x+d)**4/(c*x**2+b*x+a)**(5/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.231047, size = 741, normalized size = 2.51 \[ \frac{2 \,{\left ({\left ({\left (\frac{4 \,{\left (4 \, c^{5} d^{4} - 8 \, b c^{4} d^{3} e + 3 \, b^{2} c^{3} d^{2} e^{2} + 12 \, a c^{4} d^{2} e^{2} + b^{3} c^{2} d e^{3} - 12 \, a b c^{3} d e^{3} - b^{4} c e^{4} + 7 \, a b^{2} c^{2} e^{4} - 8 \, a^{2} c^{3} e^{4}\right )} x}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}} + \frac{3 \,{\left (8 \, b c^{4} d^{4} - 16 \, b^{2} c^{3} d^{3} e + 6 \, b^{3} c^{2} d^{2} e^{2} + 24 \, a b c^{3} d^{2} e^{2} - 8 \, a b^{2} c^{2} d e^{3} - 32 \, a^{2} c^{3} d e^{3} - b^{5} e^{4} + 6 \, a b^{3} c e^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x + \frac{6 \,{\left (b^{2} c^{3} d^{4} + 4 \, a c^{4} d^{4} - 2 \, b^{3} c^{2} d^{3} e - 8 \, a b c^{3} d^{3} e + 12 \, a b^{2} c^{2} d^{2} e^{2} - 16 \, a^{2} b c^{2} d e^{3} - a b^{4} e^{4} + 7 \, a^{2} b^{2} c e^{4} - 4 \, a^{3} c^{2} e^{4}\right )}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )} x - \frac{b^{3} c^{2} d^{4} - 12 \, a b c^{3} d^{4} + 8 \, a b^{2} c^{2} d^{3} e + 32 \, a^{2} c^{3} d^{3} e - 48 \, a^{2} b c^{2} d^{2} e^{2} + 64 \, a^{3} c^{2} d e^{3} + 3 \, a^{2} b^{3} e^{4} - 20 \, a^{3} b c e^{4}}{b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}}\right )}}{3 \,{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}} - \frac{e^{4}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + b*x + a)^(5/2),x, algorithm="giac")

[Out]

2/3*(((4*(4*c^5*d^4 - 8*b*c^4*d^3*e + 3*b^2*c^3*d^2*e^2 + 12*a*c^4*d^2*e^2 + b^3
*c^2*d*e^3 - 12*a*b*c^3*d*e^3 - b^4*c*e^4 + 7*a*b^2*c^2*e^4 - 8*a^2*c^3*e^4)*x/(
b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4) + 3*(8*b*c^4*d^4 - 16*b^2*c^3*d^3*e + 6*b^3*
c^2*d^2*e^2 + 24*a*b*c^3*d^2*e^2 - 8*a*b^2*c^2*d*e^3 - 32*a^2*c^3*d*e^3 - b^5*e^
4 + 6*a*b^3*c*e^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))*x + 6*(b^2*c^3*d^4 + 4*
a*c^4*d^4 - 2*b^3*c^2*d^3*e - 8*a*b*c^3*d^3*e + 12*a*b^2*c^2*d^2*e^2 - 16*a^2*b*
c^2*d*e^3 - a*b^4*e^4 + 7*a^2*b^2*c*e^4 - 4*a^3*c^2*e^4)/(b^4*c^2 - 8*a*b^2*c^3
+ 16*a^2*c^4))*x - (b^3*c^2*d^4 - 12*a*b*c^3*d^4 + 8*a*b^2*c^2*d^3*e + 32*a^2*c^
3*d^3*e - 48*a^2*b*c^2*d^2*e^2 + 64*a^3*c^2*d*e^3 + 3*a^2*b^3*e^4 - 20*a^3*b*c*e
^4)/(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4))/(c*x^2 + b*x + a)^(3/2) - e^4*ln(abs(-
2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2)