3.2386 \(\int \frac{1}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=310 \[ -\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-4 c e (b d-3 a e)-3 b^2 e^2+8 c^2 d^2\right )+4 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{e^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.764452, antiderivative size = 310, 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 \[ -\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-5 a e)-3 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-4 c e (b d-3 a e)-3 b^2 e^2+8 c^2 d^2\right )+4 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{e^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 139.896, size = 309, normalized size = 1. \[ - \frac{e^{4} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} + \frac{2 \left (- 2 a c e + b^{2} e - b c d + c x \left (b e - 2 c d\right )\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{4 \left (- 2 a c e \left (b e - 2 c d\right )^{2} + \frac{c x \left (b e - 2 c d\right ) \left (- 20 a c e^{2} + 3 b^{2} e^{2} + 8 b c d e - 8 c^{2} d^{2}\right )}{2} + \left (- a c e + \frac{b^{2} e}{2} - \frac{b c d}{2}\right ) \left (- 12 a c e^{2} + 3 b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right )\right )}{3 \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)/(c*x**2+b*x+a)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 1.54461, size = 339, normalized size = 1.09 \[ \frac{2 \left (8 c^2 \left (3 a^2 e^3+5 a c d e^2 x+2 c^2 d^3 x\right )+2 b^2 c e \left (c d (e x-6 d)-11 a e^2\right )+4 b c^2 \left (5 a e^2 (d-e x)+2 c d^2 (d-3 e x)\right )+3 b^4 e^3+b^3 c e^2 (d+3 e x)\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2}+\frac{4 c (a e+c d x)-2 b^2 e+2 b c (d-e x)}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2} \left (e (b d-a e)-c d^2\right )}+\frac{e^4 \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^{5/2}}-\frac{e^4 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*b^2*e + 4*c*(a*e + c*d*x) + 2*b*c*(d - e*x))/(3*(b^2 - 4*a*c)*(-(c*d^2) + e*
(b*d - a*e))*(a + x*(b + c*x))^(3/2)) + (2*(3*b^4*e^3 + b^3*c*e^2*(d + 3*e*x) +
8*c^2*(3*a^2*e^3 + 2*c^2*d^3*x + 5*a*c*d*e^2*x) + 4*b*c^2*(2*c*d^2*(d - 3*e*x) +
5*a*e^2*(d - e*x)) + 2*b^2*c*e*(-11*a*e^2 + c*d*(-6*d + e*x))))/(3*(b^2 - 4*a*c
)^2*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + x*(b + c*x)]) + (e^4*Log[d + e*x])/(c*
d^2 + e*(-(b*d) + a*e))^(5/2) - (e^4*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sq
rt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)]])/(c*d^2 + e*(-(b*d) + a*e))^
(5/2)

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Maple [B]  time = 0.018, size = 1408, normalized size = 4.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)/(c*x^2+b*x+a)^(5/2),x)

[Out]

1/3*e/(a*e^2-b*d*e+c*d^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)
/e^2)^(3/2)-2/3*e/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c*x*b+4/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/(c
*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*c^2*x*d-1/3*e/(a
*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*
d^2)/e^2)^(3/2)*b^2+2/3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)
/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(3/2)*b*c*d-16/3*e/(a*e^2-b*d*e+c*d^2)*c^2/(
4*a*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x
*b+32/3/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*d-8/3*e/(a*e^2-b*d*e+c*d^2)*c/(4*a*c-b^2)^2/(c
*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2+16/3/(a*e^2-
b*d*e+c*d^2)*c^2/(4*a*c-b^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c
*d^2)/e^2)^(1/2)*b*d+e^3/(a*e^2-b*d*e+c*d^2)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-2*e^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*x*b*c+4*e^2/(a*e^2-b*
d*e+c*d^2)^2/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/
e^2)^(1/2)*x*c^2*d-e^3/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2+2*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*
c-b^2)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b*c*d-e
^3/(a*e^2-b*d*e+c*d^2)^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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e
-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+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(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2361, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)),x, algorithm="fricas")

[Out]

[-1/6*(4*((b^3*c^2 - 12*a*b*c^3)*d^3 - 2*(b^4*c - 11*a*b^2*c^2 + 4*a^2*c^3)*d^2*
e + (b^5 - 6*a*b^3*c - 16*a^2*b*c^2)*d*e^2 - 4*(a*b^4 - 7*a^2*b^2*c + 8*a^3*c^2)
*e^3 - (16*c^5*d^3 - 24*b*c^4*d^2*e + 2*(b^2*c^3 + 20*a*c^4)*d*e^2 + (3*b^3*c^2
- 20*a*b*c^3)*e^3)*x^3 - 3*(8*b*c^4*d^3 - 12*b^2*c^3*d^2*e + (b^3*c^2 + 20*a*b*c
^3)*d*e^2 + 2*(b^4*c - 7*a*b^2*c^2 + 4*a^2*c^3)*e^3)*x^2 - 3*(2*(b^2*c^3 + 4*a*c
^4)*d^3 - 3*(b^3*c^2 + 4*a*b*c^3)*d^2*e + 8*(a*b^2*c^2 + 2*a^2*c^3)*d*e^2 + (b^5
- 6*a*b^3*c)*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a) - 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 + 1
6*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(((8*a*b*d*e -
8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2
- 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2
) - 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*
b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 +
2*d*e*x + d^2)))/(((a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*d^4 - 2*(a^2*b^5*c
- 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^3*e + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*d^
2*e^2 - 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d*e^3 + (a^4*b^4 - 8*a^5*b^2*c
+ 16*a^6*c^2)*e^4 + ((b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*d^4 - 2*(b^5*c^3 - 8*a
*b^3*c^4 + 16*a^2*b*c^5)*d^3*e + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^2*e^2 -
2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d*e^3 + (a^2*b^4*c^2 - 8*a^3*b^2*c^
3 + 16*a^4*c^4)*e^4)*x^4 + 2*((b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^4 - 2*(b^
6*c^2 - 8*a*b^4*c^3 + 16*a^2*b^2*c^4)*d^3*e + (b^7*c - 6*a*b^5*c^2 + 32*a^3*b*c^
4)*d^2*e^2 - 2*(a*b^6*c - 8*a^2*b^4*c^2 + 16*a^3*b^2*c^3)*d*e^3 + (a^2*b^5*c - 8
*a^3*b^3*c^2 + 16*a^4*b*c^3)*e^4)*x^3 + ((b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*d^
4 - 2*(b^7*c - 6*a*b^5*c^2 + 32*a^3*b*c^4)*d^3*e + (b^8 - 4*a*b^6*c - 12*a^2*b^4
*c^2 + 32*a^3*b^2*c^3 + 64*a^4*c^4)*d^2*e^2 - 2*(a*b^7 - 6*a^2*b^5*c + 32*a^4*b*
c^3)*d*e^3 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*e^4)*x^2 + 2*((a*b^5*c^2 - 8*a
^2*b^3*c^3 + 16*a^3*b*c^4)*d^4 - 2*(a*b^6*c - 8*a^2*b^4*c^2 + 16*a^3*b^2*c^3)*d^
3*e + (a*b^7 - 6*a^2*b^5*c + 32*a^4*b*c^3)*d^2*e^2 - 2*(a^2*b^6 - 8*a^3*b^4*c +
16*a^4*b^2*c^2)*d*e^3 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*e^4)*x)*sqrt(c*d^
2 - b*d*e + a*e^2)), -1/3*(2*((b^3*c^2 - 12*a*b*c^3)*d^3 - 2*(b^4*c - 11*a*b^2*c
^2 + 4*a^2*c^3)*d^2*e + (b^5 - 6*a*b^3*c - 16*a^2*b*c^2)*d*e^2 - 4*(a*b^4 - 7*a^
2*b^2*c + 8*a^3*c^2)*e^3 - (16*c^5*d^3 - 24*b*c^4*d^2*e + 2*(b^2*c^3 + 20*a*c^4)
*d*e^2 + (3*b^3*c^2 - 20*a*b*c^3)*e^3)*x^3 - 3*(8*b*c^4*d^3 - 12*b^2*c^3*d^2*e +
(b^3*c^2 + 20*a*b*c^3)*d*e^2 + 2*(b^4*c - 7*a*b^2*c^2 + 4*a^2*c^3)*e^3)*x^2 - 3
*(2*(b^2*c^3 + 4*a*c^4)*d^3 - 3*(b^3*c^2 + 4*a*b*c^3)*d^2*e + 8*(a*b^2*c^2 + 2*a
^2*c^3)*d*e^2 + (b^5 - 6*a*b^3*c)*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^
2 + b*x + a) + 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*sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x)/((c
*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a))))/(((a^2*b^4*c^2 - 8*a^3*b^2*c^3 +
16*a^4*c^4)*d^4 - 2*(a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*d^3*e + (a^2*b^6
- 6*a^3*b^4*c + 32*a^5*c^3)*d^2*e^2 - 2*(a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*d
*e^3 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*e^4 + ((b^4*c^4 - 8*a*b^2*c^5 + 16*a
^2*c^6)*d^4 - 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*d^3*e + (b^6*c^2 - 6*a*b^
4*c^3 + 32*a^3*c^5)*d^2*e^2 - 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d*e^3
+ (a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4)*e^4)*x^4 + 2*((b^5*c^3 - 8*a*b^3*c
^4 + 16*a^2*b*c^5)*d^4 - 2*(b^6*c^2 - 8*a*b^4*c^3 + 16*a^2*b^2*c^4)*d^3*e + (b^7
*c - 6*a*b^5*c^2 + 32*a^3*b*c^4)*d^2*e^2 - 2*(a*b^6*c - 8*a^2*b^4*c^2 + 16*a^3*b
^2*c^3)*d*e^3 + (a^2*b^5*c - 8*a^3*b^3*c^2 + 16*a^4*b*c^3)*e^4)*x^3 + ((b^6*c^2
- 6*a*b^4*c^3 + 32*a^3*c^5)*d^4 - 2*(b^7*c - 6*a*b^5*c^2 + 32*a^3*b*c^4)*d^3*e +
(b^8 - 4*a*b^6*c - 12*a^2*b^4*c^2 + 32*a^3*b^2*c^3 + 64*a^4*c^4)*d^2*e^2 - 2*(a
*b^7 - 6*a^2*b^5*c + 32*a^4*b*c^3)*d*e^3 + (a^2*b^6 - 6*a^3*b^4*c + 32*a^5*c^3)*
e^4)*x^2 + 2*((a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*d^4 - 2*(a*b^6*c - 8*a^
2*b^4*c^2 + 16*a^3*b^2*c^3)*d^3*e + (a*b^7 - 6*a^2*b^5*c + 32*a^4*b*c^3)*d^2*e^2
- 2*(a^2*b^6 - 8*a^3*b^4*c + 16*a^4*b^2*c^2)*d*e^3 + (a^3*b^5 - 8*a^4*b^3*c + 1
6*a^5*b*c^2)*e^4)*x)*sqrt(-c*d^2 + b*d*e - a*e^2))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/((d + e*x)*(a + b*x + c*x**2)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.32785, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(5/2)*(e*x + d)),x, algorithm="giac")

[Out]

Done