3.2387 \(\int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=473 \[ \frac{e \sqrt{a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}} \]

[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)*(d + e*x)*(a + b*x + c*x^2)^(3/2)) - (2*(6*a*c*e*(2*c*d - b*e)^2 - (
b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(b*d - 8*a*e)) - c*(2*c*
d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x))/(3*(b^2 - 4*a*c)^2*
(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(32*c^4*d^4 - 15
*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^2*c*e^3*(b*d + 5*a*e) + 4*c^2*e^2
*(3*b^2*d^2 - 36*a*b*d*e - 32*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^
2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (5*e^4*(2*c*d - b*e)*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])])/(2
*(c*d^2 - b*d*e + a*e^2)^(7/2))

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Rubi [A]  time = 1.73819, antiderivative size = 473, 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 \[ \frac{e \sqrt{a+b x+c x^2} \left (4 c^2 e^2 \left (-32 a^2 e^2-36 a b d e+3 b^2 d^2\right )+20 b^2 c e^3 (5 a e+b d)-16 c^3 d^2 e (4 b d-9 a e)-15 b^4 e^4+32 c^4 d^4\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \left (a e^2-b d e+c d^2\right )^3}-\frac{2 \left (-c x (2 c d-b e) \left (-4 c e (2 b d-7 a e)-5 b^2 e^2+8 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 c e (b d-8 a e)-5 b^2 e^2+8 c^2 d^2\right )+6 a c e (2 c d-b e)^2\right )}{3 \left (b^2-4 a c\right )^2 (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac{5 e^4 (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^2*(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)*(d + e*x)*(a + b*x + c*x^2)^(3/2)) - (2*(6*a*c*e*(2*c*d - b*e)^2 - (
b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 5*b^2*e^2 - 2*c*e*(b*d - 8*a*e)) - c*(2*c*
d - b*e)*(8*c^2*d^2 - 5*b^2*e^2 - 4*c*e*(2*b*d - 7*a*e))*x))/(3*(b^2 - 4*a*c)^2*
(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(32*c^4*d^4 - 15
*b^4*e^4 - 16*c^3*d^2*e*(4*b*d - 9*a*e) + 20*b^2*c*e^3*(b*d + 5*a*e) + 4*c^2*e^2
*(3*b^2*d^2 - 36*a*b*d*e - 32*a^2*e^2))*Sqrt[a + b*x + c*x^2])/(3*(b^2 - 4*a*c)^
2*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)) + (5*e^4*(2*c*d - b*e)*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])])/(2
*(c*d^2 - b*d*e + a*e^2)^(7/2))

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

[Out]

Timed out

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Mathematica [A]  time = 4.91143, size = 477, normalized size = 1.01 \[ -\frac{\sqrt{a+x (b+c x)} \left (\frac{2 \left (4 b c^2 \left (17 a^2 e^4-9 a c d e^2 (d-2 e x)-2 c^2 d^3 (d-4 e x)\right )-8 c^3 \left (a^2 e^3 (12 d-5 e x)+9 a c d^2 e^2 x+2 c^2 d^4 x\right )-b^3 c e^2 \left (43 a e^2+c d (3 d+10 e x)\right )+2 b^2 c^2 e \left (a e^2 (42 d-19 e x)+c d^2 (8 d-3 e x)\right )+6 b^5 e^4+b^4 c e^3 (6 e x-11 d)\right )}{\left (b^2-4 a c\right )^2 (a+x (b+c x))}+\frac{2 \left (e (a e-b d)+c d^2\right ) \left (b c \left (c d (d-2 e x)-3 a e^2\right )+2 c^2 \left (a e (2 d-e x)+c d^2 x\right )+b^3 e^2+b^2 c e (e x-2 d)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))^2}+\frac{3 e^5}{d+e x}\right )}{3 \left (e (a e-b d)+c d^2\right )^3}+\frac{5 e^4 (2 c d-b e) \log (d+e x)}{2 \left (e (a e-b d)+c d^2\right )^{7/2}}+\frac{5 e^4 (b e-2 c d) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{2 \left (e (a e-b d)+c d^2\right )^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a + x*(b + c*x)]*((3*e^5)/(d + e*x) + (2*(c*d^2 + e*(-(b*d) + a*e))*(b^3*
e^2 + b^2*c*e*(-2*d + e*x) + b*c*(-3*a*e^2 + c*d*(d - 2*e*x)) + 2*c^2*(c*d^2*x +
a*e*(2*d - e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) + (2*(6*b^5*e^4 + b^4*c*
e^3*(-11*d + 6*e*x) - 8*c^3*(2*c^2*d^4*x + 9*a*c*d^2*e^2*x + a^2*e^3*(12*d - 5*e
*x)) + 2*b^2*c^2*e*(a*e^2*(42*d - 19*e*x) + c*d^2*(8*d - 3*e*x)) + 4*b*c^2*(17*a
^2*e^4 - 2*c^2*d^3*(d - 4*e*x) - 9*a*c*d*e^2*(d - 2*e*x)) - b^3*c*e^2*(43*a*e^2
+ c*d*(3*d + 10*e*x))))/((b^2 - 4*a*c)^2*(a + x*(b + c*x)))))/(3*(c*d^2 + e*(-(b
*d) + a*e))^3) + (5*e^4*(2*c*d - b*e)*Log[d + e*x])/(2*(c*d^2 + e*(-(b*d) + a*e)
)^(7/2)) + (5*e^4*(-2*c*d + b*e)*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)]])/(2*(c*d^2 + e*(-(b*d) + a*e))^(
7/2))

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Maple [B]  time = 0.029, size = 2765, normalized size = 5.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*e^3/(a*e^2-b*d*e+c*d^2)^3/(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^2*d-5*e^3/(a*e^2-b*d*e+c*d^2)^3/((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))*c*d+20/3/(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)^(3/2)*c^3*x*d^2+10/3/(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)^(3
/2)*b*c^2*d^2+160/3/(a*e^2-b*d*e+c*d^2)^2*c^4/(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^2+80/3/(a*e^2-b*d*e+c*d^2)^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)*b*d^2+20/3*e^2/(a*e^2-b*d*e+c*d^2)^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^3-20/3*e/(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)^(3/2)*c^2*x*b
*d-64/3*c^2/(a*e^2-b*d*e+c*d^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-16/3*c^2/(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)*x+5/2*e^4/(a*e^2-b*
d*e+c*d^2)^3/((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))*b+5/6*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)^(3/2)*b^3
+5*e^3/(a*e^2-b*d*e+c*d^2)^3/(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)*c*d+5/2*e^4/(a*e^2-b*d*e+c*d^2)^3/(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^3+5/3*e/(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)^(3/2)*c*d-8/3*c/(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+20*e^2/(a*e^2-b*d*e+c*d^2)^3/(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^3*d^2-10*e^3/(a*e^2-b*d*e+c*d^2)^3
/(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*c*d+10*e^2/(a*e^2-b*d*e+c*d^2)^3/(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^2*d^2-128/3*c^3/(a*e^2-b*d*e+c*d^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-1/(a
*e^2-b*d*e+c*d^2)/(d/e+x)/(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)-5/6*e^2/(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)^(3/2)*b-5/2*e^4/(a*e^2-b*d*e+c*d^2)^3/(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-160/3*e/(a*e^2-b*d*e+c*d^2)^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*b*d+5/3*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)^(3/2)*c*x*b^2-10/3*e/(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)^(3/2)*b^2*c*d+4
0/3*e^2/(a*e^2-b*d*e+c*d^2)^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^2-80/3*e/(a*e^2-b*d*e+c*d^2)^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^2*d+5
*e^4/(a*e^2-b*d*e+c*d^2)^3/(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^2*c

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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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.42257, 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)^2),x, algorithm="fricas")

[Out]

[-1/12*(4*(2*(b^3*c^3 - 12*a*b*c^4)*d^5 - 2*(3*b^4*c^2 - 32*a*b^2*c^3 + 16*a^2*c
^4)*d^4*e + 6*(b^5*c - 7*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^2 - 2*(b^6 + 6*a*b^4*c -
84*a^2*b^2*c^2 + 112*a^3*c^3)*d^2*e^3 + 2*(7*a*b^5 - 50*a^2*b^3*c + 80*a^3*b*c^
2)*d*e^4 + 3*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^5 - (32*c^6*d^4*e - 64*b*c^5
*d^3*e^2 + 12*(b^2*c^4 + 12*a*c^5)*d^2*e^3 + 4*(5*b^3*c^3 - 36*a*b*c^4)*d*e^4 -
(15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*e^5)*x^4 - 2*(16*c^6*d^5 - 8*b*c^5*d^
4*e - 6*(7*b^2*c^4 - 12*a*c^5)*d^3*e^2 + (19*b^3*c^3 + 36*a*b*c^4)*d^2*e^3 + (15
*b^4*c^2 - 118*a*b^2*c^3 + 56*a^2*c^4)*d*e^4 - 3*(5*b^5*c - 35*a*b^3*c^2 + 52*a^
2*b*c^3)*e^5)*x^3 - 3*(16*b*c^5*d^5 - 4*(7*b^2*c^4 - 4*a*c^5)*d^4*e - 2*(b^3*c^3
- 20*a*b*c^4)*d^3*e^2 + 2*(7*b^4*c^2 - 34*a*b^2*c^3 + 56*a^2*c^4)*d^2*e^3 + 2*(
a*b^3*c^2 - 28*a^2*b*c^3)*d*e^4 - (5*b^6 - 30*a*b^4*c + 16*a^2*b^2*c^2 + 64*a^3*
c^3)*e^5)*x^2 - 2*(6*(b^2*c^4 + 4*a*c^5)*d^5 - (13*b^3*c^3 + 36*a*b*c^4)*d^4*e +
(3*b^4*c^2 + 22*a*b^2*c^3 + 88*a^2*c^4)*d^3*e^2 + 3*(3*b^5*c - 19*a*b^3*c^2 + 1
2*a^2*b*c^3)*d^2*e^3 - (5*b^6 - 42*a*b^4*c + 108*a^2*b^2*c^2 - 64*a^3*c^3)*d*e^4
- 2*(5*a*b^5 - 37*a^2*b^3*c + 64*a^3*b*c^2)*e^5)*x)*sqrt(c*d^2 - b*d*e + a*e^2)
*sqrt(c*x^2 + b*x + a) + 15*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d^2*e^4
- (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*
a^2*c^5)*d*e^5 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^6)*x^5 + (2*(b^4*c^3 -
8*a*b^2*c^4 + 16*a^2*c^5)*d^2*e^4 + 3*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*
e^5 - 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^6)*x^4 + (4*(b^5*c^2 - 8*a*b^3*
c^3 + 16*a^2*b*c^4)*d^2*e^4 + 4*(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a^3*c^4)*d*e^5 -
(b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^6)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^
4)*d^2*e^4 - (b^7 - 10*a*b^5*c + 32*a^2*b^3*c^2 - 32*a^3*b*c^3)*d*e^5 - 2*(a*b^6
- 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^6)*x^2 + (4*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3
*b*c^3)*d^2*e^4 - 2*(a*b^6 - 9*a^2*b^4*c + 24*a^3*b^2*c^2 - 16*a^4*c^3)*d*e^5 -
(a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^6)*x)*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^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^7 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c
^3 + 16*a^4*b*c^4)*d^6*e + 3*(a^2*b^6*c - 7*a^3*b^4*c^2 + 8*a^4*b^2*c^3 + 16*a^5
*c^4)*d^5*e^2 - (a^2*b^7 - 2*a^3*b^5*c - 32*a^4*b^3*c^2 + 96*a^5*b*c^3)*d^4*e^3
+ 3*(a^3*b^6 - 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*c^3)*d^3*e^4 - 3*(a^4*b^5 -
8*a^5*b^3*c + 16*a^6*b*c^2)*d^2*e^5 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*d*e^6
+ ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6*e - 3*(b^5*c^4 - 8*a*b^3*c^5 + 16*a
^2*b*c^6)*d^5*e^2 + 3*(b^6*c^3 - 7*a*b^4*c^4 + 8*a^2*b^2*c^5 + 16*a^3*c^6)*d^4*e
^3 - (b^7*c^2 - 2*a*b^5*c^3 - 32*a^2*b^3*c^4 + 96*a^3*b*c^5)*d^3*e^4 + 3*(a*b^6*
c^2 - 7*a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^2*e^5 - 3*(a^2*b^5*c^2 - 8*a
^3*b^3*c^3 + 16*a^4*b*c^4)*d*e^6 + (a^3*b^4*c^2 - 8*a^4*b^2*c^3 + 16*a^5*c^4)*e^
7)*x^5 + ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^7 - (b^5*c^4 - 8*a*b^3*c^5 + 16
*a^2*b*c^6)*d^6*e - 3*(b^6*c^3 - 9*a*b^4*c^4 + 24*a^2*b^2*c^5 - 16*a^3*c^6)*d^5*
e^2 + 5*(b^7*c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*c^4)*d^4*e^3 - (2*b^8*c - 7*a*b^6*c^
2 - 43*a^2*b^4*c^3 + 168*a^3*b^2*c^4 - 48*a^4*c^5)*d^3*e^4 + 3*(2*a*b^7*c - 15*a
^2*b^5*c^2 + 24*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^2*e^5 - (6*a^2*b^6*c - 49*a^3*b^4*
c^2 + 104*a^4*b^2*c^3 - 16*a^5*c^4)*d*e^6 + 2*(a^3*b^5*c - 8*a^4*b^3*c^2 + 16*a^
5*b*c^3)*e^7)*x^4 + (2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^7 - (5*b^6*c^3 -
42*a*b^4*c^4 + 96*a^2*b^2*c^5 - 32*a^3*c^6)*d^6*e + 3*(b^7*c^2 - 8*a*b^5*c^3 +
16*a^2*b^3*c^4)*d^5*e^2 + (b^8*c - 11*a*b^6*c^2 + 46*a^2*b^4*c^3 - 96*a^3*b^2*c^
4 + 96*a^4*c^5)*d^4*e^3 - (b^9 - 6*a*b^7*c + 6*a^2*b^5*c^2 - 16*a^3*b^3*c^3 + 96
*a^4*b*c^4)*d^3*e^4 + 3*(a*b^8 - 7*a^2*b^6*c + 10*a^3*b^4*c^2 + 32*a^5*c^4)*d^2*
e^5 - (3*a^2*b^7 - 20*a^3*b^5*c + 16*a^4*b^3*c^2 + 64*a^5*b*c^3)*d*e^6 + (a^3*b^
6 - 6*a^4*b^4*c + 32*a^6*c^3)*e^7)*x^3 + ((b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*d
^7 - (3*b^7*c^2 - 20*a*b^5*c^3 + 16*a^2*b^3*c^4 + 64*a^3*b*c^5)*d^6*e + 3*(b^8*c
- 7*a*b^6*c^2 + 10*a^2*b^4*c^3 + 32*a^4*c^5)*d^5*e^2 - (b^9 - 6*a*b^7*c + 6*a^2
*b^5*c^2 - 16*a^3*b^3*c^3 + 96*a^4*b*c^4)*d^4*e^3 + (a*b^8 - 11*a^2*b^6*c + 46*a
^3*b^4*c^2 - 96*a^4*b^2*c^3 + 96*a^5*c^4)*d^3*e^4 + 3*(a^2*b^7 - 8*a^3*b^5*c + 1
6*a^4*b^3*c^2)*d^2*e^5 - (5*a^3*b^6 - 42*a^4*b^4*c + 96*a^5*b^2*c^2 - 32*a^6*c^3
)*d*e^6 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*e^7)*x^2 + (2*(a*b^5*c^3 - 8*
a^2*b^3*c^4 + 16*a^3*b*c^5)*d^7 - (6*a*b^6*c^2 - 49*a^2*b^4*c^3 + 104*a^3*b^2*c^
4 - 16*a^4*c^5)*d^6*e + 3*(2*a*b^7*c - 15*a^2*b^5*c^2 + 24*a^3*b^3*c^3 + 16*a^4*
b*c^4)*d^5*e^2 - (2*a*b^8 - 7*a^2*b^6*c - 43*a^3*b^4*c^2 + 168*a^4*b^2*c^3 - 48*
a^5*c^4)*d^4*e^3 + 5*(a^2*b^7 - 8*a^3*b^5*c + 16*a^4*b^3*c^2)*d^3*e^4 - 3*(a^3*b
^6 - 9*a^4*b^4*c + 24*a^5*b^2*c^2 - 16*a^6*c^3)*d^2*e^5 - (a^4*b^5 - 8*a^5*b^3*c
+ 16*a^6*b*c^2)*d*e^6 + (a^5*b^4 - 8*a^6*b^2*c + 16*a^7*c^2)*e^7)*x)*sqrt(c*d^2
- b*d*e + a*e^2)), -1/6*(2*(2*(b^3*c^3 - 12*a*b*c^4)*d^5 - 2*(3*b^4*c^2 - 32*a*
b^2*c^3 + 16*a^2*c^4)*d^4*e + 6*(b^5*c - 7*a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e^2 - 2*
(b^6 + 6*a*b^4*c - 84*a^2*b^2*c^2 + 112*a^3*c^3)*d^2*e^3 + 2*(7*a*b^5 - 50*a^2*b
^3*c + 80*a^3*b*c^2)*d*e^4 + 3*(a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^5 - (32*c^
6*d^4*e - 64*b*c^5*d^3*e^2 + 12*(b^2*c^4 + 12*a*c^5)*d^2*e^3 + 4*(5*b^3*c^3 - 36
*a*b*c^4)*d*e^4 - (15*b^4*c^2 - 100*a*b^2*c^3 + 128*a^2*c^4)*e^5)*x^4 - 2*(16*c^
6*d^5 - 8*b*c^5*d^4*e - 6*(7*b^2*c^4 - 12*a*c^5)*d^3*e^2 + (19*b^3*c^3 + 36*a*b*
c^4)*d^2*e^3 + (15*b^4*c^2 - 118*a*b^2*c^3 + 56*a^2*c^4)*d*e^4 - 3*(5*b^5*c - 35
*a*b^3*c^2 + 52*a^2*b*c^3)*e^5)*x^3 - 3*(16*b*c^5*d^5 - 4*(7*b^2*c^4 - 4*a*c^5)*
d^4*e - 2*(b^3*c^3 - 20*a*b*c^4)*d^3*e^2 + 2*(7*b^4*c^2 - 34*a*b^2*c^3 + 56*a^2*
c^4)*d^2*e^3 + 2*(a*b^3*c^2 - 28*a^2*b*c^3)*d*e^4 - (5*b^6 - 30*a*b^4*c + 16*a^2
*b^2*c^2 + 64*a^3*c^3)*e^5)*x^2 - 2*(6*(b^2*c^4 + 4*a*c^5)*d^5 - (13*b^3*c^3 + 3
6*a*b*c^4)*d^4*e + (3*b^4*c^2 + 22*a*b^2*c^3 + 88*a^2*c^4)*d^3*e^2 + 3*(3*b^5*c
- 19*a*b^3*c^2 + 12*a^2*b*c^3)*d^2*e^3 - (5*b^6 - 42*a*b^4*c + 108*a^2*b^2*c^2 -
64*a^3*c^3)*d*e^4 - 2*(5*a*b^5 - 37*a^2*b^3*c + 64*a^3*b*c^2)*e^5)*x)*sqrt(-c*d
^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 15*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 1
6*a^4*c^3)*d^2*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*d*e^5 + (2*(b^4*c^3
- 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^5 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^6)*
x^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^2*e^4 + 3*(b^5*c^2 - 8*a*b^3*c^3
+ 16*a^2*b*c^4)*d*e^5 - 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^6)*x^4 + (4*
(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^2*e^4 + 4*(a*b^4*c^2 - 8*a^2*b^2*c^3 +
16*a^3*c^4)*d*e^5 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^6)*x^3 + (2*(b^6*c - 6*a*
b^4*c^2 + 32*a^3*c^4)*d^2*e^4 - (b^7 - 10*a*b^5*c + 32*a^2*b^3*c^2 - 32*a^3*b*c^
3)*d*e^5 - 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^6)*x^2 + (4*(a*b^5*c - 8*a
^2*b^3*c^2 + 16*a^3*b*c^3)*d^2*e^4 - 2*(a*b^6 - 9*a^2*b^4*c + 24*a^3*b^2*c^2 - 1
6*a^4*c^3)*d*e^5 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^6)*x)*arctan(-1/2*sq
rt(-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^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^7 -
3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^6*e + 3*(a^2*b^6*c - 7*a^3*b^4*
c^2 + 8*a^4*b^2*c^3 + 16*a^5*c^4)*d^5*e^2 - (a^2*b^7 - 2*a^3*b^5*c - 32*a^4*b^3*
c^2 + 96*a^5*b*c^3)*d^4*e^3 + 3*(a^3*b^6 - 7*a^4*b^4*c + 8*a^5*b^2*c^2 + 16*a^6*
c^3)*d^3*e^4 - 3*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d^2*e^5 + (a^5*b^4 - 8*a
^6*b^2*c + 16*a^7*c^2)*d*e^6 + ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^6*e - 3*(
b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^5*e^2 + 3*(b^6*c^3 - 7*a*b^4*c^4 + 8*a^2
*b^2*c^5 + 16*a^3*c^6)*d^4*e^3 - (b^7*c^2 - 2*a*b^5*c^3 - 32*a^2*b^3*c^4 + 96*a^
3*b*c^5)*d^3*e^4 + 3*(a*b^6*c^2 - 7*a^2*b^4*c^3 + 8*a^3*b^2*c^4 + 16*a^4*c^5)*d^
2*e^5 - 3*(a^2*b^5*c^2 - 8*a^3*b^3*c^3 + 16*a^4*b*c^4)*d*e^6 + (a^3*b^4*c^2 - 8*
a^4*b^2*c^3 + 16*a^5*c^4)*e^7)*x^5 + ((b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*d^7 -
(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*d^6*e - 3*(b^6*c^3 - 9*a*b^4*c^4 + 24*a^
2*b^2*c^5 - 16*a^3*c^6)*d^5*e^2 + 5*(b^7*c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*c^4)*d^4
*e^3 - (2*b^8*c - 7*a*b^6*c^2 - 43*a^2*b^4*c^3 + 168*a^3*b^2*c^4 - 48*a^4*c^5)*d
^3*e^4 + 3*(2*a*b^7*c - 15*a^2*b^5*c^2 + 24*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^2*e^5
- (6*a^2*b^6*c - 49*a^3*b^4*c^2 + 104*a^4*b^2*c^3 - 16*a^5*c^4)*d*e^6 + 2*(a^3*b
^5*c - 8*a^4*b^3*c^2 + 16*a^5*b*c^3)*e^7)*x^4 + (2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a
^2*b*c^6)*d^7 - (5*b^6*c^3 - 42*a*b^4*c^4 + 96*a^2*b^2*c^5 - 32*a^3*c^6)*d^6*e +
3*(b^7*c^2 - 8*a*b^5*c^3 + 16*a^2*b^3*c^4)*d^5*e^2 + (b^8*c - 11*a*b^6*c^2 + 46
*a^2*b^4*c^3 - 96*a^3*b^2*c^4 + 96*a^4*c^5)*d^4*e^3 - (b^9 - 6*a*b^7*c + 6*a^2*b
^5*c^2 - 16*a^3*b^3*c^3 + 96*a^4*b*c^4)*d^3*e^4 + 3*(a*b^8 - 7*a^2*b^6*c + 10*a^
3*b^4*c^2 + 32*a^5*c^4)*d^2*e^5 - (3*a^2*b^7 - 20*a^3*b^5*c + 16*a^4*b^3*c^2 + 6
4*a^5*b*c^3)*d*e^6 + (a^3*b^6 - 6*a^4*b^4*c + 32*a^6*c^3)*e^7)*x^3 + ((b^6*c^3 -
6*a*b^4*c^4 + 32*a^3*c^6)*d^7 - (3*b^7*c^2 - 20*a*b^5*c^3 + 16*a^2*b^3*c^4 + 64
*a^3*b*c^5)*d^6*e + 3*(b^8*c - 7*a*b^6*c^2 + 10*a^2*b^4*c^3 + 32*a^4*c^5)*d^5*e^
2 - (b^9 - 6*a*b^7*c + 6*a^2*b^5*c^2 - 16*a^3*b^3*c^3 + 96*a^4*b*c^4)*d^4*e^3 +
(a*b^8 - 11*a^2*b^6*c + 46*a^3*b^4*c^2 - 96*a^4*b^2*c^3 + 96*a^5*c^4)*d^3*e^4 +
3*(a^2*b^7 - 8*a^3*b^5*c + 16*a^4*b^3*c^2)*d^2*e^5 - (5*a^3*b^6 - 42*a^4*b^4*c +
96*a^5*b^2*c^2 - 32*a^6*c^3)*d*e^6 + 2*(a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*e
^7)*x^2 + (2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*d^7 - (6*a*b^6*c^2 - 49*
a^2*b^4*c^3 + 104*a^3*b^2*c^4 - 16*a^4*c^5)*d^6*e + 3*(2*a*b^7*c - 15*a^2*b^5*c^
2 + 24*a^3*b^3*c^3 + 16*a^4*b*c^4)*d^5*e^2 - (2*a*b^8 - 7*a^2*b^6*c - 43*a^3*b^4
*c^2 + 168*a^4*b^2*c^3 - 48*a^5*c^4)*d^4*e^3 + 5*(a^2*b^7 - 8*a^3*b^5*c + 16*a^4
*b^3*c^2)*d^3*e^4 - 3*(a^3*b^6 - 9*a^4*b^4*c + 24*a^5*b^2*c^2 - 16*a^6*c^3)*d^2*
e^5 - (a^4*b^5 - 8*a^5*b^3*c + 16*a^6*b*c^2)*d*e^6 + (a^5*b^4 - 8*a^6*b^2*c + 16
*a^7*c^2)*e^7)*x)*sqrt(-c*d^2 + b*d*e - a*e^2))]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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