3.2484 \(\int \frac{A+B x}{(d+e x) \left (a+b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=436 \[ \frac{2 \left (c x \left ((2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{e^3 (B d-A e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e
- 2*a*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)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + (b*c*d - b^2
*e + 2*a*c*e)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B*
d*e + 3*a*A*e^2)) + c*(4*c*e*(b*d - 2*a*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) +
(2*c*d - b*e)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B
*d*e + 3*a*A*e^2)))*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^3*(B*d - A*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])])/(c*d^2 - b*d*e + a*e^2)^(5/2)

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Rubi [A]  time = 1.58113, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{2 \left (c x \left ((2 c d-b e) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (4 c \left (3 a A e^2-a B d e+2 A c d^2\right )+3 b^2 e (B d-A e)-4 b c d (A e+B d)\right )+4 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac{e^3 (B d-A e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e
- 2*a*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)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + (b*c*d - b^2
*e + 2*a*c*e)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B*
d*e + 3*a*A*e^2)) + c*(4*c*e*(b*d - 2*a*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) +
(2*c*d - b*e)*(3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e) + 4*c*(2*A*c*d^2 - a*B
*d*e + 3*a*A*e^2)))*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^3*(B*d - A*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])])/(c*d^2 - b*d*e + a*e^2)^(5/2)

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Rubi in Sympy [A]  time = 155.071, size = 454, normalized size = 1.04 \[ - \frac{e^{3} \left (A e - B d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} + \frac{2 \left (A \left (- 2 a c e + b^{2} e - b c d\right ) - B a \left (b e - 2 c d\right ) + c x \left (A b e - 2 A c d - 2 B a e + B b d\right )\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (a e^{2} - b d e + c d^{2}\right )} + \frac{4 \left (- 2 a c e \left (b e - 2 c d\right ) \left (A b e - 2 A c d - 2 B a e + B b d\right ) + \frac{c x \left (- 4 c e \left (2 a e - b d\right ) \left (A b e - 2 A c d - 2 B a e + B b d\right ) + \left (A \left (- 12 a c e^{2} + 3 b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right ) - B d \left (- 4 a c e + b \left (3 b e - 4 c d\right )\right )\right ) \left (b e - 2 c d\right )\right )}{2} + \left (\frac{A \left (- 12 a c e^{2} + 3 b^{2} e^{2} + 4 b c d e - 8 c^{2} d^{2}\right )}{2} - \frac{B d \left (- 4 a c e + b \left (3 b e - 4 c d\right )\right )}{2}\right ) \left (- 2 a c e + b^{2} e - b c d\right )\right )}{3 \left (- 4 a c + b^{2}\right )^{2} \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**3*(A*e - B*d)*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*(A*(-2*a*
c*e + b**2*e - b*c*d) - B*a*(b*e - 2*c*d) + c*x*(A*b*e - 2*A*c*d - 2*B*a*e + B*b
*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)*(A*b*e - 2*A*c*d - 2*B*a*e + B*b*d) + c*x*(-4*c*e*(2*a*
e - b*d)*(A*b*e - 2*A*c*d - 2*B*a*e + B*b*d) + (A*(-12*a*c*e**2 + 3*b**2*e**2 +
4*b*c*d*e - 8*c**2*d**2) - B*d*(-4*a*c*e + b*(3*b*e - 4*c*d)))*(b*e - 2*c*d))/2
+ (A*(-12*a*c*e**2 + 3*b**2*e**2 + 4*b*c*d*e - 8*c**2*d**2)/2 - B*d*(-4*a*c*e +
b*(3*b*e - 4*c*d))/2)*(-2*a*c*e + b**2*e - b*c*d))/(3*(-4*a*c + b**2)**2*sqrt(a
+ b*x + c*x**2)*(a*e**2 - b*d*e + c*d**2)**2)

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Mathematica [A]  time = 4.68963, size = 479, normalized size = 1.1 \[ \frac{2 \left (4 b c \left (A c \left (5 a e^2 (d-e x)+2 c d^2 (d-3 e x)\right )-B \left (-2 a^2 e^3+a c d e (d+3 e x)+2 c^2 d^3 x\right )\right )+8 c^2 \left (a^2 e^2 (3 A e-3 B d+2 B e x)+a c d e x (5 A e-B d)+2 A c^2 d^3 x\right )-2 b^2 c \left (A e \left (11 a e^2+c d (6 d-e x)\right )+B d \left (c d (2 d-7 e x)-3 a e^2\right )\right )+3 b^4 e^2 (A e-B d)+b^3 c e (A e (d+3 e x)+B d (7 d-3 e x))\right )}{3 \left (b^2-4 a c\right )^2 \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2}+\frac{2 \left (2 A c (a e+c d x)+a B (b e-2 c d+2 c e x)-A b^2 e+A b c (d-e x)-b B c d x\right )}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2} \left (e (b d-a e)-c d^2\right )}+\frac{e^3 (A e-B d) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{e^3 (B d-A e) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(A*b^2*e) - b*B*c*d*x + 2*A*c*(a*e + c*d*x) + A*b*c*(d - e*x) + a*B*(-2*c*d
+ b*e + 2*c*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^2*(-(B*d) + A*e) + b^3*c*e*(B*d*(7*d - 3*e*x) + A*e*(d +
3*e*x)) + 8*c^2*(2*A*c^2*d^3*x + a*c*d*e*(-(B*d) + 5*A*e)*x + a^2*e^2*(-3*B*d +
3*A*e + 2*B*e*x)) - 2*b^2*c*(B*d*(-3*a*e^2 + c*d*(2*d - 7*e*x)) + A*e*(11*a*e^2
+ c*d*(6*d - e*x))) + 4*b*c*(A*c*(2*c*d^2*(d - 3*e*x) + 5*a*e^2*(d - e*x)) - B*(
-2*a^2*e^3 + 2*c^2*d^3*x + a*c*d*e*(d + 3*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^3*(-(B*d) + A*e)*Log[d + e*x])/(c
*d^2 + e*(-(b*d) + a*e))^(5/2) + (e^3*(B*d - A*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)]])/(c*d^2 + e*(-(
b*d) + a*e))^(5/2)

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Maple [B]  time = 0.023, size = 2996, normalized size = 6.9 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3/(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)^(3/2)*c*x*b*B*d+2*e^2/(a*e^2-b*d*e+c*d^2)^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*B*d+e^2/(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*(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))*B*d-1/3*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)^(3/2)*b^2*
A-e^3/(a*e^2-b*d*e+c*d^2)^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*A+1/3/(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)^(3/2)*b^2*B*d+16/3*B/e*c/(4*a*c-
b^2)^2/(c*x^2+b*x+a)^(1/2)*b+4/3*B/e/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*c*x+32/3*B/
e*c^2/(4*a*c-b^2)^2/(c*x^2+b*x+a)^(1/2)*x+1/3*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)^(3/2)*A+e^3/(a*e^2-b*d*e+c*d^2)^
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)*A-1/3/(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)^(3/2)*
B*d+8/3/(a*e^2-b*d*e+c*d^2)*c/(4*a*c-b^2)^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*B*d+16/3/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2)^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*d*A-8/3*e/(
a*e^2-b*d*e+c*d^2)*c/(4*a*c-b^2)^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*A-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*(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))*A+2/3*B/e/(4*a*c-b^2)/(c*x^2+b*x+a)^(3/2)*b-e^2/(a*e^2-b*d*e+c*d^2)^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*d+4/3/(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)^(3/2)*c^2*x*d*A+e^2/(a*e^2-b*d*e+c*d^2)^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*B*d+2/3/(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)^(3/2)*b*c*d*A
+32/3/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^2)^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*A+16/3/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2)^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*B*d-2*e/(a*
e^2-b*d*e+c*d^2)^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*B+4*e^2/(a*e^2-b*d*e+c*d^2)^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*A-4*e/(a*e^2-b*d*e+
c*d^2)^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*B+2*e^2/(a*e^2-b*d*e+c*d^2)^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*A-2/3*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)^(3/2)*c*x
*b*A-4/3/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)^(3/2)*c^2*x*d^2*B-2/3/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)^(3/2)*b*c*d^2*B-16/3*
e/(a*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2)^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*A-32/3/e/(a*e^2-b*d*e+c*d^2)*c^3/(4*a*c-b^2)^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^2*B-16/3/e/(a
*e^2-b*d*e+c*d^2)*c^2/(4*a*c-b^2)^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*d^2*B-2*e^3/(a*e^2-b*d*e+c*d^2)^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*A

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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((B*x + A)/((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 = 3.41027, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/6*(4*((4*(2*B*a^2 - 3*A*a*b)*c^3 + (2*B*a*b^2 + A*b^3)*c^2)*d^3 - 2*(4*A*a^2
*c^3 + (4*B*a^2*b - 11*A*a*b^2)*c^2 + (2*B*a*b^3 + A*b^4)*c)*d^2*e + (2*B*a*b^4
+ A*b^5 + 16*(2*B*a^3 - A*a^2*b)*c^2 - 2*(2*B*a^2*b^2 + 3*A*a*b^3)*c)*d*e^2 + (B
*a^2*b^3 - 4*A*a*b^4 - 32*A*a^3*c^2 - 4*(3*B*a^3*b - 7*A*a^2*b^2)*c)*e^3 + (8*(B
*b*c^4 - 2*A*c^5)*d^3 - 2*(7*B*b^2*c^3 - 4*(B*a + 3*A*b)*c^4)*d^2*e + (3*B*b^3*c
^2 - 40*A*a*c^4 + 2*(6*B*a*b - A*b^2)*c^3)*d*e^2 - (3*A*b^3*c^2 + 4*(4*B*a^2 - 5
*A*a*b)*c^3)*e^3)*x^3 + 3*(4*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - (7*B*b^3*c^2 - 4*(B*a
*b + 3*A*b^2)*c^3)*d^2*e + (2*B*b^4*c + 4*(2*B*a^2 - 5*A*a*b)*c^3 + (2*B*a*b^2 -
A*b^3)*c^2)*d*e^2 - 2*(A*b^4*c + 4*A*a^2*c^3 + (4*B*a^2*b - 7*A*a*b^2)*c^2)*e^3
)*x^2 + 3*((B*b^3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b - A*b^2)*c^3)*d^3 - (2*B*b^4*c -
12*A*a*b*c^3 + (4*B*a*b^2 - 3*A*b^3)*c^2)*d^2*e + (B*b^5 - 2*B*a*b^3*c - 16*A*a^
2*c^3 + 8*(2*B*a^2*b - A*a*b^2)*c^2)*d*e^2 - (A*b^5 + 8*B*a^3*c^2 + 2*(B*a^2*b^2
- 3*A*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*a^2*b^4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2)*d*e^3 - (A*a^2*b^4 - 8*A*a^3*b^2*c +
16*A*a^4*c^2)*e^4 + ((B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*d*e^3 - (A*b^4*c
^2 - 8*A*a*b^2*c^3 + 16*A*a^2*c^4)*e^4)*x^4 + 2*((B*b^5*c - 8*B*a*b^3*c^2 + 16*B
*a^2*b*c^3)*d*e^3 - (A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*e^4)*x^3 + ((B*b^
6 - 6*B*a*b^4*c + 32*B*a^3*c^3)*d*e^3 - (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*e^4
)*x^2 + 2*((B*a*b^5 - 8*B*a^2*b^3*c + 16*B*a^3*b*c^2)*d*e^3 - (A*a*b^5 - 8*A*a^2
*b^3*c + 16*A*a^3*b*c^2)*e^4)*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^
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 + 3
2*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 + 3
2*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*(
(4*(2*B*a^2 - 3*A*a*b)*c^3 + (2*B*a*b^2 + A*b^3)*c^2)*d^3 - 2*(4*A*a^2*c^3 + (4*
B*a^2*b - 11*A*a*b^2)*c^2 + (2*B*a*b^3 + A*b^4)*c)*d^2*e + (2*B*a*b^4 + A*b^5 +
16*(2*B*a^3 - A*a^2*b)*c^2 - 2*(2*B*a^2*b^2 + 3*A*a*b^3)*c)*d*e^2 + (B*a^2*b^3 -
4*A*a*b^4 - 32*A*a^3*c^2 - 4*(3*B*a^3*b - 7*A*a^2*b^2)*c)*e^3 + (8*(B*b*c^4 - 2
*A*c^5)*d^3 - 2*(7*B*b^2*c^3 - 4*(B*a + 3*A*b)*c^4)*d^2*e + (3*B*b^3*c^2 - 40*A*
a*c^4 + 2*(6*B*a*b - A*b^2)*c^3)*d*e^2 - (3*A*b^3*c^2 + 4*(4*B*a^2 - 5*A*a*b)*c^
3)*e^3)*x^3 + 3*(4*(B*b^2*c^3 - 2*A*b*c^4)*d^3 - (7*B*b^3*c^2 - 4*(B*a*b + 3*A*b
^2)*c^3)*d^2*e + (2*B*b^4*c + 4*(2*B*a^2 - 5*A*a*b)*c^3 + (2*B*a*b^2 - A*b^3)*c^
2)*d*e^2 - 2*(A*b^4*c + 4*A*a^2*c^3 + (4*B*a^2*b - 7*A*a*b^2)*c^2)*e^3)*x^2 + 3*
((B*b^3*c^2 - 8*A*a*c^4 + 2*(2*B*a*b - A*b^2)*c^3)*d^3 - (2*B*b^4*c - 12*A*a*b*c
^3 + (4*B*a*b^2 - 3*A*b^3)*c^2)*d^2*e + (B*b^5 - 2*B*a*b^3*c - 16*A*a^2*c^3 + 8*
(2*B*a^2*b - A*a*b^2)*c^2)*d*e^2 - (A*b^5 + 8*B*a^3*c^2 + 2*(B*a^2*b^2 - 3*A*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*a^2*b^
4 - 8*B*a^3*b^2*c + 16*B*a^4*c^2)*d*e^3 - (A*a^2*b^4 - 8*A*a^3*b^2*c + 16*A*a^4*
c^2)*e^4 + ((B*b^4*c^2 - 8*B*a*b^2*c^3 + 16*B*a^2*c^4)*d*e^3 - (A*b^4*c^2 - 8*A*
a*b^2*c^3 + 16*A*a^2*c^4)*e^4)*x^4 + 2*((B*b^5*c - 8*B*a*b^3*c^2 + 16*B*a^2*b*c^
3)*d*e^3 - (A*b^5*c - 8*A*a*b^3*c^2 + 16*A*a^2*b*c^3)*e^4)*x^3 + ((B*b^6 - 6*B*a
*b^4*c + 32*B*a^3*c^3)*d*e^3 - (A*b^6 - 6*A*a*b^4*c + 32*A*a^3*c^3)*e^4)*x^2 + 2
*((B*a*b^5 - 8*B*a^2*b^3*c + 16*B*a^3*b*c^2)*d*e^3 - (A*a*b^5 - 8*A*a^2*b^3*c +
16*A*a^3*b*c^2)*e^4)*x)*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 + 3
2*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 + 3
2*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))]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.369533, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done