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3.1212 \(\int \frac{A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx\)

Optimal. Leaf size=449 \[ -\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}+\frac{2 \left (b (c d-b e) \left (b^2 e (3 B d-5 A e)-2 b c d (A e+2 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-5 A e)+2 b^2 c d e (8 B d-A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}+\frac{e \sqrt{b x+c x^2} \left (3 b^4 e^3 (3 B d-5 A e)-2 b^3 c d e^2 (9 B d-10 A e)+4 b^2 c^2 d^2 e (3 A e+10 B d)-16 b c^3 d^3 (4 A e+B d)+32 A c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}-\frac{e^3 (B d (8 c d-3 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \]

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*(d +
 e*x)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 5*A
*e) - 2*b*c*d*(2*B*d + A*e)) + c*(16*A*c^3*d^3 - b^3*e^2*(3*B*d - 5*A*e) + 2*b^2
*c*d*e*(8*B*d - A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*x))/(3*b^4*d^2*(c*d - b*e)^2*(
d + e*x)*Sqrt[b*x + c*x^2]) + (e*(32*A*c^4*d^4 - 2*b^3*c*d*e^2*(9*B*d - 10*A*e)
+ 3*b^4*e^3*(3*B*d - 5*A*e) + 4*b^2*c^2*d^2*e*(10*B*d + 3*A*e) - 16*b*c^3*d^3*(B
*d + 4*A*e))*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)) - (e^3*(B*d*
(8*c*d - 3*b*e) - 5*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d
]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*d^(7/2)*(c*d - b*e)^(7/2))

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Rubi [A]  time = 1.74378, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}+\frac{2 \left (b (c d-b e) \left (b^2 e (3 B d-5 A e)-2 b c d (A e+2 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-5 A e)+2 b^2 c d e (8 B d-A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}+\frac{e \sqrt{b x+c x^2} \left (3 b^4 e^3 (3 B d-5 A e)-2 b^3 c d e^2 (9 B d-10 A e)+4 b^2 c^2 d^2 e (3 A e+10 B d)-16 b c^3 d^3 (4 A e+B d)+32 A c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}-\frac{e^3 (B d (8 c d-3 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*(d +
 e*x)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 5*A
*e) - 2*b*c*d*(2*B*d + A*e)) + c*(16*A*c^3*d^3 - b^3*e^2*(3*B*d - 5*A*e) + 2*b^2
*c*d*e*(8*B*d - A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*x))/(3*b^4*d^2*(c*d - b*e)^2*(
d + e*x)*Sqrt[b*x + c*x^2]) + (e*(32*A*c^4*d^4 - 2*b^3*c*d*e^2*(9*B*d - 10*A*e)
+ 3*b^4*e^3*(3*B*d - 5*A*e) + 4*b^2*c^2*d^2*e*(10*B*d + 3*A*e) - 16*b*c^3*d^3*(B
*d + 4*A*e))*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)) - (e^3*(B*d*
(8*c*d - 3*b*e) - 5*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d
]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*d^(7/2)*(c*d - b*e)^(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((B*x+A)/(e*x+d)**2/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 3.97388, size = 285, normalized size = 0.63 \[ \frac{x^{5/2} \left (\frac{(b+c x)^3 \left (\frac{2 x (6 A b e+8 A c d-3 b B d)}{b^4 d^3}+\frac{2 c^3 x^2 (A c-b B)}{b^3 (b+c x)^2 (c d-b e)^2}-\frac{2 A}{b^3 d^2}-\frac{2 c^3 x^2 \left (-b c (14 A e+5 B d)+8 A c^2 d+11 b^2 B e\right )}{b^4 (b+c x) (b e-c d)^3}+\frac{3 e^4 x^2 (B d-A e)}{d^3 (d+e x) (c d-b e)^3}\right )}{3 x^{3/2}}+\frac{e^3 (b+c x)^{5/2} (5 A e (b e-2 c d)+B d (8 c d-3 b e)) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{7/2} (b e-c d)^{7/2}}\right )}{(x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^(5/2)*(((b + c*x)^3*((-2*A)/(b^3*d^2) + (2*(-3*b*B*d + 8*A*c*d + 6*A*b*e)*x)/
(b^4*d^3) + (2*c^3*(-(b*B) + A*c)*x^2)/(b^3*(c*d - b*e)^2*(b + c*x)^2) - (2*c^3*
(8*A*c^2*d + 11*b^2*B*e - b*c*(5*B*d + 14*A*e))*x^2)/(b^4*(-(c*d) + b*e)^3*(b +
c*x)) + (3*e^4*(B*d - A*e)*x^2)/(d^3*(c*d - b*e)^3*(d + e*x))))/(3*x^(3/2)) + (e
^3*(B*d*(8*c*d - 3*b*e) + 5*A*e*(-2*c*d + b*e))*(b + c*x)^(5/2)*ArcTan[(Sqrt[-(c
*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(d^(7/2)*(-(c*d) + b*e)^(7/2))))/(
x*(b + c*x))^(5/2)

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Maple [B]  time = 0.026, size = 4295, normalized size = 9.6 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*B/d/(b*e-c*d)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)-5/(
b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*B-5*e^2/d
/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+
x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2
)^(1/2))/(d/e+x))*c*B+5*e^4/d^3/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2)*x*c*A-5*e^3/d^2/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c*B-26/3*B*e/d/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c-2/3*B/d/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*x+20/3*B/e/(b*e-c*d)/b^2/(c*(d/e+x)^2+(b*
e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2*x+16/3*B/d/(b*e-c*d)*c^2/b^3/(c*(d
/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x-160/3*B/e/(b*e-c*d)*c^3/b
^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x+20*e^2/d/(b*e-c*d
)^3/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*B-160/3*e/
d/(b*e-c*d)^2*c^3/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*
x*A+20*e^2/d/(b*e-c*d)^3/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)
^(1/2)*x*c^3*A+20/3*e/d/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-
c*d)/e^2)^(3/2)*c^2*x*A+2*B*e^2/d^2/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/
e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c-52/3*B*e/d/(b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2+20/3/e/(b*e-c*d)^2/b^2/(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^3*x*B*d+40/3*e^2/d^2/(b*e-c*d)^2*c
^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*A-160/3/e/(b*
e-c*d)^2*c^4/b^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*B*d
+1/d/(b*e-c*d)/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)
*A-10/3/(b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*
c^2*A+80/3/(b*e-c*d)^2*c^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^
2)^(1/2)*B-1/e/(b*e-c*d)/(d/e+x)/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/
e^2)^(3/2)*B+2*B*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c
*d)/e^2)^(1/2)+8/3*B/d/(b*e-c*d)*c/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e
-c*d)/e^2)^(1/2)-80/3*B/e/(b*e-c*d)*c^2/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2)+5*e/d/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(3/2)*c*A+5*e^4/d^3/(b*e-c*d)^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2)*b*A-5/2*e^4/d^3/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-
2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+
(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b*A+10/3/e/(b*e-c*d)^2/b/
(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2*B*d+20/3*e^2/d^2/(
b*e-c*d)^2*c/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A-80/3*
e/d/(b*e-c*d)^2*c^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2
)*A-80/3/e/(b*e-c*d)^2*c^3/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^
2)^(1/2)*B*d-16/3*c^2/d/(b*e-c*d)/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-
c*d)/e^2)^(3/2)*x*A+128/3*c^3/d/(b*e-c*d)/b^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
-d*(b*e-c*d)/e^2)^(1/2)*x*A-20*e/(b*e-c*d)^3/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^3*B-5/3*e^2/d^2/(b*e-c*d)^2/(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*x*A+10*e^2/d/(b*e-c*d)^3/b/(c*(d/e+x)^2+(
b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A+5/3*e/d/(b*e-c*d)^2/(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c*x*B+5/2*e^3/d^2/(b*e-c*d)^3/(-
d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c
*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x
))*b*B+5*e^3/d^2/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*
e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)
-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c*A+80/3/(b*e-c*d)^2*c^3/b^3/(c*(d/e+x)^2+(b*e
-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*A-5*e^3/d^2/(b*e-c*d)^3/(c*(d/e+x)^2+(b
*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*b*B-15*e^3/d^2/(b*e-c*d)^3/(c*(d/e+x)
^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c*A+15*e^2/d/(b*e-c*d)^3/(c*(d/e
+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c*B-5/3*e^2/d^2/(b*e-c*d)^2/(
c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*A+5/3*e/d/(b*e-c*d)^2
/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*b*B-10*e/(b*e-c*d)^3/
b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*c^2*B+160/3/(b*e-c*d
)^2*c^3/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*B-20/3/(
b*e-c*d)^2/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^2*x*B-2
0/3/(b*e-c*d)^2/b^2/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(3/2)*c^
3*x*A+10/3*B/e/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(
3/2)*c+160/3/(b*e-c*d)^2*c^4/b^4/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/
e^2)^(1/2)*x*A-8/3*c/d/(b*e-c*d)/b/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d
)/e^2)^(3/2)*A+64/3*c^2/d/(b*e-c*d)/b^3/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*
e-c*d)/e^2)^(1/2)*A-B*e^2/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e
-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c
*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))-20*e^3/d^2/(b*e-c*d)^3/b/(c*(d/e+
x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*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)^(5/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.317967, 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)^(5/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

[-1/6*(3*((8*B*b^4*c^2*d^2*e^4 + 5*A*b^5*c*e^6 - (3*B*b^5*c + 10*A*b^4*c^2)*d*e^
5)*x^3 + (8*B*b^4*c^2*d^3*e^3 + 5*A*b^6*e^6 + 5*(B*b^5*c - 2*A*b^4*c^2)*d^2*e^4
- (3*B*b^6 + 5*A*b^5*c)*d*e^5)*x^2 + (8*B*b^5*c*d^3*e^3 + 5*A*b^6*d*e^5 - (3*B*b
^6 + 10*A*b^5*c)*d^2*e^4)*x)*sqrt(c*x^2 + b*x)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2
 + b*x) + sqrt(c*d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)) + 2*(2*A*b^3*c
^3*d^5 - 6*A*b^4*c^2*d^4*e + 6*A*b^5*c*d^3*e^2 - 2*A*b^6*d^2*e^3 + (15*A*b^4*c^2
*e^5 + 16*(B*b*c^5 - 2*A*c^6)*d^4*e - 8*(5*B*b^2*c^4 - 8*A*b*c^5)*d^3*e^2 + 6*(3
*B*b^3*c^3 - 2*A*b^2*c^4)*d^2*e^3 - (9*B*b^4*c^2 + 20*A*b^3*c^3)*d*e^4)*x^4 + 2*
(15*A*b^5*c*e^5 + 8*(B*b*c^5 - 2*A*c^6)*d^5 - 8*(B*b^2*c^4 - A*b*c^5)*d^4*e - 21
*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3*e^2 + (15*B*b^4*c^2 - 19*A*b^3*c^3)*d^2*e^3 - 3*(
3*B*b^5*c + 5*A*b^4*c^2)*d*e^4)*x^3 - 3*(3*B*b^6*d*e^4 - 5*A*b^6*e^5 - 8*(B*b^2*
c^4 - 2*A*b*c^5)*d^5 + 2*(9*B*b^3*c^3 - 14*A*b^2*c^4)*d^4*e - 2*(3*B*b^4*c^2 + A
*b^3*c^3)*d^3*e^2 - 2*(B*b^5*c - 7*A*b^4*c^2)*d^2*e^3)*x^2 + 2*(5*A*b^6*d*e^4 +
3*(B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - (9*B*b^4*c^2 - 13*A*b^3*c^3)*d^4*e + 3*(3*B*b^
5*c - A*b^4*c^2)*d^3*e^2 - 3*(B*b^6 + 3*A*b^5*c)*d^2*e^3)*x)*sqrt(c*d^2 - b*d*e)
)/(((b^4*c^4*d^6*e - 3*b^5*c^3*d^5*e^2 + 3*b^6*c^2*d^4*e^3 - b^7*c*d^3*e^4)*x^3
+ (b^4*c^4*d^7 - 2*b^5*c^3*d^6*e + 2*b^7*c*d^4*e^3 - b^8*d^3*e^4)*x^2 + (b^5*c^3
*d^7 - 3*b^6*c^2*d^6*e + 3*b^7*c*d^5*e^2 - b^8*d^4*e^3)*x)*sqrt(c*d^2 - b*d*e)*s
qrt(c*x^2 + b*x)), 1/3*(3*((8*B*b^4*c^2*d^2*e^4 + 5*A*b^5*c*e^6 - (3*B*b^5*c + 1
0*A*b^4*c^2)*d*e^5)*x^3 + (8*B*b^4*c^2*d^3*e^3 + 5*A*b^6*e^6 + 5*(B*b^5*c - 2*A*
b^4*c^2)*d^2*e^4 - (3*B*b^6 + 5*A*b^5*c)*d*e^5)*x^2 + (8*B*b^5*c*d^3*e^3 + 5*A*b
^6*d*e^5 - (3*B*b^6 + 10*A*b^5*c)*d^2*e^4)*x)*sqrt(c*x^2 + b*x)*arctan(-sqrt(-c*
d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - (2*A*b^3*c^3*d^5 - 6*A*b^4*c^2
*d^4*e + 6*A*b^5*c*d^3*e^2 - 2*A*b^6*d^2*e^3 + (15*A*b^4*c^2*e^5 + 16*(B*b*c^5 -
 2*A*c^6)*d^4*e - 8*(5*B*b^2*c^4 - 8*A*b*c^5)*d^3*e^2 + 6*(3*B*b^3*c^3 - 2*A*b^2
*c^4)*d^2*e^3 - (9*B*b^4*c^2 + 20*A*b^3*c^3)*d*e^4)*x^4 + 2*(15*A*b^5*c*e^5 + 8*
(B*b*c^5 - 2*A*c^6)*d^5 - 8*(B*b^2*c^4 - A*b*c^5)*d^4*e - 21*(B*b^3*c^3 - 2*A*b^
2*c^4)*d^3*e^2 + (15*B*b^4*c^2 - 19*A*b^3*c^3)*d^2*e^3 - 3*(3*B*b^5*c + 5*A*b^4*
c^2)*d*e^4)*x^3 - 3*(3*B*b^6*d*e^4 - 5*A*b^6*e^5 - 8*(B*b^2*c^4 - 2*A*b*c^5)*d^5
 + 2*(9*B*b^3*c^3 - 14*A*b^2*c^4)*d^4*e - 2*(3*B*b^4*c^2 + A*b^3*c^3)*d^3*e^2 -
2*(B*b^5*c - 7*A*b^4*c^2)*d^2*e^3)*x^2 + 2*(5*A*b^6*d*e^4 + 3*(B*b^3*c^3 - 2*A*b
^2*c^4)*d^5 - (9*B*b^4*c^2 - 13*A*b^3*c^3)*d^4*e + 3*(3*B*b^5*c - A*b^4*c^2)*d^3
*e^2 - 3*(B*b^6 + 3*A*b^5*c)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d*e))/(((b^4*c^4*d^6*e
- 3*b^5*c^3*d^5*e^2 + 3*b^6*c^2*d^4*e^3 - b^7*c*d^3*e^4)*x^3 + (b^4*c^4*d^7 - 2*
b^5*c^3*d^6*e + 2*b^7*c*d^4*e^3 - b^8*d^3*e^4)*x^2 + (b^5*c^3*d^7 - 3*b^6*c^2*d^
6*e + 3*b^7*c*d^5*e^2 - b^8*d^4*e^3)*x)*sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x))]

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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)**2/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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