3.1205 \(\int \frac{A+B x}{(d+e x)^3 \left (b x+c x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=374 \[ -\frac{e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-5 A e)-4 b c d (2 A e+B d)+8 A c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}-\frac{3 e \left (B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )-A e \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)}-\frac{e \sqrt{b x+c x^2} \left (3 b^3 e^2 (B d-5 A e)-2 b^2 c d e (5 B d-19 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3} \]
[Out]
(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e
*x)^2*Sqrt[b*x + c*x^2]) - (e*(8*A*c^2*d^2 - b^2*e*(B*d - 5*A*e) - 4*b*c*d*(B*d
+ 2*A*e))*Sqrt[b*x + c*x^2])/(2*b^2*d^2*(c*d - b*e)^2*(d + e*x)^2) - (e*(16*A*c^
3*d^3 - 2*b^2*c*d*e*(5*B*d - 19*A*e) + 3*b^3*e^2*(B*d - 5*A*e) - 8*b*c^2*d^2*(B*
d + 3*A*e))*Sqrt[b*x + c*x^2])/(4*b^2*d^3*(c*d - b*e)^3*(d + e*x)) - (3*e*(B*d*(
8*c^2*d^2 - 4*b*c*d*e + b^2*e^2) - A*e*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2))*Ar
cTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8
*d^(7/2)*(c*d - b*e)^(7/2))
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Rubi [A] time = 1.44014, antiderivative size = 374, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192
\[ -\frac{e \sqrt{b x+c x^2} \left (b^2 (-e) (B d-5 A e)-4 b c d (2 A e+B d)+8 A c^2 d^2\right )}{2 b^2 d^2 (d+e x)^2 (c d-b e)^2}-\frac{3 e \left (B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )-A e \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )\right ) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{7/2} (c d-b e)^{7/2}}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{b^2 d \sqrt{b x+c x^2} (d+e x)^2 (c d-b e)}-\frac{e \sqrt{b x+c x^2} \left (3 b^3 e^2 (B d-5 A e)-2 b^2 c d e (5 B d-19 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )}{4 b^2 d^3 (d+e x) (c d-b e)^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/((d + e*x)^3*(b*x + c*x^2)^(3/2)),x]
[Out]
(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^2*d*(c*d - b*e)*(d + e
*x)^2*Sqrt[b*x + c*x^2]) - (e*(8*A*c^2*d^2 - b^2*e*(B*d - 5*A*e) - 4*b*c*d*(B*d
+ 2*A*e))*Sqrt[b*x + c*x^2])/(2*b^2*d^2*(c*d - b*e)^2*(d + e*x)^2) - (e*(16*A*c^
3*d^3 - 2*b^2*c*d*e*(5*B*d - 19*A*e) + 3*b^3*e^2*(B*d - 5*A*e) - 8*b*c^2*d^2*(B*
d + 3*A*e))*Sqrt[b*x + c*x^2])/(4*b^2*d^3*(c*d - b*e)^3*(d + e*x)) - (3*e*(B*d*(
8*c^2*d^2 - 4*b*c*d*e + b^2*e^2) - A*e*(16*c^2*d^2 - 16*b*c*d*e + 5*b^2*e^2))*Ar
cTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(8
*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)**3/(c*x**2+b*x)**(3/2),x)
[Out]
Timed out
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Mathematica [A] time = 2.6068, size = 277, normalized size = 0.74 \[ \frac{x^{3/2} \left (\frac{(b+c x)^2 \left (\frac{8 c^3 x (A c-b B)}{b^2 (b+c x) (b e-c d)^3}-\frac{8 A}{b^2 d^3}+\frac{e^2 x (7 A e (b e-2 c d)+B d (10 c d-3 b e))}{d^3 (d+e x) (c d-b e)^3}+\frac{2 e^2 x (B d-A e)}{d^2 (d+e x)^2 (c d-b e)^2}\right )}{\sqrt{x}}-\frac{3 e (b+c x)^{3/2} \left (A e \left (5 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (b^2 e^2-4 b c d e+8 c^2 d^2\right )\right ) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{7/2} (b e-c d)^{7/2}}\right )}{4 (x (b+c x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/((d + e*x)^3*(b*x + c*x^2)^(3/2)),x]
[Out]
(x^(3/2)*(((b + c*x)^2*((-8*A)/(b^2*d^3) + (8*c^3*(-(b*B) + A*c)*x)/(b^2*(-(c*d)
+ b*e)^3*(b + c*x)) + (2*e^2*(B*d - A*e)*x)/(d^2*(c*d - b*e)^2*(d + e*x)^2) + (
e^2*(B*d*(10*c*d - 3*b*e) + 7*A*e*(-2*c*d + b*e))*x)/(d^3*(c*d - b*e)^3*(d + e*x
))))/Sqrt[x] - (3*e*(-(B*d*(8*c^2*d^2 - 4*b*c*d*e + b^2*e^2)) + A*e*(16*c^2*d^2
- 16*b*c*d*e + 5*b^2*e^2))*(b + c*x)^(3/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(
Sqrt[d]*Sqrt[b + c*x])])/(d^(7/2)*(-(c*d) + b*e)^(7/2))))/(4*(x*(b + c*x))^(3/2)
)
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Maple [B] time = 0.022, size = 3735, normalized size = 10. \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/(e*x+d)^3/(c*x^2+b*x)^(3/2),x)
[Out]
15/2*e/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))*b*c*B-13*e/d^2/(b*e-c*d)^2*c^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*A+30/(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^2*B-15/2/(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^2*B+30/(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^4*A+15/4*e^
2/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^2*
B-30*e/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
^2*A-8*e/d^2/(b*e-c*d)^2*c/(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(
1/2)*A+1/2/e/d/(b*e-c*d)/(d/e+x)^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+5/2/e/(b*e-c*d)^2/(d/e+x)/(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+13/d/(b*e-c*d)^2*c^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)*A-5/4/d/(b*e-c*d)^2/(d/e+x)/(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-5/2/d/(b*e-c*d)^2/(d/e+x)/(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-9/2*B/d/(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))*c-3*B*e/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)*b-19*B/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)^(1/2)*c^2+B/e/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)^(1/2)+45/(
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^3*B-1
5/4*e^3/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^2*A+3/2*e*c/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))*A+15/(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^3*A-15/4*e^3/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*b*A+15/4*e^2/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*B-45*e/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^3*A-30/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^4*B*d-8*
B/e*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)^(1/2
)*x-3*B*e/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)*x*c+25*B/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)*x*c^2-38*B/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)^(1/2)*x*c^3+3/2*B*e/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))*b-4*B/e*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)^(1/2)+75/4*e^2/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*c*A-75/4*e/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)*b*c*B+45/2*e^2/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^2*A-45/2
*e/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)*x*c^2
*B-1/2/e^2/(b*e-c*d)/(d/e+x)^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+17*B/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
)^(1/2)*c-15/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^3*B*d+26/d/(b*e-c*d)^2*c^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*A+15/8*e^3/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^2*A-15/8*e^2/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^2*B+15/2*e/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^2*A+5/4*e/d^2/(b*e-c*d)^2/(d/e+x)/(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-15/2*e^2/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*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)^(3/2)*(e*x + d)^3),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.308316, 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)^(3/2)*(e*x + d)^3),x, algorithm="fricas")
[Out]
[-1/8*(3*(8*B*b^2*c^2*d^5*e - 5*A*b^4*d^2*e^4 - 4*(B*b^3*c + 4*A*b^2*c^2)*d^4*e^
2 + (B*b^4 + 16*A*b^3*c)*d^3*e^3 + (8*B*b^2*c^2*d^3*e^3 - 5*A*b^4*e^6 - 4*(B*b^3
*c + 4*A*b^2*c^2)*d^2*e^4 + (B*b^4 + 16*A*b^3*c)*d*e^5)*x^2 + 2*(8*B*b^2*c^2*d^4
*e^2 - 5*A*b^4*d*e^5 - 4*(B*b^3*c + 4*A*b^2*c^2)*d^3*e^3 + (B*b^4 + 16*A*b^3*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*(8*A*b*c^3*d^5 - 24*A*b^2*c
^2*d^4*e + 24*A*b^3*c*d^3*e^2 - 8*A*b^4*d^2*e^3 - (15*A*b^3*c*e^5 + 8*(B*b*c^3 -
2*A*c^4)*d^3*e^2 + 2*(5*B*b^2*c^2 + 12*A*b*c^3)*d^2*e^3 - (3*B*b^3*c + 38*A*b^2
*c^2)*d*e^4)*x^3 - (15*A*b^4*e^5 + 16*(B*b*c^3 - 2*A*c^4)*d^4*e + 4*(3*B*b^2*c^2
+ 10*A*b*c^3)*d^3*e^2 + 5*(B*b^3*c - 8*A*b^2*c^2)*d^2*e^3 - (3*B*b^4 + 13*A*b^3
*c)*d*e^4)*x^2 - (8*A*b*c^3*d^4*e + 25*A*b^4*d*e^4 + 8*(B*b*c^3 - 2*A*c^4)*d^5 +
12*(B*b^3*c + 2*A*b^2*c^2)*d^3*e^2 - (5*B*b^4 + 56*A*b^3*c)*d^2*e^3)*x)*sqrt(c*
d^2 - b*d*e))/((b^2*c^3*d^8 - 3*b^3*c^2*d^7*e + 3*b^4*c*d^6*e^2 - b^5*d^5*e^3 +
(b^2*c^3*d^6*e^2 - 3*b^3*c^2*d^5*e^3 + 3*b^4*c*d^4*e^4 - b^5*d^3*e^5)*x^2 + 2*(b
^2*c^3*d^7*e - 3*b^3*c^2*d^6*e^2 + 3*b^4*c*d^5*e^3 - b^5*d^4*e^4)*x)*sqrt(c*d^2
- b*d*e)*sqrt(c*x^2 + b*x)), 1/4*(3*(8*B*b^2*c^2*d^5*e - 5*A*b^4*d^2*e^4 - 4*(B*
b^3*c + 4*A*b^2*c^2)*d^4*e^2 + (B*b^4 + 16*A*b^3*c)*d^3*e^3 + (8*B*b^2*c^2*d^3*e
^3 - 5*A*b^4*e^6 - 4*(B*b^3*c + 4*A*b^2*c^2)*d^2*e^4 + (B*b^4 + 16*A*b^3*c)*d*e^
5)*x^2 + 2*(8*B*b^2*c^2*d^4*e^2 - 5*A*b^4*d*e^5 - 4*(B*b^3*c + 4*A*b^2*c^2)*d^3*
e^3 + (B*b^4 + 16*A*b^3*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)) - (8*A*b*c^3*d^5 - 24*A*b^2*c^2*d^4*e +
24*A*b^3*c*d^3*e^2 - 8*A*b^4*d^2*e^3 - (15*A*b^3*c*e^5 + 8*(B*b*c^3 - 2*A*c^4)*
d^3*e^2 + 2*(5*B*b^2*c^2 + 12*A*b*c^3)*d^2*e^3 - (3*B*b^3*c + 38*A*b^2*c^2)*d*e^
4)*x^3 - (15*A*b^4*e^5 + 16*(B*b*c^3 - 2*A*c^4)*d^4*e + 4*(3*B*b^2*c^2 + 10*A*b*
c^3)*d^3*e^2 + 5*(B*b^3*c - 8*A*b^2*c^2)*d^2*e^3 - (3*B*b^4 + 13*A*b^3*c)*d*e^4)
*x^2 - (8*A*b*c^3*d^4*e + 25*A*b^4*d*e^4 + 8*(B*b*c^3 - 2*A*c^4)*d^5 + 12*(B*b^3
*c + 2*A*b^2*c^2)*d^3*e^2 - (5*B*b^4 + 56*A*b^3*c)*d^2*e^3)*x)*sqrt(-c*d^2 + b*d
*e))/((b^2*c^3*d^8 - 3*b^3*c^2*d^7*e + 3*b^4*c*d^6*e^2 - b^5*d^5*e^3 + (b^2*c^3*
d^6*e^2 - 3*b^3*c^2*d^5*e^3 + 3*b^4*c*d^4*e^4 - b^5*d^3*e^5)*x^2 + 2*(b^2*c^3*d^
7*e - 3*b^3*c^2*d^6*e^2 + 3*b^4*c*d^5*e^3 - b^5*d^4*e^4)*x)*sqrt(-c*d^2 + b*d*e)
*sqrt(c*x^2 + b*x))]
_______________________________________________________________________________________
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)**3/(c*x**2+b*x)**(3/2),x)
[Out]
Timed out
_______________________________________________________________________________________
GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/((c*x^2 + b*x)^(3/2)*(e*x + d)^3),x, algorithm="giac")
[Out]
Exception raised: TypeError