3.1252 \(\int \frac{A+B x}{\sqrt{d+e x} \left (b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=418 \[ -\frac{\sqrt{d+e x} (-3 A b e-8 A c d+4 b B d)}{4 b^2 d^2 x (b+c x)^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right )}{4 b^5 d^{5/2}}+\frac{c \sqrt{d+e x} \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 (c d-b e)}+\frac{c^{3/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}+\frac{c \sqrt{d+e x} \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 (b+c x) (c d-b e)^2}-\frac{A \sqrt{d+e x}}{2 b d x^2 (b+c x)^2} \]

[Out]

(c*(12*A*c^2*d^2 + b^2*e*(4*B*d - 3*A*e) - b*c*d*(6*B*d + 7*A*e))*Sqrt[d + e*x])
/(4*b^3*d^2*(c*d - b*e)*(b + c*x)^2) - (A*Sqrt[d + e*x])/(2*b*d*x^2*(b + c*x)^2)
- ((4*b*B*d - 8*A*c*d - 3*A*b*e)*Sqrt[d + e*x])/(4*b^2*d^2*x*(b + c*x)^2) + (c*
(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 3*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e
*(19*B*d + 6*A*e))*Sqrt[d + e*x])/(4*b^4*d^2*(c*d - b*e)^2*(b + c*x)) - ((48*A*c
^2*d^2 - b^2*e*(4*B*d - 3*A*e) - 12*b*c*d*(2*B*d - A*e))*ArcTanh[Sqrt[d + e*x]/S
qrt[d]])/(4*b^5*d^(5/2)) + (c^(3/2)*(48*A*c^3*d^2 - 35*b^3*B*e^2 - 12*b*c^2*d*(2
*B*d + 9*A*e) + 7*b^2*c*e*(8*B*d + 9*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[
c*d - b*e]])/(4*b^5*(c*d - b*e)^(5/2))

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Rubi [A]  time = 2.24785, antiderivative size = 418, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{\sqrt{d+e x} (-3 A b e-8 A c d+4 b B d)}{4 b^2 d^2 x (b+c x)^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right )}{4 b^5 d^{5/2}}+\frac{c \sqrt{d+e x} \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )}{4 b^3 d^2 (b+c x)^2 (c d-b e)}+\frac{c^{3/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}}+\frac{c \sqrt{d+e x} \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )}{4 b^4 d^2 (b+c x) (c d-b e)^2}-\frac{A \sqrt{d+e x}}{2 b d x^2 (b+c x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^3),x]

[Out]

(c*(12*A*c^2*d^2 + b^2*e*(4*B*d - 3*A*e) - b*c*d*(6*B*d + 7*A*e))*Sqrt[d + e*x])
/(4*b^3*d^2*(c*d - b*e)*(b + c*x)^2) - (A*Sqrt[d + e*x])/(2*b*d*x^2*(b + c*x)^2)
- ((4*b*B*d - 8*A*c*d - 3*A*b*e)*Sqrt[d + e*x])/(4*b^2*d^2*x*(b + c*x)^2) + (c*
(24*A*c^3*d^3 - b^3*e^2*(4*B*d - 3*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e
*(19*B*d + 6*A*e))*Sqrt[d + e*x])/(4*b^4*d^2*(c*d - b*e)^2*(b + c*x)) - ((48*A*c
^2*d^2 - b^2*e*(4*B*d - 3*A*e) - 12*b*c*d*(2*B*d - A*e))*ArcTanh[Sqrt[d + e*x]/S
qrt[d]])/(4*b^5*d^(5/2)) + (c^(3/2)*(48*A*c^3*d^2 - 35*b^3*B*e^2 - 12*b*c^2*d*(2
*B*d + 9*A*e) + 7*b^2*c*e*(8*B*d + 9*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[
c*d - b*e]])/(4*b^5*(c*d - b*e)^(5/2))

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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)/(c*x**2+b*x)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 2.25856, size = 290, normalized size = 0.69 \[ \frac{b \sqrt{d+e x} \left (\frac{c^2 \left (-b c (15 A e+8 B d)+12 A c^2 d+11 b^2 B e\right )}{(b+c x) (c d-b e)^2}+\frac{2 b c^2 (b B-A c)}{(b+c x)^2 (b e-c d)}+\frac{3 A b e+12 A c d-4 b B d}{d^2 x}-\frac{2 A b}{d x^2}\right )+\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (b^2 e (4 B d-3 A e)+12 b c d (2 B d-A e)-48 A c^2 d^2\right )}{d^{5/2}}+\frac{c^{3/2} \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{(c d-b e)^{5/2}}}{4 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(Sqrt[d + e*x]*(b*x + c*x^2)^3),x]

[Out]

(b*Sqrt[d + e*x]*((-2*A*b)/(d*x^2) + (-4*b*B*d + 12*A*c*d + 3*A*b*e)/(d^2*x) + (
2*b*c^2*(b*B - A*c))/((-(c*d) + b*e)*(b + c*x)^2) + (c^2*(12*A*c^2*d + 11*b^2*B*
e - b*c*(8*B*d + 15*A*e)))/((c*d - b*e)^2*(b + c*x))) + ((-48*A*c^2*d^2 + b^2*e*
(4*B*d - 3*A*e) + 12*b*c*d*(2*B*d - A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(5/2
) + (c^(3/2)*(48*A*c^3*d^2 - 35*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 9*A*e) + 7*b^2*c
*e*(8*B*d + 9*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(c*d - b*e
)^(5/2))/(4*b^5)

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Maple [B]  time = 0.04, size = 1009, normalized size = 2.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(c*x^2+b*x)^3/(e*x+d)^(1/2),x)

[Out]

27*e*c^4/b^4/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1
/2)/((b*e-c*d)*c)^(1/2))*A*d-14*e*c^3/b^3/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)
*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d-3*e/b^4/d^(3/2)*arctan
h((e*x+d)^(1/2)/d^(1/2))*A*c-1/e/b^3/x^2/d*(e*x+d)^(3/2)*B-3/e/b^4/x^2*(e*x+d)^(
1/2)*A*c-12/b^5/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2+6/b^4/d^(1/2)*arcta
nh((e*x+d)^(1/2)/d^(1/2))*B*c+3/4/b^3/x^2/d^2*(e*x+d)^(3/2)*A+1/e/b^3/x^2*(e*x+d
)^(1/2)*B-3/4*e^2/b^3/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A+e/b^3/d^(3/2)*arc
tanh((e*x+d)^(1/2)/d^(1/2))*B-63/4*e^2*c^3/b^3/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e
-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A+35/4*e^2*c^2/b^2/(b
^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*
c)^(1/2))*B+3/e/b^4/x^2/d*(e*x+d)^(3/2)*A*c-12*c^5/b^5/(b^2*e^2-2*b*c*d*e+c^2*d^
2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d^2+6*c^4/b
^4/(b^2*e^2-2*b*c*d*e+c^2*d^2)/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/((b*e-
c*d)*c)^(1/2))*B*d^2-15/4*e^2*c^4/b^3/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*
(e*x+d)^(3/2)*A+11/4*e^2*c^3/b^2/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*e+c^2*d^2)*(e*x+
d)^(3/2)*B-17/4*e^2*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*A+13/4*e^2*c^2
/b^2/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*B+3*e*c^5/b^4/(c*e*x+b*e)^2/(b^2*e^2-
2*b*c*d*e+c^2*d^2)*(e*x+d)^(3/2)*A*d-2*e*c^4/b^3/(c*e*x+b*e)^2/(b^2*e^2-2*b*c*d*
e+c^2*d^2)*(e*x+d)^(3/2)*B*d+3*e*c^4/b^4/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*A
*d-2*e*c^3/b^3/(c*e*x+b*e)^2/(b*e-c*d)*(e*x+d)^(1/2)*B*d-5/4/b^3/x^2/d*(e*x+d)^(
1/2)*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)^3*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 33.8215, 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*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

[-1/8*(((24*(B*b*c^5 - 2*A*c^6)*d^4 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^3*e + 7*(5
*B*b^3*c^3 - 9*A*b^2*c^4)*d^2*e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^4 - 4*(
14*B*b^3*c^3 - 27*A*b^2*c^4)*d^3*e + 7*(5*B*b^4*c^2 - 9*A*b^3*c^3)*d^2*e^2)*x^3
+ (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^3*e + 7*
(5*B*b^5*c - 9*A*b^4*c^2)*d^2*e^2)*x^2)*sqrt(d)*sqrt(c/(c*d - b*e))*log((c*e*x +
2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*d - b*e)))/(c*x + b)) + 2*(
2*A*b^4*c^2*d^3 - 4*A*b^5*c*d^2*e + 2*A*b^6*d*e^2 - (3*A*b^4*c^2*e^3 - 12*(B*b^2
*c^4 - 2*A*b*c^5)*d^3 + (19*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(2*B*b^4*c^2 - 3
*A*b^3*c^3)*d*e^2)*x^3 - (6*A*b^5*c*e^3 - 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + (29
*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e - 2*(4*B*b^5*c - 5*A*b^4*c^2)*d*e^2)*x^2 - (3*A
*b^6*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3 + (8*B*b^5*c - 13*A*b^4*c^2)*d^2*e -
2*(2*B*b^6 - A*b^5*c)*d*e^2)*x)*sqrt(e*x + d)*sqrt(d) - ((3*A*b^4*c^2*e^4 - 24*(
B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^3*e - (16*B*b^3*c^3 - 2
7*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e
^4 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (1
6*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2*(2*B*b^5*c - 3*A*b^4*c^2)*d*e^3)*x^3 + (
3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^3*c^3)
*d^3*e - (16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^
2)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*x + d)*d)/x))/(((b^5*c^4*d^4 - 2*b^6*c^3*
d^3*e + b^7*c^2*d^2*e^2)*x^4 + 2*(b^6*c^3*d^4 - 2*b^7*c^2*d^3*e + b^8*c*d^2*e^2)
*x^3 + (b^7*c^2*d^4 - 2*b^8*c*d^3*e + b^9*d^2*e^2)*x^2)*sqrt(d)), -1/8*(2*((24*(
B*b*c^5 - 2*A*c^6)*d^4 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^3*e + 7*(5*B*b^3*c^3 -
9*A*b^2*c^4)*d^2*e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^4 - 4*(14*B*b^3*c^3
- 27*A*b^2*c^4)*d^3*e + 7*(5*B*b^4*c^2 - 9*A*b^3*c^3)*d^2*e^2)*x^3 + (24*(B*b^3*
c^3 - 2*A*b^2*c^4)*d^4 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^3*e + 7*(5*B*b^5*c -
9*A*b^4*c^2)*d^2*e^2)*x^2)*sqrt(d)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt
(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) + 2*(2*A*b^4*c^2*d^3 - 4*A*b^5*c*d^2*e + 2*A
*b^6*d*e^2 - (3*A*b^4*c^2*e^3 - 12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + (19*B*b^3*c^3 -
36*A*b^2*c^4)*d^2*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^2)*x^3 - (6*A*b^5*c*e^3
- 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e - 2*(4
*B*b^5*c - 5*A*b^4*c^2)*d*e^2)*x^2 - (3*A*b^6*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*
d^3 + (8*B*b^5*c - 13*A*b^4*c^2)*d^2*e - 2*(2*B*b^6 - A*b^5*c)*d*e^2)*x)*sqrt(e*
x + d)*sqrt(d) - ((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^
4 - 21*A*b*c^5)*d^3*e - (16*B*b^3*c^3 - 27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 -
3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + 4
*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2
*(2*B*b^5*c - 3*A*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c
^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^3*c^3)*d^3*e - (16*B*b^5*c - 27*A*b^4*c^2)*d^
2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*log(((e*x + 2*d)*sqrt(d) - 2*sqrt(e*
x + d)*d)/x))/(((b^5*c^4*d^4 - 2*b^6*c^3*d^3*e + b^7*c^2*d^2*e^2)*x^4 + 2*(b^6*c
^3*d^4 - 2*b^7*c^2*d^3*e + b^8*c*d^2*e^2)*x^3 + (b^7*c^2*d^4 - 2*b^8*c*d^3*e + b
^9*d^2*e^2)*x^2)*sqrt(d)), -1/8*(((24*(B*b*c^5 - 2*A*c^6)*d^4 - 4*(14*B*b^2*c^4
- 27*A*b*c^5)*d^3*e + 7*(5*B*b^3*c^3 - 9*A*b^2*c^4)*d^2*e^2)*x^4 + 2*(24*(B*b^2*
c^4 - 2*A*b*c^5)*d^4 - 4*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^3*e + 7*(5*B*b^4*c^2 -
9*A*b^3*c^3)*d^2*e^2)*x^3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 - 4*(14*B*b^4*c^2
- 27*A*b^3*c^3)*d^3*e + 7*(5*B*b^5*c - 9*A*b^4*c^2)*d^2*e^2)*x^2)*sqrt(-d)*sqrt(
c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*
d - b*e)))/(c*x + b)) + 2*(2*A*b^4*c^2*d^3 - 4*A*b^5*c*d^2*e + 2*A*b^6*d*e^2 - (
3*A*b^4*c^2*e^3 - 12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + (19*B*b^3*c^3 - 36*A*b^2*c^4)
*d^2*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^2)*x^3 - (6*A*b^5*c*e^3 - 18*(B*b^3*c
^3 - 2*A*b^2*c^4)*d^3 + (29*B*b^4*c^2 - 55*A*b^3*c^3)*d^2*e - 2*(4*B*b^5*c - 5*A
*b^4*c^2)*d*e^2)*x^2 - (3*A*b^6*e^3 - 4*(B*b^4*c^2 - 2*A*b^3*c^3)*d^3 + (8*B*b^5
*c - 13*A*b^4*c^2)*d^2*e - 2*(2*B*b^6 - A*b^5*c)*d*e^2)*x)*sqrt(e*x + d)*sqrt(-d
) - 2*((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 + 4*(11*B*b^2*c^4 - 21*A*b*
c^5)*d^3*e - (16*B*b^3*c^3 - 27*A*b^2*c^4)*d^2*e^2 - 2*(2*B*b^4*c^2 - 3*A*b^3*c^
3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2*c^4 - 2*A*b*c^5)*d^4 + 4*(11*B*b^3*
c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c^3)*d^2*e^2 - 2*(2*B*b^5*c
- 3*A*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 + 4
*(11*B*b^4*c^2 - 21*A*b^3*c^3)*d^3*e - (16*B*b^5*c - 27*A*b^4*c^2)*d^2*e^2 - 2*(
2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*arctan(d/(sqrt(e*x + d)*sqrt(-d))))/(((b^5*c^4*
d^4 - 2*b^6*c^3*d^3*e + b^7*c^2*d^2*e^2)*x^4 + 2*(b^6*c^3*d^4 - 2*b^7*c^2*d^3*e
+ b^8*c*d^2*e^2)*x^3 + (b^7*c^2*d^4 - 2*b^8*c*d^3*e + b^9*d^2*e^2)*x^2)*sqrt(-d)
), -1/4*(((24*(B*b*c^5 - 2*A*c^6)*d^4 - 4*(14*B*b^2*c^4 - 27*A*b*c^5)*d^3*e + 7*
(5*B*b^3*c^3 - 9*A*b^2*c^4)*d^2*e^2)*x^4 + 2*(24*(B*b^2*c^4 - 2*A*b*c^5)*d^4 - 4
*(14*B*b^3*c^3 - 27*A*b^2*c^4)*d^3*e + 7*(5*B*b^4*c^2 - 9*A*b^3*c^3)*d^2*e^2)*x^
3 + (24*(B*b^3*c^3 - 2*A*b^2*c^4)*d^4 - 4*(14*B*b^4*c^2 - 27*A*b^3*c^3)*d^3*e +
7*(5*B*b^5*c - 9*A*b^4*c^2)*d^2*e^2)*x^2)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-
(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) + (2*A*b^4*c^2*d^3 - 4*A*b^5
*c*d^2*e + 2*A*b^6*d*e^2 - (3*A*b^4*c^2*e^3 - 12*(B*b^2*c^4 - 2*A*b*c^5)*d^3 + (
19*B*b^3*c^3 - 36*A*b^2*c^4)*d^2*e - 2*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^2)*x^3 -
(6*A*b^5*c*e^3 - 18*(B*b^3*c^3 - 2*A*b^2*c^4)*d^3 + (29*B*b^4*c^2 - 55*A*b^3*c^3
)*d^2*e - 2*(4*B*b^5*c - 5*A*b^4*c^2)*d*e^2)*x^2 - (3*A*b^6*e^3 - 4*(B*b^4*c^2 -
2*A*b^3*c^3)*d^3 + (8*B*b^5*c - 13*A*b^4*c^2)*d^2*e - 2*(2*B*b^6 - A*b^5*c)*d*e
^2)*x)*sqrt(e*x + d)*sqrt(-d) - ((3*A*b^4*c^2*e^4 - 24*(B*b*c^5 - 2*A*c^6)*d^4 +
4*(11*B*b^2*c^4 - 21*A*b*c^5)*d^3*e - (16*B*b^3*c^3 - 27*A*b^2*c^4)*d^2*e^2 - 2
*(2*B*b^4*c^2 - 3*A*b^3*c^3)*d*e^3)*x^4 + 2*(3*A*b^5*c*e^4 - 24*(B*b^2*c^4 - 2*A
*b*c^5)*d^4 + 4*(11*B*b^3*c^3 - 21*A*b^2*c^4)*d^3*e - (16*B*b^4*c^2 - 27*A*b^3*c
^3)*d^2*e^2 - 2*(2*B*b^5*c - 3*A*b^4*c^2)*d*e^3)*x^3 + (3*A*b^6*e^4 - 24*(B*b^3*
c^3 - 2*A*b^2*c^4)*d^4 + 4*(11*B*b^4*c^2 - 21*A*b^3*c^3)*d^3*e - (16*B*b^5*c - 2
7*A*b^4*c^2)*d^2*e^2 - 2*(2*B*b^6 - 3*A*b^5*c)*d*e^3)*x^2)*arctan(d/(sqrt(e*x +
d)*sqrt(-d))))/(((b^5*c^4*d^4 - 2*b^6*c^3*d^3*e + b^7*c^2*d^2*e^2)*x^4 + 2*(b^6*
c^3*d^4 - 2*b^7*c^2*d^3*e + b^8*c*d^2*e^2)*x^3 + (b^7*c^2*d^4 - 2*b^8*c*d^3*e +
b^9*d^2*e^2)*x^2)*sqrt(-d))]

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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)/(c*x**2+b*x)**3/(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.324878, 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)^3*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Done