3.1244 \(\int \frac{A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=261 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-3 A b e-4 A c d+2 b B d)}{b^3 d^{5/2}}-\frac{e \left (b^2 (-e) (2 B d-3 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right )}{b^2 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{c (A b e-2 A c d+b B d)}{b^2 d (b+c x) \sqrt{d+e x} (c d-b e)}+\frac{c^{3/2} \left (7 A b c e-4 A c^2 d-5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{5/2}}-\frac{A}{b d x (b+c x) \sqrt{d+e x}} \]

[Out]

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

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Rubi [A]  time = 1.25902, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-3 A b e-4 A c d+2 b B d)}{b^3 d^{5/2}}-\frac{e \left (b^2 (-e) (2 B d-3 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right )}{b^2 d^2 \sqrt{d+e x} (c d-b e)^2}+\frac{c (A b e-2 A c d+b B d)}{b^2 d (b+c x) \sqrt{d+e x} (c d-b e)}-\frac{c^{3/2} \left (-b c (7 A e+2 B d)+4 A c^2 d+5 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{5/2}}-\frac{A}{b d x (b+c x) \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

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

[Out]

Timed out

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Mathematica [A]  time = 1.08746, size = 209, normalized size = 0.8 \[ -\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-3 A b e-4 A c d+2 b B d)}{b^3 d^{5/2}}+\sqrt{d+e x} \left (\frac{c^2 (b B-A c)}{b^2 (b+c x) (c d-b e)^2}-\frac{A}{b^2 d^2 x}+\frac{2 e^2 (B d-A e)}{d^2 (d+e x) (c d-b e)^2}\right )-\frac{c^{3/2} \left (-b c (7 A e+2 B d)+4 A c^2 d+5 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.04, size = 427, normalized size = 1.6 \[ -2\,{\frac{{e}^{3}A}{{d}^{2} \left ( be-cd \right ) ^{2}\sqrt{ex+d}}}+2\,{\frac{{e}^{2}B}{d \left ( be-cd \right ) ^{2}\sqrt{ex+d}}}-{\frac{{c}^{3}eA}{ \left ( be-cd \right ) ^{2}{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{e{c}^{2}B}{ \left ( be-cd \right ) ^{2}b \left ( cex+be \right ) }\sqrt{ex+d}}-7\,{\frac{{c}^{3}eA}{ \left ( be-cd \right ) ^{2}{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{{c}^{4}Ad}{ \left ( be-cd \right ) ^{2}{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+5\,{\frac{e{c}^{2}B}{ \left ( be-cd \right ) ^{2}b\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-2\,{\frac{{c}^{3}Bd}{ \left ( be-cd \right ) ^{2}{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{c\sqrt{ex+d}}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }-{\frac{A}{{b}^{2}{d}^{2}x}\sqrt{ex+d}}+3\,{\frac{Ae}{{b}^{2}{d}^{5/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }+4\,{\frac{Ac}{{b}^{3}{d}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{B}{{b}^{2}{d}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*e^3/d^2/(b*e-c*d)^2/(e*x+d)^(1/2)*A+2*e^2/d/(b*e-c*d)^2/(e*x+d)^(1/2)*B-e*c^3
/(b*e-c*d)^2/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A+e*c^2/(b*e-c*d)^2/b*(e*x+d)^(1/2)/(
c*e*x+b*e)*B-7*e*c^3/(b*e-c*d)^2/b^2/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d)^(1/2)/
((b*e-c*d)*c)^(1/2))*A+4*c^4/(b*e-c*d)^2/b^3/((b*e-c*d)*c)^(1/2)*arctan(c*(e*x+d
)^(1/2)/((b*e-c*d)*c)^(1/2))*A*d+5*e*c^2/(b*e-c*d)^2/b/((b*e-c*d)*c)^(1/2)*arcta
n(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B-2*c^3/(b*e-c*d)^2/b^2/((b*e-c*d)*c)^(1/
2)*arctan(c*(e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2))*B*d-1/b^2/d^2*A*(e*x+d)^(1/2)/x+3
*e/b^2/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A+4/b^3/d^(3/2)*arctanh((e*x+d)^(1
/2)/d^(1/2))*A*c-2/b^2/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B

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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)^2*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 17.5641, 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)^2*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/2*(((2*(B*b*c^3 - 2*A*c^4)*d^3 - (5*B*b^2*c^2 - 7*A*b*c^3)*d^2*e)*x^2 + (2*(B
*b^2*c^2 - 2*A*b*c^3)*d^3 - (5*B*b^3*c - 7*A*b^2*c^2)*d^2*e)*x)*sqrt(e*x + d)*sq
rt(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)) + ((3*A*b^3*c*e^3 - 2*(B*b*c^3 - 2*A*c^4)*d^3 +
(4*B*b^2*c^2 - 5*A*b*c^3)*d^2*e - 2*(B*b^3*c + A*b^2*c^2)*d*e^2)*x^2 + (3*A*b^4
*e^3 - 2*(B*b^2*c^2 - 2*A*b*c^3)*d^3 + (4*B*b^3*c - 5*A*b^2*c^2)*d^2*e - 2*(B*b^
4 + A*b^3*c)*d*e^2)*x)*sqrt(e*x + d)*log(((e*x + 2*d)*sqrt(d) + 2*sqrt(e*x + d)*
d)/x) - 2*(A*b^2*c^2*d^3 - 2*A*b^3*c*d^2*e + A*b^4*d*e^2 + (3*A*b^3*c*e^3 - (B*b
^2*c^2 - 2*A*b*c^3)*d^2*e - 2*(B*b^3*c + A*b^2*c^2)*d*e^2)*x^2 - (A*b^2*c^2*d^2*
e - 3*A*b^4*e^3 + (B*b^2*c^2 - 2*A*b*c^3)*d^3 + (2*B*b^4 + A*b^3*c)*d*e^2)*x)*sq
rt(d))/(((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5*c*d^2*e^2)*x^2 + (b^4*c^2*d^4 - 2*
b^5*c*d^3*e + b^6*d^2*e^2)*x)*sqrt(e*x + d)*sqrt(d)), 1/2*(2*((2*(B*b*c^3 - 2*A*
c^4)*d^3 - (5*B*b^2*c^2 - 7*A*b*c^3)*d^2*e)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^3
- (5*B*b^3*c - 7*A*b^2*c^2)*d^2*e)*x)*sqrt(e*x + d)*sqrt(d)*sqrt(-c/(c*d - b*e)
)*arctan(-(c*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) + ((3*A*b^3*c*e^3
- 2*(B*b*c^3 - 2*A*c^4)*d^3 + (4*B*b^2*c^2 - 5*A*b*c^3)*d^2*e - 2*(B*b^3*c + A*b
^2*c^2)*d*e^2)*x^2 + (3*A*b^4*e^3 - 2*(B*b^2*c^2 - 2*A*b*c^3)*d^3 + (4*B*b^3*c -
5*A*b^2*c^2)*d^2*e - 2*(B*b^4 + A*b^3*c)*d*e^2)*x)*sqrt(e*x + d)*log(((e*x + 2*
d)*sqrt(d) + 2*sqrt(e*x + d)*d)/x) - 2*(A*b^2*c^2*d^3 - 2*A*b^3*c*d^2*e + A*b^4*
d*e^2 + (3*A*b^3*c*e^3 - (B*b^2*c^2 - 2*A*b*c^3)*d^2*e - 2*(B*b^3*c + A*b^2*c^2)
*d*e^2)*x^2 - (A*b^2*c^2*d^2*e - 3*A*b^4*e^3 + (B*b^2*c^2 - 2*A*b*c^3)*d^3 + (2*
B*b^4 + A*b^3*c)*d*e^2)*x)*sqrt(d))/(((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5*c*d^2
*e^2)*x^2 + (b^4*c^2*d^4 - 2*b^5*c*d^3*e + b^6*d^2*e^2)*x)*sqrt(e*x + d)*sqrt(d)
), 1/2*(((2*(B*b*c^3 - 2*A*c^4)*d^3 - (5*B*b^2*c^2 - 7*A*b*c^3)*d^2*e)*x^2 + (2*
(B*b^2*c^2 - 2*A*b*c^3)*d^3 - (5*B*b^3*c - 7*A*b^2*c^2)*d^2*e)*x)*sqrt(e*x + d)*
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*((3*A*b^3*c*e^3 - 2*(B*b*c^3 - 2*A*c^4)*
d^3 + (4*B*b^2*c^2 - 5*A*b*c^3)*d^2*e - 2*(B*b^3*c + A*b^2*c^2)*d*e^2)*x^2 + (3*
A*b^4*e^3 - 2*(B*b^2*c^2 - 2*A*b*c^3)*d^3 + (4*B*b^3*c - 5*A*b^2*c^2)*d^2*e - 2*
(B*b^4 + A*b^3*c)*d*e^2)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sqrt(-d))) - 2
*(A*b^2*c^2*d^3 - 2*A*b^3*c*d^2*e + A*b^4*d*e^2 + (3*A*b^3*c*e^3 - (B*b^2*c^2 -
2*A*b*c^3)*d^2*e - 2*(B*b^3*c + A*b^2*c^2)*d*e^2)*x^2 - (A*b^2*c^2*d^2*e - 3*A*b
^4*e^3 + (B*b^2*c^2 - 2*A*b*c^3)*d^3 + (2*B*b^4 + A*b^3*c)*d*e^2)*x)*sqrt(-d))/(
((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5*c*d^2*e^2)*x^2 + (b^4*c^2*d^4 - 2*b^5*c*d^
3*e + b^6*d^2*e^2)*x)*sqrt(e*x + d)*sqrt(-d)), (((2*(B*b*c^3 - 2*A*c^4)*d^3 - (5
*B*b^2*c^2 - 7*A*b*c^3)*d^2*e)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^3 - (5*B*b^3*c
- 7*A*b^2*c^2)*d^2*e)*x)*sqrt(e*x + d)*sqrt(-d)*sqrt(-c/(c*d - b*e))*arctan(-(c
*d - b*e)*sqrt(-c/(c*d - b*e))/(sqrt(e*x + d)*c)) - ((3*A*b^3*c*e^3 - 2*(B*b*c^3
- 2*A*c^4)*d^3 + (4*B*b^2*c^2 - 5*A*b*c^3)*d^2*e - 2*(B*b^3*c + A*b^2*c^2)*d*e^
2)*x^2 + (3*A*b^4*e^3 - 2*(B*b^2*c^2 - 2*A*b*c^3)*d^3 + (4*B*b^3*c - 5*A*b^2*c^2
)*d^2*e - 2*(B*b^4 + A*b^3*c)*d*e^2)*x)*sqrt(e*x + d)*arctan(d/(sqrt(e*x + d)*sq
rt(-d))) - (A*b^2*c^2*d^3 - 2*A*b^3*c*d^2*e + A*b^4*d*e^2 + (3*A*b^3*c*e^3 - (B*
b^2*c^2 - 2*A*b*c^3)*d^2*e - 2*(B*b^3*c + A*b^2*c^2)*d*e^2)*x^2 - (A*b^2*c^2*d^2
*e - 3*A*b^4*e^3 + (B*b^2*c^2 - 2*A*b*c^3)*d^3 + (2*B*b^4 + A*b^3*c)*d*e^2)*x)*s
qrt(-d))/(((b^3*c^3*d^4 - 2*b^4*c^2*d^3*e + b^5*c*d^2*e^2)*x^2 + (b^4*c^2*d^4 -
2*b^5*c*d^3*e + b^6*d^2*e^2)*x)*sqrt(e*x + d)*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)/(e*x+d)**(3/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.29967, size = 675, normalized size = 2.59 \[ -\frac{{\left (2 \, B b c^{3} d - 4 \, A c^{4} d - 5 \, B b^{2} c^{2} e + 7 \, A b c^{3} e\right )} \arctan \left (\frac{\sqrt{x e + d} c}{\sqrt{-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}\right )} \sqrt{-c^{2} d + b c e}} + \frac{{\left (x e + d\right )}^{2} B b c^{2} d^{2} e - 2 \,{\left (x e + d\right )}^{2} A c^{3} d^{2} e -{\left (x e + d\right )} B b c^{2} d^{3} e + 2 \,{\left (x e + d\right )} A c^{3} d^{3} e + 2 \,{\left (x e + d\right )}^{2} B b^{2} c d e^{2} + 2 \,{\left (x e + d\right )}^{2} A b c^{2} d e^{2} - 4 \,{\left (x e + d\right )} B b^{2} c d^{2} e^{2} - 3 \,{\left (x e + d\right )} A b c^{2} d^{2} e^{2} + 2 \, B b^{2} c d^{3} e^{2} - 3 \,{\left (x e + d\right )}^{2} A b^{2} c e^{3} + 2 \,{\left (x e + d\right )} B b^{3} d e^{3} + 7 \,{\left (x e + d\right )} A b^{2} c d e^{3} - 2 \, B b^{3} d^{2} e^{3} - 2 \, A b^{2} c d^{2} e^{3} - 3 \,{\left (x e + d\right )} A b^{3} e^{4} + 2 \, A b^{3} d e^{4}}{{\left (b^{2} c^{2} d^{4} - 2 \, b^{3} c d^{3} e + b^{4} d^{2} e^{2}\right )}{\left ({\left (x e + d\right )}^{\frac{5}{2}} c - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} c d + \sqrt{x e + d} c d^{2} +{\left (x e + d\right )}^{\frac{3}{2}} b e - \sqrt{x e + d} b d e\right )}} + \frac{{\left (2 \, B b d - 4 \, A c d - 3 \, A b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/((c*x^2 + b*x)^2*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

-(2*B*b*c^3*d - 4*A*c^4*d - 5*B*b^2*c^2*e + 7*A*b*c^3*e)*arctan(sqrt(x*e + d)*c/
sqrt(-c^2*d + b*c*e))/((b^3*c^2*d^2 - 2*b^4*c*d*e + b^5*e^2)*sqrt(-c^2*d + b*c*e
)) + ((x*e + d)^2*B*b*c^2*d^2*e - 2*(x*e + d)^2*A*c^3*d^2*e - (x*e + d)*B*b*c^2*
d^3*e + 2*(x*e + d)*A*c^3*d^3*e + 2*(x*e + d)^2*B*b^2*c*d*e^2 + 2*(x*e + d)^2*A*
b*c^2*d*e^2 - 4*(x*e + d)*B*b^2*c*d^2*e^2 - 3*(x*e + d)*A*b*c^2*d^2*e^2 + 2*B*b^
2*c*d^3*e^2 - 3*(x*e + d)^2*A*b^2*c*e^3 + 2*(x*e + d)*B*b^3*d*e^3 + 7*(x*e + d)*
A*b^2*c*d*e^3 - 2*B*b^3*d^2*e^3 - 2*A*b^2*c*d^2*e^3 - 3*(x*e + d)*A*b^3*e^4 + 2*
A*b^3*d*e^4)/((b^2*c^2*d^4 - 2*b^3*c*d^3*e + b^4*d^2*e^2)*((x*e + d)^(5/2)*c - 2
*(x*e + d)^(3/2)*c*d + sqrt(x*e + d)*c*d^2 + (x*e + d)^(3/2)*b*e - sqrt(x*e + d)
*b*d*e)) + (2*B*b*d - 4*A*c*d - 3*A*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqr
t(-d)*d^2)