3.268 \(\int \frac{1}{x^2 \left (a x^2+b x^3\right )^{3/2}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{315 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{64 a^{11/2}}+\frac{315 b^3 \sqrt{a x^2+b x^3}}{64 a^5 x^2}-\frac{105 b^2 \sqrt{a x^2+b x^3}}{32 a^4 x^3}+\frac{21 b \sqrt{a x^2+b x^3}}{8 a^3 x^4}-\frac{9 \sqrt{a x^2+b x^3}}{4 a^2 x^5}+\frac{2}{a x^3 \sqrt{a x^2+b x^3}} \]

[Out]

2/(a*x^3*Sqrt[a*x^2 + b*x^3]) - (9*Sqrt[a*x^2 + b*x^3])/(4*a^2*x^5) + (21*b*Sqrt
[a*x^2 + b*x^3])/(8*a^3*x^4) - (105*b^2*Sqrt[a*x^2 + b*x^3])/(32*a^4*x^3) + (315
*b^3*Sqrt[a*x^2 + b*x^3])/(64*a^5*x^2) - (315*b^4*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2
+ b*x^3]])/(64*a^(11/2))

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Rubi [A]  time = 0.405005, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ -\frac{315 b^4 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{64 a^{11/2}}+\frac{315 b^3 \sqrt{a x^2+b x^3}}{64 a^5 x^2}-\frac{105 b^2 \sqrt{a x^2+b x^3}}{32 a^4 x^3}+\frac{21 b \sqrt{a x^2+b x^3}}{8 a^3 x^4}-\frac{9 \sqrt{a x^2+b x^3}}{4 a^2 x^5}+\frac{2}{a x^3 \sqrt{a x^2+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a*x^2 + b*x^3)^(3/2)),x]

[Out]

2/(a*x^3*Sqrt[a*x^2 + b*x^3]) - (9*Sqrt[a*x^2 + b*x^3])/(4*a^2*x^5) + (21*b*Sqrt
[a*x^2 + b*x^3])/(8*a^3*x^4) - (105*b^2*Sqrt[a*x^2 + b*x^3])/(32*a^4*x^3) + (315
*b^3*Sqrt[a*x^2 + b*x^3])/(64*a^5*x^2) - (315*b^4*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2
+ b*x^3]])/(64*a^(11/2))

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Rubi in Sympy [A]  time = 40.5994, size = 156, normalized size = 0.94 \[ \frac{2}{a x^{3} \sqrt{a x^{2} + b x^{3}}} - \frac{9 \sqrt{a x^{2} + b x^{3}}}{4 a^{2} x^{5}} + \frac{21 b \sqrt{a x^{2} + b x^{3}}}{8 a^{3} x^{4}} - \frac{105 b^{2} \sqrt{a x^{2} + b x^{3}}}{32 a^{4} x^{3}} + \frac{315 b^{3} \sqrt{a x^{2} + b x^{3}}}{64 a^{5} x^{2}} - \frac{315 b^{4} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{64 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**3+a*x**2)**(3/2),x)

[Out]

2/(a*x**3*sqrt(a*x**2 + b*x**3)) - 9*sqrt(a*x**2 + b*x**3)/(4*a**2*x**5) + 21*b*
sqrt(a*x**2 + b*x**3)/(8*a**3*x**4) - 105*b**2*sqrt(a*x**2 + b*x**3)/(32*a**4*x*
*3) + 315*b**3*sqrt(a*x**2 + b*x**3)/(64*a**5*x**2) - 315*b**4*atanh(sqrt(a)*x/s
qrt(a*x**2 + b*x**3))/(64*a**(11/2))

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Mathematica [A]  time = 0.0766567, size = 106, normalized size = 0.64 \[ \frac{\sqrt{a} \left (-16 a^4+24 a^3 b x-42 a^2 b^2 x^2+105 a b^3 x^3+315 b^4 x^4\right )-315 b^4 x^4 \sqrt{a+b x} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{64 a^{11/2} x^3 \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a*x^2 + b*x^3)^(3/2)),x]

[Out]

(Sqrt[a]*(-16*a^4 + 24*a^3*b*x - 42*a^2*b^2*x^2 + 105*a*b^3*x^3 + 315*b^4*x^4) -
315*b^4*x^4*Sqrt[a + b*x]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(64*a^(11/2)*x^3*Sqrt
[x^2*(a + b*x)])

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Maple [A]  time = 0.025, size = 100, normalized size = 0.6 \[ -{\frac{bx+a}{64\,x} \left ( 315\,\sqrt{bx+a}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){x}^{4}{b}^{4}-24\,{a}^{7/2}xb+42\,{a}^{5/2}{x}^{2}{b}^{2}-105\,{a}^{3/2}{x}^{3}{b}^{3}-315\,{x}^{4}{b}^{4}\sqrt{a}+16\,{a}^{9/2} \right ) \left ( b{x}^{3}+a{x}^{2} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^3+a*x^2)^(3/2),x)

[Out]

-1/64*(b*x+a)*(315*(b*x+a)^(1/2)*arctanh((b*x+a)^(1/2)/a^(1/2))*x^4*b^4-24*a^(7/
2)*x*b+42*a^(5/2)*x^2*b^2-105*a^(3/2)*x^3*b^3-315*x^4*b^4*a^(1/2)+16*a^(9/2))/x/
(b*x^3+a*x^2)^(3/2)/a^(11/2)

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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(1/((b*x^3 + a*x^2)^(3/2)*x^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232443, size = 1, normalized size = 0.01 \[ \left [\frac{315 \,{\left (b^{5} x^{6} + a b^{4} x^{5}\right )} \sqrt{a} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) + 2 \,{\left (315 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} - 42 \, a^{3} b^{2} x^{2} + 24 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt{b x^{3} + a x^{2}}}{128 \,{\left (a^{6} b x^{6} + a^{7} x^{5}\right )}}, -\frac{315 \,{\left (b^{5} x^{6} + a b^{4} x^{5}\right )} \sqrt{-a} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) -{\left (315 \, a b^{4} x^{4} + 105 \, a^{2} b^{3} x^{3} - 42 \, a^{3} b^{2} x^{2} + 24 \, a^{4} b x - 16 \, a^{5}\right )} \sqrt{b x^{3} + a x^{2}}}{64 \,{\left (a^{6} b x^{6} + a^{7} x^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x^2)^(3/2)*x^2),x, algorithm="fricas")

[Out]

[1/128*(315*(b^5*x^6 + a*b^4*x^5)*sqrt(a)*log(((b*x^2 + 2*a*x)*sqrt(a) - 2*sqrt(
b*x^3 + a*x^2)*a)/x^2) + 2*(315*a*b^4*x^4 + 105*a^2*b^3*x^3 - 42*a^3*b^2*x^2 + 2
4*a^4*b*x - 16*a^5)*sqrt(b*x^3 + a*x^2))/(a^6*b*x^6 + a^7*x^5), -1/64*(315*(b^5*
x^6 + a*b^4*x^5)*sqrt(-a)*arctan(a*x/(sqrt(b*x^3 + a*x^2)*sqrt(-a))) - (315*a*b^
4*x^4 + 105*a^2*b^3*x^3 - 42*a^3*b^2*x^2 + 24*a^4*b*x - 16*a^5)*sqrt(b*x^3 + a*x
^2))/(a^6*b*x^6 + a^7*x^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \left (x^{2} \left (a + b x\right )\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**3+a*x**2)**(3/2),x)

[Out]

Integral(1/(x**2*(x**2*(a + b*x))**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a x^{2}\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x^2)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a*x^2)^(3/2)*x^2), x)