3.259 \(\int \frac{1}{x^3 \sqrt{a x^2+b x^3}} \, dx\)

Optimal. Leaf size=115 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{7/2}}-\frac{5 b^2 \sqrt{a x^2+b x^3}}{8 a^3 x^2}+\frac{5 b \sqrt{a x^2+b x^3}}{12 a^2 x^3}-\frac{\sqrt{a x^2+b x^3}}{3 a x^4} \]

[Out]

-Sqrt[a*x^2 + b*x^3]/(3*a*x^4) + (5*b*Sqrt[a*x^2 + b*x^3])/(12*a^2*x^3) - (5*b^2
*Sqrt[a*x^2 + b*x^3])/(8*a^3*x^2) + (5*b^3*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^
3]])/(8*a^(7/2))

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Rubi [A]  time = 0.23733, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{5 b^3 \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{8 a^{7/2}}-\frac{5 b^2 \sqrt{a x^2+b x^3}}{8 a^3 x^2}+\frac{5 b \sqrt{a x^2+b x^3}}{12 a^2 x^3}-\frac{\sqrt{a x^2+b x^3}}{3 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[a*x^2 + b*x^3]/(3*a*x^4) + (5*b*Sqrt[a*x^2 + b*x^3])/(12*a^2*x^3) - (5*b^2
*Sqrt[a*x^2 + b*x^3])/(8*a^3*x^2) + (5*b^3*ArcTanh[(Sqrt[a]*x)/Sqrt[a*x^2 + b*x^
3]])/(8*a^(7/2))

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Rubi in Sympy [A]  time = 24.3666, size = 105, normalized size = 0.91 \[ - \frac{\sqrt{a x^{2} + b x^{3}}}{3 a x^{4}} + \frac{5 b \sqrt{a x^{2} + b x^{3}}}{12 a^{2} x^{3}} - \frac{5 b^{2} \sqrt{a x^{2} + b x^{3}}}{8 a^{3} x^{2}} + \frac{5 b^{3} \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{8 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a*x**2 + b*x**3)/(3*a*x**4) + 5*b*sqrt(a*x**2 + b*x**3)/(12*a**2*x**3) - 5
*b**2*sqrt(a*x**2 + b*x**3)/(8*a**3*x**2) + 5*b**3*atanh(sqrt(a)*x/sqrt(a*x**2 +
b*x**3))/(8*a**(7/2))

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Mathematica [A]  time = 0.0623039, size = 96, normalized size = 0.83 \[ \frac{15 b^3 x^3 \sqrt{a+b x} \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )-\sqrt{a} \left (8 a^3-2 a^2 b x+5 a b^2 x^2+15 b^3 x^3\right )}{24 a^{7/2} x^2 \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-(Sqrt[a]*(8*a^3 - 2*a^2*b*x + 5*a*b^2*x^2 + 15*b^3*x^3)) + 15*b^3*x^3*Sqrt[a +
b*x]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/(24*a^(7/2)*x^2*Sqrt[x^2*(a + b*x)])

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Maple [A]  time = 0.013, size = 95, normalized size = 0.8 \[ -{\frac{1}{24\,{x}^{2}}\sqrt{bx+a} \left ( 15\,\sqrt{bx+a}{a}^{3/2}{x}^{2}{b}^{2}-15\,{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ){x}^{3}a{b}^{3}-10\,\sqrt{bx+a}{a}^{5/2}xb+8\,\sqrt{bx+a}{a}^{7/2} \right ){\frac{1}{\sqrt{b{x}^{3}+a{x}^{2}}}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/24/x^2*(b*x+a)^(1/2)*(15*(b*x+a)^(1/2)*a^(3/2)*x^2*b^2-15*arctanh((b*x+a)^(1/
2)/a^(1/2))*x^3*a*b^3-10*(b*x+a)^(1/2)*a^(5/2)*x*b+8*(b*x+a)^(1/2)*a^(7/2))/(b*x
^3+a*x^2)^(1/2)/a^(9/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240169, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{a} b^{3} x^{4} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} + 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) - 2 \,{\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{48 \, a^{4} x^{4}}, \frac{15 \, \sqrt{-a} b^{3} x^{4} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) -{\left (15 \, a b^{2} x^{2} - 10 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt{b x^{3} + a x^{2}}}{24 \, a^{4} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/48*(15*sqrt(a)*b^3*x^4*log(((b*x^2 + 2*a*x)*sqrt(a) + 2*sqrt(b*x^3 + a*x^2)*a
)/x^2) - 2*(15*a*b^2*x^2 - 10*a^2*b*x + 8*a^3)*sqrt(b*x^3 + a*x^2))/(a^4*x^4), 1
/24*(15*sqrt(-a)*b^3*x^4*arctan(a*x/(sqrt(b*x^3 + a*x^2)*sqrt(-a))) - (15*a*b^2*
x^2 - 10*a^2*b*x + 8*a^3)*sqrt(b*x^3 + a*x^2))/(a^4*x^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: NotImplementedError