3.2939 \(\int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx\)

Optimal. Leaf size=71 \[ -\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac{b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right )}{3 \sqrt{a} x^3} \]

[Out]

-Sqrt[a + b*(c*x^2)^(3/2)]/(3*x^3) - (b*(c*x^2)^(3/2)*ArcTanh[Sqrt[a + b*(c*x^2)
^(3/2)]/Sqrt[a]])/(3*Sqrt[a]*x^3)

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Rubi [A]  time = 0.124638, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{3 x^3}-\frac{b \left (c x^2\right )^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{\sqrt{a}}\right )}{3 \sqrt{a} x^3} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*(c*x^2)^(3/2)]/x^4,x]

[Out]

-Sqrt[a + b*(c*x^2)^(3/2)]/(3*x^3) - (b*(c*x^2)^(3/2)*ArcTanh[Sqrt[a + b*(c*x^2)
^(3/2)]/Sqrt[a]])/(3*Sqrt[a]*x^3)

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Rubi in Sympy [A]  time = 10.3422, size = 63, normalized size = 0.89 \[ - \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{3 x^{3}} - \frac{b \left (c x^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{\sqrt{a}} \right )}}{3 \sqrt{a} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+b*(c*x**2)**(3/2))**(1/2)/x**4,x)

[Out]

-sqrt(a + b*(c*x**2)**(3/2))/(3*x**3) - b*(c*x**2)**(3/2)*atanh(sqrt(a + b*(c*x*
*2)**(3/2))/sqrt(a))/(3*sqrt(a)*x**3)

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Mathematica [A]  time = 0.0423885, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \left (c x^2\right )^{3/2}}}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]  Integrate[Sqrt[a + b*(c*x^2)^(3/2)]/x^4,x]

[Out]

Integrate[Sqrt[a + b*(c*x^2)^(3/2)]/x^4, x]

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Maple [F]  time = 0.049, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}}\sqrt{a+b \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*(c*x^2)^(3/2))^(1/2)/x^4,x)

[Out]

int((a+b*(c*x^2)^(3/2))^(1/2)/x^4,x)

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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(sqrt((c*x^2)^(3/2)*b + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226451, size = 1, normalized size = 0.01 \[ \left [\frac{b c x^{3} \sqrt{\frac{c}{a}} \log \left (\frac{\sqrt{c x^{2}} b c^{2} x^{3} + 2 \, a c x - 2 \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a} \sqrt{c x^{2}} a \sqrt{\frac{c}{a}}}{x^{4}}\right ) - 2 \, \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{6 \, x^{3}}, \frac{b c x^{3} \sqrt{-\frac{c}{a}} \arctan \left (\frac{a x \sqrt{-\frac{c}{a}}}{\sqrt{\sqrt{c x^{2}} b c x^{2} + a} \sqrt{c x^{2}}}\right ) - \sqrt{\sqrt{c x^{2}} b c x^{2} + a}}{3 \, x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)/x^4,x, algorithm="fricas")

[Out]

[1/6*(b*c*x^3*sqrt(c/a)*log((sqrt(c*x^2)*b*c^2*x^3 + 2*a*c*x - 2*sqrt(sqrt(c*x^2
)*b*c*x^2 + a)*sqrt(c*x^2)*a*sqrt(c/a))/x^4) - 2*sqrt(sqrt(c*x^2)*b*c*x^2 + a))/
x^3, 1/3*(b*c*x^3*sqrt(-c/a)*arctan(a*x*sqrt(-c/a)/(sqrt(sqrt(c*x^2)*b*c*x^2 + a
)*sqrt(c*x^2))) - sqrt(sqrt(c*x^2)*b*c*x^2 + a))/x^3]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + b \left (c x^{2}\right )^{\frac{3}{2}}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*(c*x**2)**(3/2))**(1/2)/x**4,x)

[Out]

Integral(sqrt(a + b*(c*x**2)**(3/2))/x**4, x)

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GIAC/XCAS [A]  time = 0.221922, size = 74, normalized size = 1.04 \[ \frac{1}{3} \, b c^{\frac{3}{2}}{\left (\frac{\arctan \left (\frac{\sqrt{b c^{\frac{3}{2}} x^{3} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b c^{\frac{3}{2}} x^{3} + a}}{b c^{\frac{3}{2}} x^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt((c*x^2)^(3/2)*b + a)/x^4,x, algorithm="giac")

[Out]

1/3*b*c^(3/2)*(arctan(sqrt(b*c^(3/2)*x^3 + a)/sqrt(-a))/sqrt(-a) - sqrt(b*c^(3/2
)*x^3 + a)/(b*c^(3/2)*x^3))