\(\int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx\) [1822]

Optimal result

Integrand size = 38, antiderivative size = 123 \[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {4 \left (-2+a x^2\right ) \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{15 x^2}-\frac {2 b \left (-3+2 a x^2\right ) \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{15 a x^3} \]

[Out]

4/15*(a*x^2-2)*(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)/x^2-2/15*b*(2*a*x^2-3)*(-a/b^2+a^2*x^2/b^2)^(1/2)*
(a*x^2+b*x*(-a/b^2+a^2*x^2/b^2)^(1/2))^(1/2)/a/x^3

Rubi [F]

\[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.50 \[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {2 \left (-3+a x^2-b x \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}{15 x^2 \sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}} \]

[In]

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

[Out]

(2*(-3 + a*x^2 - b*x*Sqrt[(a*(-1 + a*x^2))/b^2]))/(15*x^2*Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])])

Maple [F]

\[\int \frac {1}{x^{3} \sqrt {a \,x^{2}+b x \sqrt {-\frac {a}{b^{2}}+\frac {a^{2} x^{2}}{b^{2}}}}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64 \[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\frac {2 \, {\left (2 \, a^{2} x^{3} - 4 \, a x - {\left (2 \, a b x^{2} - 3 \, b\right )} \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}\right )} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}}{15 \, a x^{3}} \]

[In]

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

[Out]

2/15*(2*a^2*x^3 - 4*a*x - (2*a*b*x^2 - 3*b)*sqrt((a^2*x^2 - a)/b^2))*sqrt(a*x^2 + b*x*sqrt((a^2*x^2 - a)/b^2))
/(a*x^3)

Sympy [F]

\[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {1}{x^{3} \sqrt {x \left (a x + b \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}}\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {1}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int { \frac {1}{\sqrt {a x^{2} + \sqrt {\frac {a^{2} x^{2}}{b^{2}} - \frac {a}{b^{2}}} b x} x^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^3 \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}} \, dx=\int \frac {1}{x^3\,\sqrt {a\,x^2+b\,x\,\sqrt {\frac {a^2\,x^2}{b^2}-\frac {a}{b^2}}}} \,d x \]

[In]

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

[Out]

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