Optimal. Leaf size=39 \[ a^3 \log (x)-\frac {3 a^2 b}{2 x^2}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{6 x^6} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {263, 266, 43} \[ -\frac {3 a^2 b}{2 x^2}+a^3 \log (x)-\frac {3 a b^2}{4 x^4}-\frac {b^3}{6 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 263
Rule 266
Rubi steps
\begin {align*} \int \frac {\left (a+\frac {b}{x^2}\right )^3}{x} \, dx &=\int \frac {\left (b+a x^2\right )^3}{x^7} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(b+a x)^3}{x^4} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {b^3}{x^4}+\frac {3 a b^2}{x^3}+\frac {3 a^2 b}{x^2}+\frac {a^3}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac {b^3}{6 x^6}-\frac {3 a b^2}{4 x^4}-\frac {3 a^2 b}{2 x^2}+a^3 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 39, normalized size = 1.00 \[ a^3 \log (x)-\frac {3 a^2 b}{2 x^2}-\frac {3 a b^2}{4 x^4}-\frac {b^3}{6 x^6} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 39, normalized size = 1.00 \[ \frac {12 \, a^{3} x^{6} \log \relax (x) - 18 \, a^{2} b x^{4} - 9 \, a b^{2} x^{2} - 2 \, b^{3}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 47, normalized size = 1.21 \[ \frac {1}{2} \, a^{3} \log \left (x^{2}\right ) - \frac {11 \, a^{3} x^{6} + 18 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 34, normalized size = 0.87 \[ a^{3} \ln \relax (x )-\frac {3 a^{2} b}{2 x^{2}}-\frac {3 a \,b^{2}}{4 x^{4}}-\frac {b^{3}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.89, size = 39, normalized size = 1.00 \[ \frac {1}{2} \, a^{3} \log \left (x^{2}\right ) - \frac {18 \, a^{2} b x^{4} + 9 \, a b^{2} x^{2} + 2 \, b^{3}}{12 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.05, size = 36, normalized size = 0.92 \[ a^3\,\ln \relax (x)-\frac {\frac {3\,a^2\,b\,x^4}{2}+\frac {3\,a\,b^2\,x^2}{4}+\frac {b^3}{6}}{x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.26, size = 37, normalized size = 0.95 \[ a^{3} \log {\relax (x )} + \frac {- 18 a^{2} b x^{4} - 9 a b^{2} x^{2} - 2 b^{3}}{12 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________