Optimal. Leaf size=18 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0059163, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {261} \[ -\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rubi steps
\begin{align*} \int \frac{\left (a+\frac{b}{x^2}\right )^{5/2}}{x^3} \, dx &=-\frac{\left (a+\frac{b}{x^2}\right )^{7/2}}{7 b}\\ \end{align*}
Mathematica [A] time = 0.0108239, size = 28, normalized size = 1.56 \[ -\frac{\left (a+\frac{b}{x^2}\right )^{5/2} \left (a x^2+b\right )}{7 b x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.004, size = 29, normalized size = 1.6 \begin{align*} -{\frac{a{x}^{2}+b}{7\,b{x}^{2}} \left ({\frac{a{x}^{2}+b}{{x}^{2}}} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.00338, size = 19, normalized size = 1.06 \begin{align*} -\frac{{\left (a + \frac{b}{x^{2}}\right )}^{\frac{7}{2}}}{7 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.52103, size = 108, normalized size = 6. \begin{align*} -\frac{{\left (a^{3} x^{6} + 3 \, a^{2} b x^{4} + 3 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{\frac{a x^{2} + b}{x^{2}}}}{7 \, b x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.72643, size = 88, normalized size = 4.89 \begin{align*} \begin{cases} - \frac{a^{3} \sqrt{a + \frac{b}{x^{2}}}}{7 b} - \frac{3 a^{2} \sqrt{a + \frac{b}{x^{2}}}}{7 x^{2}} - \frac{3 a b \sqrt{a + \frac{b}{x^{2}}}}{7 x^{4}} - \frac{b^{2} \sqrt{a + \frac{b}{x^{2}}}}{7 x^{6}} & \text{for}\: b \neq 0 \\- \frac{a^{\frac{5}{2}}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.2684, size = 163, normalized size = 9.06 \begin{align*} \frac{2 \,{\left (7 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{12} a^{\frac{7}{2}} \mathrm{sgn}\left (x\right ) + 35 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{8} a^{\frac{7}{2}} b^{2} \mathrm{sgn}\left (x\right ) + 21 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{4} a^{\frac{7}{2}} b^{4} \mathrm{sgn}\left (x\right ) + a^{\frac{7}{2}} b^{6} \mathrm{sgn}\left (x\right )\right )}}{7 \,{\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]