Optimal. Leaf size=24 \[ -\frac{2 x^{-n}}{3 b n \sqrt{b x^n}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.003785, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {15, 30} \[ -\frac{2 x^{-n}}{3 b n \sqrt{b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{x \left (b x^n\right )^{3/2}} \, dx &=\frac{x^{n/2} \int x^{-1-\frac{3 n}{2}} \, dx}{b \sqrt{b x^n}}\\ &=-\frac{2 x^{-n}}{3 b n \sqrt{b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0028107, size = 16, normalized size = 0.67 \[ -\frac{2}{3 n \left (b x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.001, size = 13, normalized size = 0.5 \begin{align*} -{\frac{2}{3\,n} \left ( b{x}^{n} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.82914, size = 46, normalized size = 1.92 \begin{align*} -\frac{2 \, \sqrt{b x^{n}}}{3 \, b^{2} n x^{2 \, n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.54714, size = 26, normalized size = 1.08 \begin{align*} \begin{cases} - \frac{2}{3 b^{\frac{3}{2}} n \left (x^{n}\right )^{\frac{3}{2}}} & \text{for}\: n \neq 0 \\\frac{\log{\left (x \right )}}{b^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (b x^{n}\right )^{\frac{3}{2}} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]