Optimal. Leaf size=28 \[ -\frac{2 x^{-n-3}}{3 b (n+2) \sqrt{b x^n}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0064715, antiderivative size = 28, 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} \[ -\frac{2 x^{-n-3}}{3 b (n+2) \sqrt{b x^n}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 30
Rubi steps
\begin{align*} \int \frac{1}{x^4 \left (b x^n\right )^{3/2}} \, dx &=\frac{x^{n/2} \int x^{-4-\frac{3 n}{2}} \, dx}{b \sqrt{b x^n}}\\ &=-\frac{2 x^{-3-n}}{3 b (2+n) \sqrt{b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0070921, size = 22, normalized size = 0.79 \[ \frac{1}{\left (-\frac{3 n}{2}-3\right ) x^3 \left (b x^n\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 18, normalized size = 0.6 \begin{align*} -{\frac{2}{3\,{x}^{3} \left ( 2+n \right ) } \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]