Optimal. Leaf size=18 \[ \frac {x^2 \left (b x^n\right )^p}{n p+2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \frac {x^2 \left (b x^n\right )^p}{n p+2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 30
Rubi steps
\begin {align*} \int x \left (b x^n\right )^p \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{1+n p} \, dx\\ &=\frac {x^2 \left (b x^n\right )^p}{2+n p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 18, normalized size = 1.00 \[ \frac {x^2 \left (b x^n\right )^p}{n p+2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.88, size = 22, normalized size = 1.22 \[ \frac {x^{2} e^{\left (n p \log \relax (x) + p \log \relax (b)\right )}}{n p + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.17, size = 22, normalized size = 1.22 \[ \frac {x^{2} e^{\left (n p \log \relax (x) + p \log \relax (b)\right )}}{n p + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 19, normalized size = 1.06 \[ \frac {x^{2} \left (b \,x^{n}\right )^{p}}{n p +2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.57, size = 19, normalized size = 1.06 \[ \frac {b^{p} x^{2} {\left (x^{n}\right )}^{p}}{n p + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.20, size = 18, normalized size = 1.00 \[ \frac {x^2\,{\left (b\,x^n\right )}^p}{n\,p+2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \begin {cases} \frac {b^{p} x^{2} \left (x^{n}\right )^{p}}{n p + 2} & \text {for}\: n \neq - \frac {2}{p} \\\int x \left (b x^{- \frac {2}{p}}\right )^{p}\, dx & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________