Optimal. Leaf size=18 \[ \frac {x^2 \left (b x^n\right )^p}{2+n p} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, 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}
\begin {gather*} \frac {x^2 \left (b x^n\right )^p}{n p+2} \end {gather*}
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 = 0.00, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^2 \left (b x^n\right )^p}{2+n p} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.02, size = 19, normalized size = 1.06
method | result | size |
gosper | \(\frac {x^{2} \left (b \,x^{n}\right )^{p}}{n p +2}\) | \(19\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 19, normalized size = 1.06 \begin {gather*} \frac {b^{p} x^{2} {\left (x^{n}\right )}^{p}}{n p + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 22, normalized size = 1.22 \begin {gather*} \frac {x^{2} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \frac {x^{2} \left (b 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 1.02, size = 22, normalized size = 1.22 \begin {gather*} \frac {x^{2} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.06, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^2\,{\left (b\,x^n\right )}^p}{n\,p+2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________