Optimal. Leaf size=16 \[ \frac {x \left (b x^n\right )^p}{1+n p} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {15, 30}
\begin {gather*} \frac {x \left (b x^n\right )^p}{n p+1} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 15
Rule 30
Rubi steps
\begin {align*} \int \left (b x^n\right )^p \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{n p} \, dx\\ &=\frac {x \left (b x^n\right )^p}{1+n p}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} \frac {x \left (b x^n\right )^p}{1+n p} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.03, size = 17, normalized size = 1.06
method | result | size |
gosper | \(\frac {x \left (b \,x^{n}\right )^{p}}{n p +1}\) | \(17\) |
norman | \(\frac {x \,{\mathrm e}^{p \ln \left (b \,{\mathrm e}^{n \ln \left (x \right )}\right )}}{n p +1}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 17, normalized size = 1.06 \begin {gather*} \frac {b^{p} x {\left (x^{n}\right )}^{p}}{n p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.39, size = 20, normalized size = 1.25 \begin {gather*} \frac {x e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 1} \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 \left (b x^{n}\right )^{p}}{n p + 1} & \text {for}\: n \neq - \frac {1}{p} \\\int \left (b x^{- \frac {1}{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 = 1.27, size = 20, normalized size = 1.25 \begin {gather*} \frac {x e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.76, size = 16, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (b\,x^n\right )}^p}{n\,p+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________