Optimal. Leaf size=18 \[ \frac{x^3 \left (b x^n\right )^p}{n p+3} \]
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Rubi [A] time = 0.0060111, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {15, 30} \[ \frac{x^3 \left (b x^n\right )^p}{n p+3} \]
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rubi steps
\begin{align*} \int x^2 \left (b x^n\right )^p \, dx &=\left (x^{-n p} \left (b x^n\right )^p\right ) \int x^{2+n p} \, dx\\ &=\frac{x^3 \left (b x^n\right )^p}{3+n p}\\ \end{align*}
Mathematica [A] time = 0.0025774, size = 18, normalized size = 1. \[ \frac{x^3 \left (b x^n\right )^p}{n p+3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 1.1 \begin{align*}{\frac{{x}^{3} \left ( b{x}^{n} \right ) ^{p}}{np+3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983508, size = 26, normalized size = 1.44 \begin{align*} \frac{b^{p} x^{3}{\left (x^{n}\right )}^{p}}{n p + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.27108, size = 55, normalized size = 3.06 \begin{align*} \frac{x^{3} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.13197, size = 30, normalized size = 1.67 \begin{align*} \frac{x^{3} e^{\left (n p \log \left (x\right ) + p \log \left (b\right )\right )}}{n p + 3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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