Optimal. Leaf size=38 \[ \frac {x^{m+1} \left (c x^2\right )^p (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {15, 37} \[ \frac {x^{m+1} \left (c x^2\right )^p (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int x^m \left (c x^2\right )^p (a+b x)^{-2-m-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{m+2 p} (a+b x)^{-2-m-2 p} \, dx\\ &=\frac {x^{1+m} \left (c x^2\right )^p (a+b x)^{-1-m-2 p}}{a (1+m+2 p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 38, normalized size = 1.00 \[ \frac {x^{m+1} \left (c x^2\right )^p (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 49, normalized size = 1.29 \[ \frac {{\left (b x^{2} + a x\right )} {\left (b x + a\right )}^{-m - 2 \, p - 2} x^{m} e^{\left (p \log \relax (c) + 2 \, p \log \relax (x)\right )}}{a m + 2 \, a p + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-m - 2 \, p - 2} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 39, normalized size = 1.03 \[ \frac {x^{m +1} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-m -2 p -1}}{\left (m +2 p +1\right ) a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-m - 2 \, p - 2} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 50, normalized size = 1.32 \[ \frac {x\,x^m\,{\left (c\,x^2\right )}^p}{a\,{\left (a+b\,x\right )}^m\,{\left (a+b\,x\right )}^{2\,p}\,\left (a+b\,x\right )\,\left (m+2\,p+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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