Optimal. Leaf size=39 \[ \frac {x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 20, 37} \begin {gather*} \frac {x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 20
Rule 37
Rubi steps
\begin {align*} \int (d 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^{2 p} (d x)^m (a+b x)^{-2-m-2 p} \, dx\\ &=\left (x^{-m-2 p} (d x)^m \left (c x^2\right )^p\right ) \int x^{m+2 p} (a+b x)^{-2-m-2 p} \, dx\\ &=\frac {x (d x)^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 = 0.02, size = 39, normalized size = 1.00 \begin {gather*} \frac {x \left (c x^2\right )^p (d x)^m (a+b x)^{-m-2 p-1}}{a (m+2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (c x^2\right )^p (a+b x)^{-2-m-2 p} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.36, size = 57, normalized size = 1.46 \begin {gather*} \frac {{\left (b x^{2} + a x\right )} {\left (b x + a\right )}^{-m - 2 \, p - 2} \left (d x\right )^{m} e^{\left (2 \, p \log \left (d x\right ) + p \log \left (\frac {c}{d^{2}}\right )\right )}}{a m + 2 \, a p + a} \end {gather*}
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 \begin {gather*} \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-m - 2 \, p - 2} \left (d x\right )^{m}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 40, normalized size = 1.03 \begin {gather*} \frac {x \left (c \,x^{2}\right )^{p} \left (d x \right )^{m} \left (b x +a \right )^{-m -2 p -1}}{\left (m +2 p +1\right ) a} \end {gather*}
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 \begin {gather*} \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-m - 2 \, p - 2} \left (d x\right )^{m}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 39, normalized size = 1.00 \begin {gather*} \frac {x\,{\left (d\,x\right )}^m\,{\left (c\,x^2\right )}^p}{a\,{\left (a+b\,x\right )}^{m+2\,p+1}\,\left (m+2\,p+1\right )} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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