Optimal. Leaf size=33 \[ \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (p+1)}}{2 a (p+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 37} \[ \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (p+1)}}{2 a (p+1)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int x \left (c x^2\right )^p (a+b x)^{-3-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{1+2 p} (a+b x)^{-3-2 p} \, dx\\ &=\frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 (1+p)}}{2 a (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.97 \[ \frac {x^2 \left (c x^2\right )^p (a+b x)^{-2 p-2}}{a (2 p+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 38, normalized size = 1.15 \[ \frac {{\left (b x^{3} + a x^{2}\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 3}}{2 \, {\left (a p + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.07, size = 72, normalized size = 2.18 \[ \frac {\left (c x^{2}\right )^{p} b x^{3} e^{\left (-2 \, p \log \left (b x + a\right ) - 3 \, \log \left (b x + a\right )\right )} + \left (c x^{2}\right )^{p} a x^{2} e^{\left (-2 \, p \log \left (b x + a\right ) - 3 \, \log \left (b x + a\right )\right )}}{2 \, {\left (a p + a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 32, normalized size = 0.97 \[ \frac {x^{2} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-2 p -2}}{2 \left (p +1\right ) a} \]
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 \[ \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 3} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.22, size = 33, normalized size = 1.00 \[ \frac {x^2\,{\left (c\,x^2\right )}^p}{2\,a\,\left (p+1\right )\,{\left (a+b\,x\right )}^{2\,p+2}} \]
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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