Optimal. Leaf size=33 \[ \frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 (p+2)}}{2 a (p+2)} \]
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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {15, 37} \[ \frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 (p+2)}}{2 a (p+2)} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int x^3 \left (c x^2\right )^p (a+b x)^{-5-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{3+2 p} (a+b x)^{-5-2 p} \, dx\\ &=\frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 (2+p)}}{2 a (2+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 32, normalized size = 0.97 \[ \frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 p-4}}{a (2 p+4)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 40, normalized size = 1.21 \[ \frac {{\left (b x^{5} + a x^{4}\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 5}}{2 \, {\left (a p + 2 \, a\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.26, size = 74, normalized size = 2.24 \[ \frac {\left (c x^{2}\right )^{p} b x^{5} e^{\left (-2 \, p \log \left (b x + a\right ) - 5 \, \log \left (b x + a\right )\right )} + \left (c x^{2}\right )^{p} a x^{4} e^{\left (-2 \, p \log \left (b x + a\right ) - 5 \, \log \left (b x + a\right )\right )}}{2 \, {\left (a p + 2 \, a\right )}} \]
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 \[ \frac {x^{4} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-2 p -4}}{2 \left (p +2\right ) a} \]
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 \[ \int \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 5} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 33, normalized size = 1.00 \[ \frac {x^4\,{\left (c\,x^2\right )}^p}{2\,a\,\left (p+2\right )\,{\left (a+b\,x\right )}^{2\,p+4}} \]
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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