Optimal. Leaf size=32 \[ \frac {x^3 \left (c x^2\right )^p (a+b x)^{-3-2 p}}{a (3+2 p)} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, 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}
\begin {gather*} \frac {x^3 \left (c x^2\right )^p (a+b x)^{-2 p-3}}{a (2 p+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int x^2 \left (c x^2\right )^p (a+b x)^{-4-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{2+2 p} (a+b x)^{-4-2 p} \, dx\\ &=\frac {x^3 \left (c x^2\right )^p (a+b x)^{-3-2 p}}{a (3+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 34, normalized size = 1.06 \begin {gather*} \frac {x^3 \left (c x^2\right )^p (a+b x)^{1-2 (2+p)}}{a (3+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 33, normalized size = 1.03
method | result | size |
gosper | \(\frac {x^{3} \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-3-2 p}}{a \left (3+2 p \right )}\) | \(33\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.04, size = 40, normalized size = 1.25 \begin {gather*} \frac {{\left (b x^{4} + a x^{3}\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 4}}{2 \, a p + 3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 34, normalized size = 1.06 \begin {gather*} \frac {x^3\,{\left (c\,x^2\right )}^p}{a\,\left (2\,p+3\right )\,{\left (a+b\,x\right )}^{2\,p+3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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