Optimal. Leaf size=33 \[ \frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 (2+p)}}{2 a (2+p)} \]
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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}
\begin {gather*} \frac {x^4 \left (c x^2\right )^p (a+b x)^{-2 (p+2)}}{2 a (p+2)} \end {gather*}
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 = 0.04, size = 32, normalized size = 0.97 \begin {gather*} \frac {x^4 \left (c x^2\right )^p (a+b x)^{-4-2 p}}{a (4+2 p)} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.14, size = 32, normalized size = 0.97
method | result | size |
gosper | \(\frac {x^{4} \left (b x +a \right )^{-4-2 p} \left (c \,x^{2}\right )^{p}}{2 a \left (2+p \right )}\) | \(32\) |
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 = 0.31, size = 40, normalized size = 1.21 \begin {gather*} \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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs.
\(2 (33) = 66\).
time = 0.01, size = 81, normalized size = 2.45 \begin {gather*} \frac {a x^{4} \mathrm {e}^{-2 p \ln \left (a+b x\right )-5 \ln \left (a+b x\right )} \mathrm {e}^{p \ln \left (c x^{2}\right )}+b x^{5} \mathrm {e}^{-2 p \ln \left (a+b x\right )-5 \ln \left (a+b x\right )} \mathrm {e}^{p \ln \left (c x^{2}\right )}}{2 a p+4 a} \end {gather*}
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 \begin {gather*} \frac {x^4\,{\left (c\,x^2\right )}^p}{2\,a\,\left (p+2\right )\,{\left (a+b\,x\right )}^{2\,p+4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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