Optimal. Leaf size=26 \[ \frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]
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Rubi [A] time = 0.01, antiderivative size = 26, 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} \[ \frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int \frac {\left (c x^2\right )^p (a+b x)^{-1-2 p}}{x} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{-1+2 p} (a+b x)^{-1-2 p} \, dx\\ &=\frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ \frac {\left (c x^2\right )^p (a+b x)^{-2 p}}{2 a p} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 31, normalized size = 1.19 \[ \frac {{\left (b x + a\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 1}}{2 \, a p} \]
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 \[ \int \frac {\left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 1}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.96 \[ \frac {\left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-2 p}}{2 a p} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 27, normalized size = 1.04 \[ \frac {c^{p} e^{\left (-2 \, p \log \left (b x + a\right ) + 2 \, p \log \relax (x)\right )}}{2 \, a p} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 26, normalized size = 1.00 \[ \frac {{\left (c\,x^2\right )}^p}{2\,a\,p\,{\left (a+b\,x\right )}^{2\,p}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 59.69, size = 264, normalized size = 10.15 \[ \begin {cases} - \frac {b^{- 2 p} c^{p} x^{- 2 p} \left (x^{2}\right )^{p}}{b x} & \text {for}\: a = 0 \\\frac {0^{- 2 p - 1} c^{p} \left (x^{2}\right )^{p}}{2 p} & \text {for}\: a = - b x \\\frac {c^{p} \left (0^{\frac {1}{p}}\right )^{- 2 p - 1} \left (x^{2}\right )^{p}}{2 p} & \text {for}\: a = 0^{\frac {1}{p}} - b x \\\frac {\log {\relax (x )}}{a} - \frac {\log {\left (\frac {a}{b} + x \right )}}{a} & \text {for}\: p = 0 \\\frac {a^{2} c^{p} \left (x^{2}\right )^{p}}{2 a^{3} p \left (a + b x\right )^{2 p} + 4 a^{2} b p x \left (a + b x\right )^{2 p} + 2 a b^{2} p x^{2} \left (a + b x\right )^{2 p}} + \frac {a b c^{p} x \left (x^{2}\right )^{p}}{2 a^{3} p \left (a + b x\right )^{2 p} + 4 a^{2} b p x \left (a + b x\right )^{2 p} + 2 a b^{2} p x^{2} \left (a + b x\right )^{2 p}} + \frac {b c^{p} x \left (x^{2}\right )^{p}}{2 a^{2} p \left (a + b x\right )^{2 p} + 2 a b p x \left (a + b x\right )^{2 p}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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