Optimal. Leaf size=30 \[ \frac {x \left (c x^2\right )^p (a+b x)^{-1-2 p}}{a (1+2 p)} \]
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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {15, 37}
\begin {gather*} \frac {x \left (c x^2\right )^p (a+b x)^{-2 p-1}}{a (2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 37
Rubi steps
\begin {align*} \int \left (c x^2\right )^p (a+b x)^{-2-2 p} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{2 p} (a+b x)^{-2-2 p} \, dx\\ &=\frac {x \left (c x^2\right )^p (a+b x)^{-1-2 p}}{a (1+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 28, normalized size = 0.93 \begin {gather*} \frac {x \left (c x^2\right )^p (a+b x)^{-1-2 p}}{a+2 a p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 31, normalized size = 1.03
method | result | size |
gosper | \(\frac {x \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-1-2 p}}{a \left (1+2 p \right )}\) | \(31\) |
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 = 1.17, size = 36, normalized size = 1.20 \begin {gather*} \frac {{\left (b x^{2} + a x\right )} \left (c x^{2}\right )^{p} {\left (b x + a\right )}^{-2 \, p - 2}}{2 \, a p + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} - \frac {\left (b x\right )^{- 2 p} \left (c x^{2}\right )^{p}}{b^{2} x} & \text {for}\: a = 0 \\\frac {0^{- 2 p - 2} x \left (c x^{2}\right )^{p}}{2 p + 1} & \text {for}\: a = - b x \\\frac {x \left (c x^{2}\right )^{p} \left (0^{\frac {1}{p}}\right )^{- 2 p - 2}}{2 p + 1} & \text {for}\: a = 0^{\frac {1}{p}} - b x \\\int \frac {1}{\sqrt {c x^{2}} \left (a + b x\right )}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a^{3} x \left (c x^{2}\right )^{p}}{2 a^{5} p \left (a + b x\right )^{2 p} + a^{5} \left (a + b x\right )^{2 p} + 8 a^{4} b p x \left (a + b x\right )^{2 p} + 4 a^{4} b x \left (a + b x\right )^{2 p} + 12 a^{3} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 6 a^{3} b^{2} x^{2} \left (a + b x\right )^{2 p} + 8 a^{2} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 4 a^{2} b^{3} x^{3} \left (a + b x\right )^{2 p} + 2 a b^{4} p x^{4} \left (a + b x\right )^{2 p} + a b^{4} x^{4} \left (a + b x\right )^{2 p}} + \frac {2 a^{2} b x^{2} \left (c x^{2}\right )^{p}}{2 a^{5} p \left (a + b x\right )^{2 p} + a^{5} \left (a + b x\right )^{2 p} + 8 a^{4} b p x \left (a + b x\right )^{2 p} + 4 a^{4} b x \left (a + b x\right )^{2 p} + 12 a^{3} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 6 a^{3} b^{2} x^{2} \left (a + b x\right )^{2 p} + 8 a^{2} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 4 a^{2} b^{3} x^{3} \left (a + b x\right )^{2 p} + 2 a b^{4} p x^{4} \left (a + b x\right )^{2 p} + a b^{4} x^{4} \left (a + b x\right )^{2 p}} + \frac {a b^{2} x^{3} \left (c x^{2}\right )^{p}}{2 a^{5} p \left (a + b x\right )^{2 p} + a^{5} \left (a + b x\right )^{2 p} + 8 a^{4} b p x \left (a + b x\right )^{2 p} + 4 a^{4} b x \left (a + b x\right )^{2 p} + 12 a^{3} b^{2} p x^{2} \left (a + b x\right )^{2 p} + 6 a^{3} b^{2} x^{2} \left (a + b x\right )^{2 p} + 8 a^{2} b^{3} p x^{3} \left (a + b x\right )^{2 p} + 4 a^{2} b^{3} x^{3} \left (a + b x\right )^{2 p} + 2 a b^{4} p x^{4} \left (a + b x\right )^{2 p} + a b^{4} x^{4} \left (a + b x\right )^{2 p}} + \frac {b x^{2} \left (c x^{2}\right )^{p}}{2 a^{3} p \left (a + b x\right )^{2 p} + a^{3} \left (a + b x\right )^{2 p} + 4 a^{2} b p x \left (a + b x\right )^{2 p} + 2 a^{2} b x \left (a + b x\right )^{2 p} + 2 a b^{2} p x^{2} \left (a + b x\right )^{2 p} + a b^{2} x^{2} \left (a + b x\right )^{2 p}} & \text {otherwise} \end {cases} \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 = 0.20, size = 32, normalized size = 1.07 \begin {gather*} \frac {x\,{\left (c\,x^2\right )}^p}{a\,\left (2\,p+1\right )\,{\left (a+b\,x\right )}^{2\,p+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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