Optimal. Leaf size=33 \[ -\frac {\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x} \]
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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 37}
\begin {gather*} -\frac {\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x} \end {gather*}
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)^{-2 p}}{x^2} \, dx &=\left (x^{-2 p} \left (c x^2\right )^p\right ) \int x^{-2+2 p} (a+b x)^{-2 p} \, dx\\ &=-\frac {\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (1-2 p) x}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 32, normalized size = 0.97 \begin {gather*} \frac {\left (c x^2\right )^p (a+b x)^{1-2 p}}{a (-1+2 p) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 38, normalized size = 1.15
method | result | size |
gosper | \(\frac {\left (b x +a \right ) \left (c \,x^{2}\right )^{p} \left (b x +a \right )^{-2 p}}{x a \left (2 p -1\right )}\) | \(38\) |
risch | \(\frac {\left (b x +a \right ) \left (b x +a \right )^{-2 p} {\mathrm e}^{\frac {p \left (-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i c \,x^{2}\right )^{2}-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i c \,x^{2}\right ) \mathrm {csgn}\left (i c \right )-i \pi \mathrm {csgn}\left (i c \,x^{2}\right )^{3}+i \pi \mathrm {csgn}\left (i c \,x^{2}\right )^{2} \mathrm {csgn}\left (i c \right )+2 \ln \left (c \right )+4 \ln \left (x \right )\right )}{2}}}{\left (2 p -1\right ) a x}\) | \(171\) |
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 = 0.57, size = 37, normalized size = 1.12 \begin {gather*} \frac {{\left (b x + a\right )} \left (c x^{2}\right )^{p}}{{\left (2 \, a p - a\right )} {\left (b x + a\right )}^{2 \, p} x} \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 {\sqrt {c x^{2}}}{b x^{2}} & \text {for}\: a = 0 \wedge p = \frac {1}{2} \\- \frac {\left (b x\right )^{- 2 p} \left (c x^{2}\right )^{p}}{x} & \text {for}\: a = 0 \\\int \frac {\sqrt {c x^{2}}}{x^{2} \left (a + b x\right )}\, dx & \text {for}\: p = \frac {1}{2} \\\frac {a \left (c x^{2}\right )^{p}}{2 a p x \left (a + b x\right )^{2 p} - a x \left (a + b x\right )^{2 p}} + \frac {b x \left (c x^{2}\right )^{p}}{2 a p x \left (a + b x\right )^{2 p} - a x \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 = 0.24, size = 32, normalized size = 0.97 \begin {gather*} \frac {{\left (c\,x^2\right )}^p\,{\left (a+b\,x\right )}^{1-2\,p}}{a\,x\,\left (2\,p-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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