Optimal. Leaf size=96 \[ \frac {1}{3} a x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right )-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^2 (2 p+1)} \]
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Rubi [A] time = 0.11, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {6153, 6148, 764, 261, 364} \[ \frac {1}{3} a x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right )-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^2 (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 261
Rule 364
Rule 764
Rule 6148
Rule 6153
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(a x)} x \left (c-a^2 c x^2\right )^p \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int e^{\tanh ^{-1}(a x)} x \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x (1+a x) \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx+\left (a \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int x^2 \left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p}{a^2 (1+2 p)}+\frac {1}{3} a x^3 \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 88, normalized size = 0.92 \[ \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{3} a x^3 \, _2F_1\left (\frac {3}{2},\frac {1}{2}-p;\frac {5}{2};a^2 x^2\right )-\frac {\left (1-a^2 x^2\right )^{p+\frac {1}{2}}}{2 a^2 \left (p+\frac {1}{2}\right )}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p} x}{a x - 1}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p} x}{\sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x +1\right ) x \left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}}\, dx \]
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 \[ a c^{p} \int \frac {x^{2} e^{\left (p \log \left (a x + 1\right ) + p \log \left (-a x + 1\right )\right )}}{\sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} + \frac {{\left (a^{2} c^{p} x^{2} - c^{p}\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} a^{2} {\left (2 \, p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 17.59, size = 314, normalized size = 3.27 \[ - \frac {a a^{2 p} c^{p} x^{3} x^{2 p} e^{i \pi p} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {3}{2} \\ p + 1, p + \frac {5}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + 1\right )} - \frac {a a^{2 p} c^{p} x^{3} x^{2 p} e^{i \pi p} \Gamma \left (- p - \frac {3}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {3}{2} \\ \frac {1}{2}, - p - \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, 1 \\ p + 2 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2} x^{2 p} e^{i \pi p} \Gamma \left (- p - 1\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} 1, - p - 1 \\ \frac {1}{2} \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (- p\right ) \Gamma \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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