Optimal. Leaf size=113 \[ -\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {1}{2},\frac {1}{2}-p;\frac {1}{2};a^2 x^2\right )}{x}-\frac {a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,p+\frac {1}{2};p+\frac {3}{2};1-a^2 x^2\right )}{2 p+1} \]
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Rubi [A] time = 0.17, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6153, 6148, 764, 364, 266, 65} \[ -\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {1}{2},\frac {1}{2}-p;\frac {1}{2};a^2 x^2\right )}{x}-\frac {a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,p+\frac {1}{2};p+\frac {3}{2};1-a^2 x^2\right )}{2 p+1} \]
Antiderivative was successfully verified.
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Rule 65
Rule 266
Rule 364
Rule 764
Rule 6148
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \frac {e^{\tanh ^{-1}(a x)} \left (1-a^2 x^2\right )^p}{x^2} \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \frac {(1+a x) \left (1-a^2 x^2\right )^{-\frac {1}{2}+p}}{x^2} \, dx\\ &=\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \frac {\left (1-a^2 x^2\right )^{-\frac {1}{2}+p}}{x^2} \, dx+\left (a \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \frac {\left (1-a^2 x^2\right )^{-\frac {1}{2}+p}}{x} \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {1}{2},\frac {1}{2}-p;\frac {1}{2};a^2 x^2\right )}{x}+\frac {1}{2} \left (a \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \operatorname {Subst}\left (\int \frac {\left (1-a^2 x\right )^{-\frac {1}{2}+p}}{x} \, dx,x,x^2\right )\\ &=-\frac {\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (-\frac {1}{2},\frac {1}{2}-p;\frac {1}{2};a^2 x^2\right )}{x}-\frac {a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,\frac {1}{2}+p;\frac {3}{2}+p;1-a^2 x^2\right )}{1+2 p}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 105, normalized size = 0.93 \[ \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {a \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \, _2F_1\left (1,p+\frac {1}{2};p+\frac {3}{2};1-a^2 x^2\right )}{2 \left (p+\frac {1}{2}\right )}-\frac {\, _2F_1\left (-\frac {1}{2},\frac {1}{2}-p;\frac {1}{2};a^2 x^2\right )}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} {\left (-a^{2} c x^{2} + c\right )}^{p}}{a x^{3} - x^{2}}, x\right ) \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
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 \[ \int \frac {\left (a x +1\right ) \left (-a^{2} c \,x^{2}+c \right )^{p}}{\sqrt {-a^{2} x^{2}+1}\, x^{2}}\, dx \]
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 \[ \int \frac {{\left (a x + 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,\left (a\,x+1\right )}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 12.52, size = 294, normalized size = 2.60 \[ - \frac {a a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p \\ p + 1, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } \Gamma \left (1 - p\right ) \Gamma \left (p + 1\right )} - \frac {a a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma \left (- p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p \\ \frac {1}{2}, 1 - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (1 - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p - \frac {1}{2} \\ p + \frac {1}{2}, p + 1 \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } x \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p} c^{p} x^{2 p} e^{i \pi p} \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, \frac {1}{2} - p \\ \frac {1}{2}, \frac {3}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } x \Gamma \left (\frac {3}{2} - p\right ) \Gamma \left (p + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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