Optimal. Leaf size=217 \[ \frac{3 a^2 x^3 \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2} (3-2 p),\frac{3}{2}-p,\frac{1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}+\frac{a (5-2 p) x^2 \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \text{Hypergeometric2F1}\left (1-p,\frac{3}{2}-p,2-p,a^2 x^2\right )}{2 (1-p)}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^p}{\sqrt{1-a^2 x^2}}+\frac{x \left (c-\frac{c}{a^2 x^2}\right )^p}{(1-2 p) \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.330087, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6160, 6148, 1809, 1808, 364, 807} \[ \frac{3 a^2 x^3 \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (\frac{1}{2} (3-2 p),\frac{3}{2}-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac{a (5-2 p) x^2 \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \, _2F_1\left (1-p,\frac{3}{2}-p;2-p;a^2 x^2\right )}{2 (1-p)}-\frac{a x^2 \left (c-\frac{c}{a^2 x^2}\right )^p}{\sqrt{1-a^2 x^2}}+\frac{x \left (c-\frac{c}{a^2 x^2}\right )^p}{(1-2 p) \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6148
Rule 1809
Rule 1808
Rule 364
Rule 807
Rubi steps
\begin{align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac{c}{a^2 x^2}\right )^p \, dx &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int e^{3 \tanh ^{-1}(a x)} x^{-2 p} \left (1-a^2 x^2\right )^p \, dx\\ &=\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{-2 p} (1+a x)^3 \left (1-a^2 x^2\right )^{-\frac{3}{2}+p} \, dx\\ &=-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^p x^2}{\sqrt{1-a^2 x^2}}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{-2 p} \left (1-a^2 x^2\right )^{-\frac{3}{2}+p} \left (-a^2-a^3 (5-2 p) x-3 a^4 x^2\right ) \, dx}{a^2}\\ &=-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^p x^2}{\sqrt{1-a^2 x^2}}-\frac{\left (\left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{-2 p} \left (-a^2-a^3 (5-2 p) x\right ) \left (1-a^2 x^2\right )^{-\frac{3}{2}+p} \, dx}{a^2}+\left (3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{2-2 p} \left (1-a^2 x^2\right )^{-\frac{3}{2}+p} \, dx\\ &=\frac{\left (c-\frac{c}{a^2 x^2}\right )^p x}{(1-2 p) \sqrt{1-a^2 x^2}}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^p x^2}{\sqrt{1-a^2 x^2}}+\frac{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^3 \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (\frac{1}{2} (3-2 p),\frac{3}{2}-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\left (a (5-2 p) \left (c-\frac{c}{a^2 x^2}\right )^p x^{2 p} \left (1-a^2 x^2\right )^{-p}\right ) \int x^{1-2 p} \left (1-a^2 x^2\right )^{-\frac{3}{2}+p} \, dx\\ &=\frac{\left (c-\frac{c}{a^2 x^2}\right )^p x}{(1-2 p) \sqrt{1-a^2 x^2}}-\frac{a \left (c-\frac{c}{a^2 x^2}\right )^p x^2}{\sqrt{1-a^2 x^2}}+\frac{3 a^2 \left (c-\frac{c}{a^2 x^2}\right )^p x^3 \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (\frac{1}{2} (3-2 p),\frac{3}{2}-p;\frac{1}{2} (5-2 p);a^2 x^2\right )}{3-2 p}+\frac{a (5-2 p) \left (c-\frac{c}{a^2 x^2}\right )^p x^2 \left (1-a^2 x^2\right )^{-p} \, _2F_1\left (1-p,\frac{3}{2}-p;2-p;a^2 x^2\right )}{2 (1-p)}\\ \end{align*}
Mathematica [A] time = 0.15768, size = 175, normalized size = 0.81 \[ \frac{1}{2} x \left (1-a^2 x^2\right )^{-p} \left (c-\frac{c}{a^2 x^2}\right )^p \left (-\frac{3 a x \text{Hypergeometric2F1}\left (1-p,\frac{3}{2}-p,2-p,a^2 x^2\right )}{p-1}+\frac{6 a^2 x^2 \text{Hypergeometric2F1}\left (\frac{3}{2}-p,\frac{3}{2}-p,\frac{5}{2}-p,a^2 x^2\right )}{3-2 p}+\frac{a^3 x^3 \text{Hypergeometric2F1}\left (\frac{3}{2}-p,2-p,3-p,a^2 x^2\right )}{2-p}+\frac{2 \left (1-a^2 x^2\right )^{p-\frac{1}{2}}}{1-2 p}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.298, size = 0, normalized size = 0. \begin{align*} \int{ \left ( ax+1 \right ) ^{3} \left ( c-{\frac{c}{{a}^{2}{x}^{2}}} \right ) ^{p} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x + 1\right )} \left (\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}\right )^{p}}{a^{2} x^{2} - 2 \, a x + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )\right )^{p} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x + 1\right )}^{3}{\left (c - \frac{c}{a^{2} x^{2}}\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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