Optimal. Leaf size=187 \[ a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{2}-p,\frac{3}{2},a^2 x^2\right )-\frac{3 a \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^p \text{Hypergeometric2F1}\left (1,p+\frac{1}{2},p+\frac{3}{2},1-a^2 x^2\right )}{2 p+1}+\frac{4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt{1-a^2 x^2}}-\frac{\left (c-a^2 c x^2\right )^p}{x \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.331396, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {6153, 6148, 1807, 1652, 446, 79, 65, 12, 245} \[ a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{1}{2},\frac{3}{2}-p;\frac{3}{2};a^2 x^2\right )-\frac{3 a \sqrt{1-a^2 x^2} \left (c-a^2 c x^2\right )^p \, _2F_1\left (1,p+\frac{1}{2};p+\frac{3}{2};1-a^2 x^2\right )}{2 p+1}+\frac{4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt{1-a^2 x^2}}-\frac{\left (c-a^2 c x^2\right )^p}{x \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6153
Rule 6148
Rule 1807
Rule 1652
Rule 446
Rule 79
Rule 65
Rule 12
Rule 245
Rubi steps
\begin{align*} \int \frac{e^{3 \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^{3 \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)^3 \left (1-a^2 x^2\right )^{-\frac{3}{2}+p}}{x^2} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^p}{x \sqrt{1-a^2 x^2}}-\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{3}{2}+p} \left (-3 a-a^2 (5-2 p) x-a^3 x^2\right )}{x} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^p}{x \sqrt{1-a^2 x^2}}+\left (\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int a^2 (5-2 p) \left (1-a^2 x^2\right )^{-\frac{3}{2}+p} \, 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{3}{2}+p} \left (-3 a-a^3 x^2\right )}{x} \, dx\\ &=-\frac{\left (c-a^2 c x^2\right )^p}{x \sqrt{1-a^2 x^2}}-\frac{1}{2} \left (\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{3}{2}+p} \left (-3 a-a^3 x\right )}{x} \, dx,x,x^2\right )+\left (a^2 (5-2 p) \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p\right ) \int \left (1-a^2 x^2\right )^{-\frac{3}{2}+p} \, dx\\ &=\frac{4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt{1-a^2 x^2}}-\frac{\left (c-a^2 c x^2\right )^p}{x \sqrt{1-a^2 x^2}}+a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{1}{2},\frac{3}{2}-p;\frac{3}{2};a^2 x^2\right )+\frac{1}{2} \left (3 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{4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt{1-a^2 x^2}}-\frac{\left (c-a^2 c x^2\right )^p}{x \sqrt{1-a^2 x^2}}+a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \, _2F_1\left (\frac{1}{2},\frac{3}{2}-p;\frac{3}{2};a^2 x^2\right )-\frac{3 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*}
Mathematica [A] time = 0.250108, size = 133, normalized size = 0.71 \[ \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a \left (\frac{\left (1-a^2 x^2\right )^{p-\frac{1}{2}} \left (3 \text{Hypergeometric2F1}\left (1,p-\frac{1}{2},p+\frac{1}{2},1-a^2 x^2\right )+1\right )}{1-2 p}+3 a x \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{2}-p,\frac{3}{2},a^2 x^2\right )\right )-\frac{\text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{3}{2}-p,\frac{1}{2},a^2 x^2\right )}{x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.396, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ax+1 \right ) ^{3} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{p}}{{x}^{2}} \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 (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{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 (-a^{2} c x^{2} + c\right )}^{p}}{a^{2} x^{4} - 2 \, a x^{3} + x^{2}}, 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 (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{3}}{x^{2} \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 (-a^{2} c x^{2} + c\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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