Optimal. Leaf size=96 \[ -\frac{7 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{3/2}}-\frac{x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{7}{a c \sqrt{c-\frac{c}{a x}}}+\frac{7}{3 a \left (c-\frac{c}{a x}\right )^{3/2}} \]
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Rubi [A] time = 0.150278, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6133, 25, 514, 375, 78, 51, 63, 208} \[ -\frac{7 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{3/2}}-\frac{x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{7}{a c \sqrt{c-\frac{c}{a x}}}+\frac{7}{3 a \left (c-\frac{c}{a x}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6133
Rule 25
Rule 514
Rule 375
Rule 78
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{2 \tanh ^{-1}(a x)}}{\left (c-\frac{c}{a x}\right )^{3/2}} \, dx &=\int \frac{1+a x}{\left (c-\frac{c}{a x}\right )^{3/2} (1-a x)} \, dx\\ &=-\frac{c \int \frac{1+a x}{\left (c-\frac{c}{a x}\right )^{5/2} x} \, dx}{a}\\ &=-\frac{c \int \frac{a+\frac{1}{x}}{\left (c-\frac{c}{a x}\right )^{5/2}} \, dx}{a}\\ &=\frac{c \operatorname{Subst}\left (\int \frac{a+x}{x^2 \left (c-\frac{c x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{(7 c) \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{5/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{7}{3 a \left (c-\frac{c}{a x}\right )^{3/2}}-\frac{x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{x \left (c-\frac{c x}{a}\right )^{3/2}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{7}{3 a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{7}{a c \sqrt{c-\frac{c}{a x}}}-\frac{x}{\left (c-\frac{c}{a x}\right )^{3/2}}+\frac{7 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )}{2 a c}\\ &=\frac{7}{3 a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{7}{a c \sqrt{c-\frac{c}{a x}}}-\frac{x}{\left (c-\frac{c}{a x}\right )^{3/2}}-\frac{7 \operatorname{Subst}\left (\int \frac{1}{a-\frac{a x^2}{c}} \, dx,x,\sqrt{c-\frac{c}{a x}}\right )}{c^2}\\ &=\frac{7}{3 a \left (c-\frac{c}{a x}\right )^{3/2}}+\frac{7}{a c \sqrt{c-\frac{c}{a x}}}-\frac{x}{\left (c-\frac{c}{a x}\right )^{3/2}}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt{c-\frac{c}{a x}}}{\sqrt{c}}\right )}{a c^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0278301, size = 55, normalized size = 0.57 \[ \frac{x \left (7 \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},1-\frac{1}{a x}\right )-3 a x\right )}{3 c (a x-1) \sqrt{c-\frac{c}{a x}}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.128, size = 260, normalized size = 2.7 \begin{align*} -{\frac{x}{6\,{c}^{2} \left ( ax-1 \right ) ^{3}}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( 42\,{a}^{7/2}\sqrt{ \left ( ax-1 \right ) x}{x}^{3}+21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{3}{a}^{3}-36\,{a}^{5/2} \left ( \left ( ax-1 \right ) x \right ) ^{3/2}x-126\,{a}^{5/2}\sqrt{ \left ( ax-1 \right ) x}{x}^{2}-63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ){x}^{2}{a}^{2}+28\,{a}^{3/2} \left ( \left ( ax-1 \right ) x \right ) ^{3/2}+126\,{a}^{3/2}\sqrt{ \left ( ax-1 \right ) x}x+63\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) xa-42\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}-21\,\ln \left ( 1/2\,{\frac{2\,\sqrt{ \left ( ax-1 \right ) x}\sqrt{a}+2\,ax-1}{\sqrt{a}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax-1 \right ) x}}}{\frac{1}{\sqrt{a}}}} \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 )}^{2}}{{\left (a^{2} x^{2} - 1\right )}{\left (c - \frac{c}{a x}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89251, size = 513, normalized size = 5.34 \begin{align*} \left [\frac{21 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{c} \log \left (-2 \, a c x + 2 \, a \sqrt{c} x \sqrt{\frac{a c x - c}{a x}} + c\right ) - 2 \,{\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{6 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}, \frac{21 \,{\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} \sqrt{\frac{a c x - c}{a x}}}{c}\right ) -{\left (3 \, a^{3} x^{3} - 28 \, a^{2} x^{2} + 21 \, a x\right )} \sqrt{\frac{a c x - c}{a x}}}{3 \,{\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}}\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{a x}{a c x \sqrt{c - \frac{c}{a x}} - 2 c \sqrt{c - \frac{c}{a x}} + \frac{c \sqrt{c - \frac{c}{a x}}}{a x}}\, dx - \int \frac{1}{a c x \sqrt{c - \frac{c}{a x}} - 2 c \sqrt{c - \frac{c}{a x}} + \frac{c \sqrt{c - \frac{c}{a x}}}{a x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28289, size = 193, normalized size = 2.01 \begin{align*} \frac{1}{3} \, a c{\left (\frac{2 \,{\left (2 \, c + \frac{9 \,{\left (a c x - c\right )}}{a x}\right )} x}{{\left (a c x - c\right )} a c^{2} \sqrt{\frac{a c x - c}{a x}}} + \frac{21 \, \arctan \left (\frac{\sqrt{\frac{a c x - c}{a x}}}{\sqrt{-c}}\right )}{a^{2} \sqrt{-c} c^{2}} - \frac{3 \, \sqrt{\frac{a c x - c}{a x}}}{a^{2}{\left (c - \frac{a c x - c}{a x}\right )} c^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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