Optimal. Leaf size=183 \[ \frac{(a x+1)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{6 (a x+1)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{2 (a x+1)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{8 (a x+1)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]
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Rubi [A] time = 0.182818, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6192, 6193, 43} \[ \frac{(a x+1)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{6 (a x+1)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}+\frac{2 (a x+1)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}}-\frac{8 (a x+1)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 x^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^{7/2} \, dx &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int e^{\coth ^{-1}(a x)} \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7 \, dx}{\left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int (-1+a x)^3 (1+a x)^4 \, dx}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=\frac{\left (c-a^2 c x^2\right )^{7/2} \int \left (-8 (1+a x)^4+12 (1+a x)^5-6 (1+a x)^6+(1+a x)^7\right ) \, dx}{a^7 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ &=-\frac{8 (1+a x)^5 \left (c-a^2 c x^2\right )^{7/2}}{5 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}+\frac{2 (1+a x)^6 \left (c-a^2 c x^2\right )^{7/2}}{a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}-\frac{6 (1+a x)^7 \left (c-a^2 c x^2\right )^{7/2}}{7 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}+\frac{(1+a x)^8 \left (c-a^2 c x^2\right )^{7/2}}{8 a^8 \left (1-\frac{1}{a^2 x^2}\right )^{7/2} x^7}\\ \end{align*}
Mathematica [A] time = 0.0518518, size = 71, normalized size = 0.39 \[ -\frac{c^3 (a x+1)^5 \left (35 a^3 x^3-135 a^2 x^2+185 a x-93\right ) \sqrt{c-a^2 c x^2}}{280 a^2 x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 100, normalized size = 0.6 \begin{align*}{\frac{x \left ( 35\,{a}^{7}{x}^{7}+40\,{x}^{6}{a}^{6}-140\,{x}^{5}{a}^{5}-168\,{x}^{4}{a}^{4}+210\,{x}^{3}{a}^{3}+280\,{a}^{2}{x}^{2}-140\,ax-280 \right ) }{280\, \left ( ax-1 \right ) ^{3} \left ( ax+1 \right ) ^{4}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{7}{2}}}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \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^{2} c x^{2} + c\right )}^{\frac{7}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56936, size = 212, normalized size = 1.16 \begin{align*} -\frac{{\left (35 \, a^{7} c^{3} x^{8} + 40 \, a^{6} c^{3} x^{7} - 140 \, a^{5} c^{3} x^{6} - 168 \, a^{4} c^{3} x^{5} + 210 \, a^{3} c^{3} x^{4} + 280 \, a^{2} c^{3} x^{3} - 140 \, a c^{3} x^{2} - 280 \, c^{3} x\right )} \sqrt{-a^{2} c}}{280 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{2} c x^{2} + c\right )}^{\frac{7}{2}}}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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