Optimal. Leaf size=83 \[ -\frac{16 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{35 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{35 a c \left (c-a^2 c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.093848, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {6136, 6135} \[ -\frac{16 (1-2 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{35 a c^2 \sqrt{c-a^2 c x^2}}-\frac{2 (1-6 a x) e^{\frac{1}{2} \tanh ^{-1}(a x)}}{35 a c \left (c-a^2 c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6136
Rule 6135
Rubi steps
\begin{align*} \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{35 a c \left (c-a^2 c x^2\right )^{3/2}}+\frac{24 \int \frac{e^{\frac{1}{2} \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{35 c}\\ &=-\frac{2 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-6 a x)}{35 a c \left (c-a^2 c x^2\right )^{3/2}}-\frac{16 e^{\frac{1}{2} \tanh ^{-1}(a x)} (1-2 a x)}{35 a c^2 \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0366685, size = 80, normalized size = 0.96 \[ -\frac{2 \sqrt{1-a^2 x^2} \left (16 a^3 x^3-8 a^2 x^2-22 a x+9\right )}{35 a c^2 (1-a x)^{7/4} (a x+1)^{5/4} \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.03, size = 71, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) \left ( 16\,{x}^{3}{a}^{3}-8\,{a}^{2}{x}^{2}-22\,ax+9 \right ) }{35\,a}\sqrt{{(ax+1){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}} \left ( -{a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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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{\sqrt{\frac{a x + 1}{\sqrt{-a^{2} x^{2} + 1}}}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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