Optimal. Leaf size=107 \[ \frac{(a c-d)^2 \tan ^{-1}\left (\frac{a^2 c x+d}{\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}}\right )}{d^2 \sqrt{a^2 c^2-d^2}}-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-2 d) \sin ^{-1}(a x)}{d^2} \]
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Rubi [A] time = 0.240769, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {853, 1654, 844, 216, 725, 204} \[ \frac{(a c-d)^2 \tan ^{-1}\left (\frac{a^2 c x+d}{\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}}\right )}{d^2 \sqrt{a^2 c^2-d^2}}-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-2 d) \sin ^{-1}(a x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 853
Rule 1654
Rule 844
Rule 216
Rule 725
Rule 204
Rubi steps
\begin{align*} \int \frac{\left (1-a^2 x^2\right )^{3/2}}{(1-a x)^2 (c+d x)} \, dx &=\int \frac{(1+a x)^2}{(c+d x) \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{\int \frac{-a^2 d^2+a^3 (a c-2 d) d x}{(c+d x) \sqrt{1-a^2 x^2}} \, dx}{a^2 d^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a (a c-2 d)) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{d^2}+\frac{(a c-d)^2 \int \frac{1}{(c+d x) \sqrt{1-a^2 x^2}} \, dx}{d^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-2 d) \sin ^{-1}(a x)}{d^2}-\frac{(a c-d)^2 \operatorname{Subst}\left (\int \frac{1}{-a^2 c^2+d^2-x^2} \, dx,x,\frac{d+a^2 c x}{\sqrt{1-a^2 x^2}}\right )}{d^2}\\ &=-\frac{\sqrt{1-a^2 x^2}}{d}-\frac{(a c-2 d) \sin ^{-1}(a x)}{d^2}+\frac{(a c-d)^2 \tan ^{-1}\left (\frac{d+a^2 c x}{\sqrt{a^2 c^2-d^2} \sqrt{1-a^2 x^2}}\right )}{d^2 \sqrt{a^2 c^2-d^2}}\\ \end{align*}
Mathematica [C] time = 0.307408, size = 148, normalized size = 1.38 \[ -\frac{\frac{i (d-a c)^2 \log \left (\frac{2 d^3 \left (\sqrt{1-a^2 x^2} \sqrt{a^2 c^2-d^2}+i a^2 c x+i d\right )}{(d-a c)^2 \sqrt{a^2 c^2-d^2} (c+d x)}\right )}{\sqrt{a^2 c^2-d^2}}+d \sqrt{1-a^2 x^2}+(a c-2 d) \sin ^{-1}(a x)}{d^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.289, size = 1178, normalized size = 11. \begin{align*} \text{result too large to display} \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} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a x - 1\right )}^{2}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.10313, size = 679, normalized size = 6.35 \begin{align*} \left [-\frac{{\left (a c - d\right )} \sqrt{-\frac{a c - d}{a c + d}} \log \left (\frac{a^{2} c d x + d^{2} -{\left (a^{2} c^{2} - d^{2}\right )} \sqrt{-a^{2} x^{2} + 1} -{\left (a c d + d^{2} +{\left (a^{3} c^{2} + a^{2} c d\right )} x + \sqrt{-a^{2} x^{2} + 1}{\left (a c d + d^{2}\right )}\right )} \sqrt{-\frac{a c - d}{a c + d}}}{d x + c}\right ) - 2 \,{\left (a c - 2 \, d\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt{-a^{2} x^{2} + 1} d}{d^{2}}, \frac{2 \,{\left (a c - d\right )} \sqrt{\frac{a c - d}{a c + d}} \arctan \left (\frac{{\left (d x - \sqrt{-a^{2} x^{2} + 1} c + c\right )} \sqrt{\frac{a c - d}{a c + d}}}{{\left (a c - d\right )} x}\right ) + 2 \,{\left (a c - 2 \, d\right )} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt{-a^{2} x^{2} + 1} d}{d^{2}}\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 (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}{\left (c + d x\right ) \left (a x - 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.46747, size = 281, normalized size = 2.63 \begin{align*} -{\left (\frac{{\left (a x - 1\right )} \sqrt{-\frac{2}{a x - 1} - 1} \mathrm{sgn}\left (\frac{1}{a x - 1}\right ) \mathrm{sgn}\left (a\right )}{a d} - \frac{2 \,{\left (a c \mathrm{sgn}\left (\frac{1}{a x - 1}\right ) \mathrm{sgn}\left (a\right ) - 2 \, d \mathrm{sgn}\left (\frac{1}{a x - 1}\right ) \mathrm{sgn}\left (a\right )\right )} \arctan \left (\sqrt{-\frac{2}{a x - 1} - 1}\right )}{a d^{2}} + \frac{2 \,{\left (a^{2} c^{2} \mathrm{sgn}\left (\frac{1}{a x - 1}\right ) \mathrm{sgn}\left (a\right ) - 2 \, a c d \mathrm{sgn}\left (\frac{1}{a x - 1}\right ) \mathrm{sgn}\left (a\right ) + d^{2} \mathrm{sgn}\left (\frac{1}{a x - 1}\right ) \mathrm{sgn}\left (a\right )\right )} \arctan \left (\frac{a c \sqrt{-\frac{2}{a x - 1} - 1} + d \sqrt{-\frac{2}{a x - 1} - 1}}{\sqrt{a^{2} c^{2} - d^{2}}}\right )}{\sqrt{a^{2} c^{2} - d^{2}} a d^{2}}\right )}{\left | a \right |} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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