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 = 0.24, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 204
Rule 216
Rule 725
Rule 844
Rule 853
Rule 1654
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*}
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Mathematica [C] time = 0.30, 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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fricas [A] time = 1.47, size = 318, normalized size = 2.97 \[ \left [-\frac {{\left (a c - d\right )} \sqrt {-\frac {a c - d}{a c + d}} \log \left (\frac {a^{2} c d x + d^{2} - {\left (a^{2} c^{2} - d^{2}\right )} \sqrt {-a^{2} x^{2} + 1} - {\left (a c d + d^{2} + {\left (a^{3} c^{2} + a^{2} c d\right )} x + \sqrt {-a^{2} x^{2} + 1} {\left (a c d + d^{2}\right )}\right )} \sqrt {-\frac {a c - d}{a c + d}}}{d x + c}\right ) - 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}, \frac {2 \, {\left (a c - d\right )} \sqrt {\frac {a c - d}{a c + d}} \arctan \left (\frac {{\left (d x - \sqrt {-a^{2} x^{2} + 1} c + c\right )} \sqrt {\frac {a c - d}{a c + d}}}{{\left (a c - d\right )} x}\right ) + 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.59, size = 208, normalized size = 1.94 \[ -{\left (\frac {{\left (a x - 1\right )} \sqrt {-\frac {2}{a x - 1} - 1} \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\relax (a)}{a d} - \frac {2 \, {\left (a c \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\relax (a) - 2 \, d \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\relax (a)\right )} \arctan \left (\sqrt {-\frac {2}{a x - 1} - 1}\right )}{a d^{2}} + \frac {2 \, {\left (a^{2} c^{2} \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\relax (a) - 2 \, a c d \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\relax (a) + d^{2} \mathrm {sgn}\left (\frac {1}{a x - 1}\right ) \mathrm {sgn}\relax (a)\right )} \arctan \left (\frac {a c \sqrt {-\frac {2}{a x - 1} - 1} + d \sqrt {-\frac {2}{a x - 1} - 1}}{\sqrt {a^{2} c^{2} - d^{2}}}\right )}{\sqrt {a^{2} c^{2} - d^{2}} a d^{2}}\right )} {\left | a \right |} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 1178, normalized size = 11.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a x - 1\right )}^{2} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 148, normalized size = 1.38 \[ -\frac {\sqrt {1-a^2\,x^2}}{d}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\left (2\,a\,\sqrt {-a^2}-\frac {a^2\,c\,\sqrt {-a^2}}{d}\right )}{a^2\,d}-\frac {\left (\ln \left (\sqrt {1-\frac {a^2\,c^2}{d^2}}\,\sqrt {1-a^2\,x^2}+\frac {a^2\,c\,x}{d}+1\right )-\ln \left (c+d\,x\right )\right )\,\left (a^2\,c^2-2\,a\,c\,d+d^2\right )}{d^3\,\sqrt {1-\frac {a^2\,c^2}{d^2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}{\left (c + d x\right ) \left (a x - 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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