Optimal. Leaf size=268 \[ -\frac {\sqrt {1-a^2 x^2}}{a^6 c^2 (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^6 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^6 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {23 \sqrt {1-a^2 x^2} \log (1-a x)}{16 a^6 c^2 \sqrt {c-a^2 c x^2}}+\frac {7 \sqrt {1-a^2 x^2} \log (a x+1)}{16 a^6 c^2 \sqrt {c-a^2 c x^2}}-\frac {x \sqrt {1-a^2 x^2}}{a^5 c^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.25, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6153, 6150, 88} \[ -\frac {x \sqrt {1-a^2 x^2}}{a^5 c^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^6 c^2 (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^6 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^6 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {23 \sqrt {1-a^2 x^2} \log (1-a x)}{16 a^6 c^2 \sqrt {c-a^2 c x^2}}+\frac {7 \sqrt {1-a^2 x^2} \log (a x+1)}{16 a^6 c^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rule 6153
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^5}{(1-a x)^3 (1+a x)^2} \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (-\frac {1}{a^5}-\frac {1}{4 a^5 (-1+a x)^3}-\frac {1}{a^5 (-1+a x)^2}-\frac {23}{16 a^5 (-1+a x)}-\frac {1}{8 a^5 (1+a x)^2}+\frac {7}{16 a^5 (1+a x)}\right ) \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=-\frac {x \sqrt {1-a^2 x^2}}{a^5 c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^6 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^6 c^2 (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^6 c^2 (1+a x) \sqrt {c-a^2 c x^2}}-\frac {23 \sqrt {1-a^2 x^2} \log (1-a x)}{16 a^6 c^2 \sqrt {c-a^2 c x^2}}+\frac {7 \sqrt {1-a^2 x^2} \log (1+a x)}{16 a^6 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 87, normalized size = 0.32 \[ \frac {\sqrt {1-a^2 x^2} \left (2 \left (-8 a x+\frac {8}{a x-1}+\frac {1}{a x+1}+\frac {1}{(a x-1)^2}\right )-23 \log (1-a x)+7 \log (a x+1)\right )}{16 a^6 c^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{5}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 182, normalized size = 0.68 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (16 x^{4} a^{4}+23 \ln \left (a x -1\right ) x^{3} a^{3}-7 a^{3} x^{3} \ln \left (a x +1\right )-16 x^{3} a^{3}-23 \ln \left (a x -1\right ) x^{2} a^{2}+7 \ln \left (a x +1\right ) x^{2} a^{2}-34 a^{2} x^{2}-23 \ln \left (a x -1\right ) x a +7 a x \ln \left (a x +1\right )+18 a x +23 \ln \left (a x -1\right )-7 \ln \left (a x +1\right )+12\right )}{16 \left (a^{2} x^{2}-1\right ) c^{3} a^{6} \left (a x -1\right )^{2} \left (a x +1\right )} \]
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 \[ a \int -\frac {x^{6}}{{\left (a^{4} c^{\frac {5}{2}} x^{4} - 2 \, a^{2} c^{\frac {5}{2}} x^{2} + c^{\frac {5}{2}}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )}}\,{d x} + \frac {1}{4 \, {\left (a^{10} c^{\frac {5}{2}} x^{4} - 2 \, a^{8} c^{\frac {5}{2}} x^{2} + a^{6} c^{\frac {5}{2}}\right )}} + \frac {1}{a^{8} c^{\frac {5}{2}} x^{2} - a^{6} c^{\frac {5}{2}}} - \frac {\log \left (-a^{2} c x^{2} + c\right )}{2 \, a^{6} c^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{5} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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