Optimal. Leaf size=261 \[ \frac {\sqrt {1-a^2 x^2}}{2 a^6 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {9 \sqrt {1-a^2 x^2} \log (1-a x)}{4 a^6 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (a x+1)}{4 a^6 c \sqrt {c-a^2 c x^2}}+\frac {2 x \sqrt {1-a^2 x^2}}{a^5 c \sqrt {c-a^2 c x^2}}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a^4 c \sqrt {c-a^2 c x^2}}+\frac {x^3 \sqrt {1-a^2 x^2}}{3 a^3 c \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.25, antiderivative size = 261, 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} \[ \frac {x^3 \sqrt {1-a^2 x^2}}{3 a^3 c \sqrt {c-a^2 c x^2}}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a^4 c \sqrt {c-a^2 c x^2}}+\frac {2 x \sqrt {1-a^2 x^2}}{a^5 c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{2 a^6 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {9 \sqrt {1-a^2 x^2} \log (1-a x)}{4 a^6 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (a x+1)}{4 a^6 c \sqrt {c-a^2 c x^2}} \]
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 )^{3/2}} \, dx &=\frac {\sqrt {1-a^2 x^2} \int \frac {e^{\tanh ^{-1}(a x)} x^5}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \frac {x^5}{(1-a x)^2 (1+a x)} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2} \int \left (\frac {2}{a^5}+\frac {x}{a^4}+\frac {x^2}{a^3}+\frac {1}{2 a^5 (-1+a x)^2}+\frac {9}{4 a^5 (-1+a x)}-\frac {1}{4 a^5 (1+a x)}\right ) \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=\frac {2 x \sqrt {1-a^2 x^2}}{a^5 c \sqrt {c-a^2 c x^2}}+\frac {x^2 \sqrt {1-a^2 x^2}}{2 a^4 c \sqrt {c-a^2 c x^2}}+\frac {x^3 \sqrt {1-a^2 x^2}}{3 a^3 c \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{2 a^6 c (1-a x) \sqrt {c-a^2 c x^2}}+\frac {9 \sqrt {1-a^2 x^2} \log (1-a x)}{4 a^6 c \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2} \log (1+a x)}{4 a^6 c \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 87, normalized size = 0.33 \[ \frac {\sqrt {1-a^2 x^2} \left (4 a^3 x^3+6 a^2 x^2+24 a x+\frac {6}{1-a x}+27 \log (1-a x)-3 \log (a x+1)\right )}{12 a^6 c \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{5}}{a^{5} c^{2} x^{5} - a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} + 2 \, a^{2} c^{2} x^{2} + a c^{2} x - c^{2}}, x\right ) \]
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 = 119, normalized size = 0.46 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (4 x^{4} a^{4}+2 x^{3} a^{3}+18 a^{2} x^{2}+27 \ln \left (a x -1\right ) x a -3 a x \ln \left (a x +1\right )-24 a x -27 \ln \left (a x -1\right )+3 \ln \left (a x +1\right )-6\right )}{12 \left (a^{2} x^{2}-1\right ) c^{2} a^{6} \left (a x -1\right )} \]
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 \[ -a \int -\frac {x^{6}}{{\left (a^{2} c^{\frac {3}{2}} x^{2} - c^{\frac {3}{2}}\right )} {\left (a x + 1\right )} {\left (a x - 1\right )}}\,{d x} - \frac {1}{2 \, {\left (a^{8} c^{\frac {3}{2}} x^{2} - a^{6} c^{\frac {3}{2}}\right )}} + \frac {\log \left (-a^{2} c x^{2} + c\right )}{a^{6} c^{\frac {3}{2}}} - \frac {\sqrt {a^{4} c x^{4} - 2 \, a^{2} c x^{2} + c}}{2 \, a^{6} c^{2}} \]
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 \[ \int \frac {x^5\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{5} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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