Optimal. Leaf size=232 \[ -\frac {3 \sqrt {1-a^2 x^2}}{4 a^5 c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a^5 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^5 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {11 \sqrt {1-a^2 x^2} \log (1-a x)}{16 a^5 c^2 \sqrt {c-a^2 c x^2}}-\frac {5 \sqrt {1-a^2 x^2} \log (a x+1)}{16 a^5 c^2 \sqrt {c-a^2 c x^2}} \]
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Rubi [A] time = 0.24, antiderivative size = 232, 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} \[ -\frac {3 \sqrt {1-a^2 x^2}}{4 a^5 c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a^5 c^2 (a x+1) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^5 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {11 \sqrt {1-a^2 x^2} \log (1-a x)}{16 a^5 c^2 \sqrt {c-a^2 c x^2}}-\frac {5 \sqrt {1-a^2 x^2} \log (a x+1)}{16 a^5 c^2 \sqrt {c-a^2 c x^2}} \]
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^4}{\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^4}{\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^4}{(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}{4 a^4 (-1+a x)^3}-\frac {3}{4 a^4 (-1+a x)^2}-\frac {11}{16 a^4 (-1+a x)}+\frac {1}{8 a^4 (1+a x)^2}-\frac {5}{16 a^4 (1+a x)}\right ) \, dx}{c^2 \sqrt {c-a^2 c x^2}}\\ &=\frac {\sqrt {1-a^2 x^2}}{8 a^5 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {3 \sqrt {1-a^2 x^2}}{4 a^5 c^2 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a^5 c^2 (1+a x) \sqrt {c-a^2 c x^2}}-\frac {11 \sqrt {1-a^2 x^2} \log (1-a x)}{16 a^5 c^2 \sqrt {c-a^2 c x^2}}-\frac {5 \sqrt {1-a^2 x^2} \log (1+a x)}{16 a^5 c^2 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 87, normalized size = 0.38 \[ \frac {\sqrt {1-a^2 x^2} \left (\frac {2 \left (5 a^2 x^2+3 a x-6\right )}{(a x-1)^2 (a x+1)}-11 \log (1-a x)-5 \log (a x+1)\right )}{16 a^5 c^2 \sqrt {c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{7} c^{3} x^{7} - a^{6} c^{3} x^{6} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} - a c^{3} x + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a x + 1\right )} x^{4}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 166, normalized size = 0.72 \[ \frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (11 \ln \left (a x -1\right ) x^{3} a^{3}+5 a^{3} x^{3} \ln \left (a x +1\right )-11 \ln \left (a x -1\right ) x^{2} a^{2}-5 \ln \left (a x +1\right ) x^{2} a^{2}-10 a^{2} x^{2}-11 \ln \left (a x -1\right ) x a -5 a x \ln \left (a x +1\right )-6 a x +11 \ln \left (a x -1\right )+5 \ln \left (a x +1\right )+12\right )}{16 \left (a^{2} x^{2}-1\right ) c^{3} a^{5} \left (a x -1\right )^{2} \left (a x +1\right )} \]
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 \[ \int \frac {{\left (a x + 1\right )} x^{4}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^4\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \]
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 \[ \int \frac {x^{4} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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