Optimal. Leaf size=74 \[ \frac {a x^5}{5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {1}{3 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{5 a^4 c^3 \left (1-a^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 88, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 819, 778, 191} \[ \frac {x^2 (a x+1)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {x}{5 a^3 c^3 \sqrt {1-a^2 x^2}}-\frac {3 a x+2}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 778
Rule 819
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^3}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^3 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {x^2 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {x (2+3 a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a^2 c^3}\\ &=\frac {x^2 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2+3 a x}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{5 a^3 c^3}\\ &=\frac {x^2 (1+a x)}{5 a^2 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2+3 a x}{15 a^4 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x}{5 a^3 c^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 68, normalized size = 0.92 \[ \frac {3 a^4 x^4-3 a^3 x^3+3 a^2 x^2+2 a x-2}{15 a^4 c^3 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 145, normalized size = 1.96 \[ -\frac {2 \, a^{5} x^{5} - 2 \, a^{4} x^{4} - 4 \, a^{3} x^{3} + 4 \, a^{2} x^{2} + 2 \, a x + {\left (3 \, a^{4} x^{4} - 3 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 2 \, a x - 2\right )} \sqrt {-a^{2} x^{2} + 1} - 2}{15 \, {\left (a^{9} c^{3} x^{5} - a^{8} c^{3} x^{4} - 2 \, a^{7} c^{3} x^{3} + 2 \, a^{6} c^{3} x^{2} + a^{5} c^{3} x - a^{4} c^{3}\right )}} \]
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 = 0.03, size = 58, normalized size = 0.78 \[ -\frac {3 x^{4} a^{4}-3 x^{3} a^{3}+3 a^{2} x^{2}+2 a x -2}{15 \left (a x -1\right ) c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a^{4}} \]
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 \[ -a \int \frac {x^{4}}{{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}\,{d x} + \frac {5 \, a^{2} x^{2} - 2}{15 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {5}{2}} a^{4} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 287, normalized size = 3.88 \[ \frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^6\,c^3\,x^2+2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {2\,\sqrt {1-a^2\,x^2}}{15\,\left (a^6\,c^3\,x^2-2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^2\,c^3\,\sqrt {-a^2}+3\,a^4\,c^3\,x^2\,\sqrt {-a^2}-a^5\,c^3\,x^3\,\sqrt {-a^2}-3\,a^3\,c^3\,x\,\sqrt {-a^2}\right )}+\frac {7\,\sqrt {1-a^2\,x^2}}{48\,\left (a^2\,c^3\,\sqrt {-a^2}+a^3\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {13\,\sqrt {1-a^2\,x^2}}{240\,\left (a^2\,c^3\,\sqrt {-a^2}-a^3\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{3}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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