Optimal. Leaf size=81 \[ \frac {x^4 (a x+1)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5 a^5 c^3 \sqrt {1-a^2 x^2}}-\frac {4}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6148, 805, 266, 43} \[ \frac {x^4 (a x+1)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}+\frac {4}{5 a^5 c^3 \sqrt {1-a^2 x^2}}-\frac {4}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 805
Rule 6148
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^4}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {\int \frac {x^4 (1+a x)}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^3}\\ &=\frac {x^4 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {4 \int \frac {x^3}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 a c^3}\\ &=\frac {x^4 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \operatorname {Subst}\left (\int \frac {x}{\left (1-a^2 x\right )^{5/2}} \, dx,x,x^2\right )}{5 a c^3}\\ &=\frac {x^4 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {2 \operatorname {Subst}\left (\int \left (\frac {1}{a^2 \left (1-a^2 x\right )^{5/2}}-\frac {1}{a^2 \left (1-a^2 x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{5 a c^3}\\ &=\frac {x^4 (1+a x)}{5 a c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {4}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {4}{5 a^5 c^3 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 68, normalized size = 0.84 \[ \frac {3 a^4 x^4+12 a^3 x^3-12 a^2 x^2-8 a x+8}{15 a^5 c^3 (a x-1)^2 (a x+1) \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 146, normalized size = 1.80 \[ \frac {8 \, a^{5} x^{5} - 8 \, a^{4} x^{4} - 16 \, a^{3} x^{3} + 16 \, a^{2} x^{2} + 8 \, a x - {\left (3 \, a^{4} x^{4} + 12 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 8 \, a x + 8\right )} \sqrt {-a^{2} x^{2} + 1} - 8}{15 \, {\left (a^{10} c^{3} x^{5} - a^{9} c^{3} x^{4} - 2 \, a^{8} c^{3} x^{3} + 2 \, a^{7} c^{3} x^{2} + a^{6} c^{3} x - a^{5} c^{3}\right )}} \]
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 \[ \int -\frac {{\left (a x + 1\right )} x^{4}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 58, normalized size = 0.72 \[ -\frac {3 x^{4} a^{4}+12 x^{3} a^{3}-12 a^{2} x^{2}-8 a x +8}{15 \left (a x -1\right ) c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a^{5}} \]
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 \[ -\int \frac {{\left (a x + 1\right )} x^{4}}{{\left (a^{2} c x^{2} - c\right )}^{3} \sqrt {-a^{2} x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.92, size = 335, normalized size = 4.14 \[ \frac {a^4\,\sqrt {1-a^2\,x^2}}{30\,\left (a^{11}\,c^3\,x^2-2\,a^{10}\,c^3\,x+a^9\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{24\,\left (a^7\,c^3\,x^2+2\,a^6\,c^3\,x+a^5\,c^3\right )}-\frac {\sqrt {1-a^2\,x^2}}{20\,\sqrt {-a^2}\,\left (a^3\,c^3\,\sqrt {-a^2}+3\,a^5\,c^3\,x^2\,\sqrt {-a^2}-a^6\,c^3\,x^3\,\sqrt {-a^2}-3\,a^4\,c^3\,x\,\sqrt {-a^2}\right )}-\frac {\sqrt {1-a^2\,x^2}}{4\,\left (a^7\,c^3\,x^2-2\,a^6\,c^3\,x+a^5\,c^3\right )}-\frac {13\,\sqrt {1-a^2\,x^2}}{48\,\left (a^3\,c^3\,\sqrt {-a^2}+a^4\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {113\,\sqrt {1-a^2\,x^2}}{240\,\left (a^3\,c^3\,\sqrt {-a^2}-a^4\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {x^{4}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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