Optimal. Leaf size=135 \[ -\frac {19 \sin ^{-1}(a x)}{2 a^5 c^3}+\frac {(a x+1)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {19 (a x+1)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 (a x+1)^2}{a^5 c^3 \sqrt {1-a^2 x^2}}+\frac {(a x+20) \sqrt {1-a^2 x^2}}{2 a^5 c^3} \]
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Rubi [A] time = 0.42, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6128, 852, 1635, 780, 216} \[ \frac {(a x+1)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {19 (a x+1)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 (a x+1)^2}{a^5 c^3 \sqrt {1-a^2 x^2}}+\frac {(a x+20) \sqrt {1-a^2 x^2}}{2 a^5 c^3}-\frac {19 \sin ^{-1}(a x)}{2 a^5 c^3} \]
Antiderivative was successfully verified.
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Rule 216
Rule 780
Rule 852
Rule 1635
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^4}{(c-a c x)^3} \, dx &=c \int \frac {x^4 \sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx\\ &=\frac {\int \frac {x^4 (c+a c x)^4}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^7}\\ &=\frac {(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {\int \frac {(c+a c x)^3 \left (\frac {4}{a^4}+\frac {5 x}{a^3}+\frac {5 x^2}{a^2}+\frac {5 x^3}{a}\right )}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{5 c^6}\\ &=\frac {(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {19 (1+a x)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {\int \frac {(c+a c x)^2 \left (\frac {45}{a^4}+\frac {30 x}{a^3}+\frac {15 x^2}{a^2}\right )}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^5}\\ &=\frac {(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {19 (1+a x)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 (1+a x)^2}{a^5 c^3 \sqrt {1-a^2 x^2}}-\frac {\int \frac {\left (\frac {135}{a^4}+\frac {15 x}{a^3}\right ) (c+a c x)}{\sqrt {1-a^2 x^2}} \, dx}{15 c^4}\\ &=\frac {(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {19 (1+a x)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 (1+a x)^2}{a^5 c^3 \sqrt {1-a^2 x^2}}+\frac {(20+a x) \sqrt {1-a^2 x^2}}{2 a^5 c^3}-\frac {19 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{2 a^4 c^3}\\ &=\frac {(1+a x)^4}{5 a^5 c^3 \left (1-a^2 x^2\right )^{5/2}}-\frac {19 (1+a x)^3}{15 a^5 c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 (1+a x)^2}{a^5 c^3 \sqrt {1-a^2 x^2}}+\frac {(20+a x) \sqrt {1-a^2 x^2}}{2 a^5 c^3}-\frac {19 \sin ^{-1}(a x)}{2 a^5 c^3}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 122, normalized size = 0.90 \[ \frac {\sqrt {a x+1} \left (-15 a^4 x^4-75 a^3 x^3+433 a^2 x^2-639 a x+308\right )+140 \sqrt {2} (a x-1) \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};\frac {1}{2} (1-a x)\right )+360 (1-a x)^{5/2} \sin ^{-1}\left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{30 a^5 c^3 (1-a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.47, size = 153, normalized size = 1.13 \[ \frac {448 \, a^{3} x^{3} - 1344 \, a^{2} x^{2} + 1344 \, a x + 570 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{4} x^{4} + 75 \, a^{3} x^{3} - 713 \, a^{2} x^{2} + 1059 \, a x - 448\right )} \sqrt {-a^{2} x^{2} + 1} - 448}{30 \, {\left (a^{8} c^{3} x^{3} - 3 \, a^{7} c^{3} x^{2} + 3 \, a^{6} c^{3} x - a^{5} c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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 = 208, normalized size = 1.54 \[ \frac {x \sqrt {-a^{2} x^{2}+1}}{2 c^{3} a^{4}}-\frac {19 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{2 c^{3} a^{4} \sqrt {a^{2}}}+\frac {4 \sqrt {-a^{2} x^{2}+1}}{c^{3} a^{5}}-\frac {41 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 c^{3} a^{7} \left (x -\frac {1}{a}\right )^{2}}-\frac {199 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 c^{3} a^{6} \left (x -\frac {1}{a}\right )}-\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 c^{3} a^{8} \left (x -\frac {1}{a}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 185, normalized size = 1.37 \[ -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{8} c^{3} x^{3} - 3 \, a^{7} c^{3} x^{2} + 3 \, a^{6} c^{3} x - a^{5} c^{3}\right )}} - \frac {41 \, \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{7} c^{3} x^{2} - 2 \, a^{6} c^{3} x + a^{5} c^{3}\right )}} - \frac {199 \, \sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{6} c^{3} x - a^{5} c^{3}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1} x}{2 \, a^{4} c^{3}} - \frac {19 \, \arcsin \left (a x\right )}{2 \, a^{5} c^{3}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1}}{a^{5} c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.83, size = 302, normalized size = 2.24 \[ \frac {4\,a^4\,\sqrt {1-a^2\,x^2}}{15\,\left (a^{11}\,c^3\,x^2-2\,a^{10}\,c^3\,x+a^9\,c^3\right )}-\frac {2\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (a^3\,c^3\,\sqrt {-a^2}+3\,a^5\,c^3\,x^2\,\sqrt {-a^2}-a^6\,c^3\,x^3\,\sqrt {-a^2}-3\,a^4\,c^3\,x\,\sqrt {-a^2}\right )}-\frac {3\,\sqrt {1-a^2\,x^2}}{a^7\,c^3\,x^2-2\,a^6\,c^3\,x+a^5\,c^3}-\frac {199\,\sqrt {1-a^2\,x^2}}{15\,\left (a^3\,c^3\,\sqrt {-a^2}-a^4\,c^3\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {4\,\sqrt {1-a^2\,x^2}}{a^5\,c^3}+\frac {x\,\sqrt {1-a^2\,x^2}}{2\,a^4\,c^3}-\frac {19\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{2\,a^4\,c^3\,\sqrt {-a^2}} \]
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 \[ - \frac {\int \frac {x^{4}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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