Optimal. Leaf size=97 \[ \frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 c^4 (1-a x)^3}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5} \]
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Rubi [A] time = 0.21, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6128, 1639, 793, 659, 651} \[ \frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 c^4 (1-a x)^3}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rule 793
Rule 1639
Rule 6128
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)} x^2}{(c-a c x)^4} \, dx &=c \int \frac {x^2 \sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^4 (1-a x)^4}+\frac {\int \frac {\left (4 a^2 c^2-3 a^3 c^2 x\right ) \sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx}{a^4 c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^3 c^4 (1-a x)^4}+\frac {23 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx}{7 a^2}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 c^4 (1-a x)^4}+\frac {23 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx}{35 a^2 c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a^3 c^4 (1-a x)^5}-\frac {12 \left (1-a^2 x^2\right )^{3/2}}{35 a^3 c^4 (1-a x)^4}+\frac {23 \left (1-a^2 x^2\right )^{3/2}}{105 a^3 c^4 (1-a x)^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 43, normalized size = 0.44 \[ -\frac {(a x+1)^{3/2} \left (-23 a^2 x^2+10 a x-2\right )}{105 a^3 c^4 (1-a x)^{7/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.57, size = 118, normalized size = 1.22 \[ \frac {2 \, a^{4} x^{4} - 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} - 8 \, a x + {\left (23 \, a^{3} x^{3} + 13 \, a^{2} x^{2} - 8 \, a x + 2\right )} \sqrt {-a^{2} x^{2} + 1} + 2}{105 \, {\left (a^{7} c^{4} x^{4} - 4 \, a^{6} c^{4} x^{3} + 6 \, a^{5} c^{4} x^{2} - 4 \, a^{4} c^{4} x + a^{3} c^{4}\right )}} \]
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 = 49, normalized size = 0.51 \[ -\frac {\left (23 a^{2} x^{2}-10 a x +2\right ) \left (a x +1\right )^{2}}{105 \left (a x -1\right )^{3} c^{4} \sqrt {-a^{2} x^{2}+1}\, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 197, normalized size = 2.03 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{7} c^{4} x^{4} - 4 \, a^{6} c^{4} x^{3} + 6 \, a^{5} c^{4} x^{2} - 4 \, a^{4} c^{4} x + a^{3} c^{4}\right )}} + \frac {29 \, \sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{6} c^{4} x^{3} - 3 \, a^{5} c^{4} x^{2} + 3 \, a^{4} c^{4} x - a^{3} c^{4}\right )}} + \frac {82 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{5} c^{4} x^{2} - 2 \, a^{4} c^{4} x + a^{3} c^{4}\right )}} + \frac {23 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{4} c^{4} x - a^{3} c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 347, normalized size = 3.58 \[ \frac {2\,\sqrt {1-a^2\,x^2}}{7\,\left (a^7\,c^4\,x^4-4\,a^6\,c^4\,x^3+6\,a^5\,c^4\,x^2-4\,a^4\,c^4\,x+a^3\,c^4\right )}+\frac {4\,\sqrt {1-a^2\,x^2}}{3\,\left (a^5\,c^4\,x^2-2\,a^4\,c^4\,x+a^3\,c^4\right )}+\frac {4\,a\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}+\frac {23\,\sqrt {1-a^2\,x^2}}{105\,\left (a\,c^4\,\sqrt {-a^2}-a^2\,c^4\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}-\frac {2\,a^2\,\sqrt {1-a^2\,x^2}}{3\,\left (a^7\,c^4\,x^2-2\,a^6\,c^4\,x+a^5\,c^4\right )}+\frac {29\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (a\,c^4\,\sqrt {-a^2}+3\,a^3\,c^4\,x^2\,\sqrt {-a^2}-a^4\,c^4\,x^3\,\sqrt {-a^2}-3\,a^2\,c^4\,x\,\sqrt {-a^2}\right )} \]
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 \[ \frac {\int \frac {x^{2}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{3}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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