Optimal. Leaf size=118 \[ \frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{35 d^4 e^3 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {850, 819, 778, 192, 191} \[ \frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 x}{35 d^4 e^3 \sqrt {d^2-e^2 x^2}}-\frac {x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 778
Rule 819
Rule 850
Rubi steps
\begin {align*} \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\int \frac {x^3 (d-e x)}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx\\ &=\frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {\int \frac {x \left (2 d^3-3 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{7 d^2 e^2}\\ &=\frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {3 \int \frac {1}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{35 e^3}\\ &=\frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{35 d^2 e^3}\\ &=\frac {x^2 (d-e x)}{7 e^2 \left (d^2-e^2 x^2\right )^{7/2}}-\frac {2 d-3 e x}{35 e^4 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x}{35 d^2 e^3 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 x}{35 d^4 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 104, normalized size = 0.88 \[ -\frac {\sqrt {d^2-e^2 x^2} \left (2 d^6+2 d^5 e x-5 d^4 e^2 x^2-5 d^3 e^3 x^3-5 d^2 e^4 x^4+2 d e^5 x^5+2 e^6 x^6\right )}{35 d^4 e^4 (d-e x)^3 (d+e x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 239, normalized size = 2.03 \[ -\frac {2 \, e^{7} x^{7} + 2 \, d e^{6} x^{6} - 6 \, d^{2} e^{5} x^{5} - 6 \, d^{3} e^{4} x^{4} + 6 \, d^{4} e^{3} x^{3} + 6 \, d^{5} e^{2} x^{2} - 2 \, d^{6} e x - 2 \, d^{7} - {\left (2 \, e^{6} x^{6} + 2 \, d e^{5} x^{5} - 5 \, d^{2} e^{4} x^{4} - 5 \, d^{3} e^{3} x^{3} - 5 \, d^{4} e^{2} x^{2} + 2 \, d^{5} e x + 2 \, d^{6}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{35 \, {\left (d^{4} e^{11} x^{7} + d^{5} e^{10} x^{6} - 3 \, d^{6} e^{9} x^{5} - 3 \, d^{7} e^{8} x^{4} + 3 \, d^{8} e^{7} x^{3} + 3 \, d^{9} e^{6} x^{2} - d^{10} e^{5} x - d^{11} e^{4}\right )}} \]
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 = 0.01, size = 92, normalized size = 0.78 \[ -\frac {\left (-e x +d \right ) \left (2 e^{6} x^{6}+2 e^{5} x^{5} d -5 e^{4} x^{4} d^{2}-5 x^{3} d^{3} e^{3}-5 x^{2} d^{4} e^{2}+2 d^{5} x e +2 d^{6}\right )}{35 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{4} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 133, normalized size = 1.13 \[ \frac {d^{2}}{7 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d e^{4}\right )}} + \frac {8 \, x}{35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} - \frac {d}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} - \frac {x}{35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2} e^{3}} - \frac {2 \, x}{35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{4} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.95, size = 161, normalized size = 1.36 \[ \frac {\sqrt {d^2-e^2\,x^2}}{56\,d\,e^4\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {1}{56\,d\,e^4}+\frac {x}{35\,d^2\,e^3}\right )}{{\left (d+e\,x\right )}^2\,{\left (d-e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}\,\left (\frac {2\,d}{35\,e^4}-\frac {11\,x}{70\,e^3}\right )}{{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^3}-\frac {2\,x\,\sqrt {d^2-e^2\,x^2}}{35\,d^4\,e^3\,\left (d+e\,x\right )\,\left (d-e\,x\right )} \]
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 \[ \int \frac {x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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