Optimal. Leaf size=91 \[ \frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e^3 \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {850, 819, 778, 191} \[ \frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x}{5 d^2 e^3 \sqrt {d^2-e^2 x^2}}-\frac {2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
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 )^{5/2}} \, dx &=\int \frac {x^3 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x \left (2 d^3-3 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{5 e^3}\\ &=\frac {x^2 (d-e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 d-3 e x}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e^3 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 82, normalized size = 0.90 \[ \frac {\sqrt {d^2-e^2 x^2} \left (-2 d^4-2 d^3 e x+3 d^2 e^2 x^2+3 d e^3 x^3+3 e^4 x^4\right )}{15 d^2 e^4 (d-e x)^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 171, normalized size = 1.88 \[ -\frac {2 \, e^{5} x^{5} + 2 \, d e^{4} x^{4} - 4 \, d^{2} e^{3} x^{3} - 4 \, d^{3} e^{2} x^{2} + 2 \, d^{4} e x + 2 \, d^{5} - {\left (3 \, e^{4} x^{4} + 3 \, d e^{3} x^{3} + 3 \, d^{2} e^{2} x^{2} - 2 \, d^{3} e x - 2 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d^{2} e^{9} x^{5} + d^{3} e^{8} x^{4} - 2 \, d^{4} e^{7} x^{3} - 2 \, d^{5} e^{6} x^{2} + d^{6} e^{5} x + d^{7} e^{4}\right )}} \]
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 = 70, normalized size = 0.77 \[ -\frac {\left (-e x +d \right ) \left (-3 x^{4} e^{4}-3 x^{3} d \,e^{3}-3 d^{2} x^{2} e^{2}+2 d^{3} x e +2 d^{4}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{2} e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 110, normalized size = 1.21 \[ \frac {d^{2}}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{4}\right )}} + \frac {2 \, x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} - \frac {d}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} - \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.84, size = 78, normalized size = 0.86 \[ \frac {\sqrt {d^2-e^2\,x^2}\,\left (-2\,d^4-2\,d^3\,e\,x+3\,d^2\,e^2\,x^2+3\,d\,e^3\,x^3+3\,e^4\,x^4\right )}{15\,d^2\,e^4\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \]
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 \[ \int \frac {x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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