Optimal. Leaf size=85 \[ -\frac {x^4 (d-e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{5 e^5 \sqrt {d^2-e^2 x^2}}+\frac {4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {850, 805, 266, 43} \[ -\frac {x^4 (d-e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}-\frac {4}{5 e^5 \sqrt {d^2-e^2 x^2}}+\frac {4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 805
Rule 850
Rubi steps
\begin {align*} \int \frac {x^4}{(d+e x) \left (d^2-e^2 x^2\right )^{5/2}} \, dx &=\int \frac {x^4 (d-e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=-\frac {x^4 (d-e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 \int \frac {x^3}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 e}\\ &=-\frac {x^4 (d-e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 \operatorname {Subst}\left (\int \frac {x}{\left (d^2-e^2 x\right )^{5/2}} \, dx,x,x^2\right )}{5 e}\\ &=-\frac {x^4 (d-e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {2 \operatorname {Subst}\left (\int \left (\frac {d^2}{e^2 \left (d^2-e^2 x\right )^{5/2}}-\frac {1}{e^2 \left (d^2-e^2 x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{5 e}\\ &=-\frac {x^4 (d-e x)}{5 d e \left (d^2-e^2 x^2\right )^{5/2}}+\frac {4 d^2}{15 e^5 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {4}{5 e^5 \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 82, normalized size = 0.96 \[ -\frac {\sqrt {d^2-e^2 x^2} \left (8 d^4+8 d^3 e x-12 d^2 e^2 x^2-12 d e^3 x^3+3 e^4 x^4\right )}{15 d e^5 (d-e x)^2 (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 168, normalized size = 1.98 \[ -\frac {8 \, e^{5} x^{5} + 8 \, d e^{4} x^{4} - 16 \, d^{2} e^{3} x^{3} - 16 \, d^{3} e^{2} x^{2} + 8 \, d^{4} e x + 8 \, d^{5} + {\left (3 \, e^{4} x^{4} - 12 \, d e^{3} x^{3} - 12 \, d^{2} e^{2} x^{2} + 8 \, d^{3} e x + 8 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (d e^{10} x^{5} + d^{2} e^{9} x^{4} - 2 \, d^{3} e^{8} x^{3} - 2 \, d^{4} e^{7} x^{2} + d^{5} e^{6} x + d^{6} e^{5}\right )}} \]
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 = 0.82 \[ -\frac {\left (-e x +d \right ) \left (3 x^{4} e^{4}-12 x^{3} d \,e^{3}-12 d^{2} x^{2} e^{2}+8 d^{3} x e +8 d^{4}\right )}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d \,e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 134, normalized size = 1.58 \[ -\frac {d^{3}}{5 \, {\left ({\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6} x + {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d e^{5}\right )}} + \frac {x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{3}} - \frac {2 \, d x}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} - \frac {d^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}} + \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.95, size = 78, normalized size = 0.92 \[ -\frac {\sqrt {d^2-e^2\,x^2}\,\left (8\,d^4+8\,d^3\,e\,x-12\,d^2\,e^2\,x^2-12\,d\,e^3\,x^3+3\,e^4\,x^4\right )}{15\,d\,e^5\,{\left (d+e\,x\right )}^3\,{\left (d-e\,x\right )}^2} \]
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 \[ \int \frac {x^{4}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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