Optimal. Leaf size=177 \[ \frac {127 d^2 (d-e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {3 d \sqrt {d^2-e^2 x^2}}{e^6}+\frac {13 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.44, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {852, 1635, 1815, 641, 217, 203} \[ \frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {127 d^2 (d-e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {3 d \sqrt {d^2-e^2 x^2}}{e^6}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {13 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 641
Rule 852
Rule 1635
Rule 1815
Rubi steps
\begin {align*} \int \frac {x^5}{(d+e x)^3 \sqrt {d^2-e^2 x^2}} \, dx &=\int \frac {x^5 (d-e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d-e x)^2 \left (-\frac {3 d^5}{e^5}+\frac {5 d^4 x}{e^4}-\frac {5 d^3 x^2}{e^3}+\frac {5 d^2 x^3}{e^2}-\frac {5 d x^4}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d-e x) \left (-\frac {37 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {30 d^3 x^2}{e^3}+\frac {15 d^2 x^3}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {127 d^2 (d-e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {-\frac {90 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {15 d^3 x^2}{e^3}}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {127 d^2 (d-e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {\int \frac {\frac {195 d^5}{e^3}-\frac {90 d^4 x}{e^2}}{\sqrt {d^2-e^2 x^2}} \, dx}{30 d^3 e^2}\\ &=\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {127 d^2 (d-e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {3 d \sqrt {d^2-e^2 x^2}}{e^6}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {\left (13 d^2\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2 e^5}\\ &=\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {127 d^2 (d-e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {3 d \sqrt {d^2-e^2 x^2}}{e^6}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {\left (13 d^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^5}\\ &=\frac {d^4 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d-e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {127 d^2 (d-e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {3 d \sqrt {d^2-e^2 x^2}}{e^6}-\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {13 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 98, normalized size = 0.55 \[ \frac {195 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (304 d^4+717 d^3 e x+479 d^2 e^2 x^2+45 d e^3 x^3-15 e^4 x^4\right )}{(d+e x)^3}}{30 e^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 190, normalized size = 1.07 \[ \frac {304 \, d^{2} e^{3} x^{3} + 912 \, d^{3} e^{2} x^{2} + 912 \, d^{4} e x + 304 \, d^{5} - 390 \, {\left (d^{2} e^{3} x^{3} + 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x + d^{5}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (15 \, e^{4} x^{4} - 45 \, d e^{3} x^{3} - 479 \, d^{2} e^{2} x^{2} - 717 \, d^{3} e x - 304 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 212, normalized size = 1.20 \[ \frac {13 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{5}}-\frac {\sqrt {-e^{2} x^{2}+d^{2}}\, x}{2 e^{5}}+\frac {\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{4}}{5 \left (x +\frac {d}{e}\right )^{3} e^{9}}-\frac {23 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{3}}{15 \left (x +\frac {d}{e}\right )^{2} e^{8}}+\frac {127 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{2}}{15 \left (x +\frac {d}{e}\right ) e^{7}}+\frac {3 \sqrt {-e^{2} x^{2}+d^{2}}\, d}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 185, normalized size = 1.05 \[ \frac {\sqrt {-e^{2} x^{2} + d^{2}} d^{4}}{5 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} - \frac {23 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{15 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {127 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{15 \, {\left (e^{7} x + d e^{6}\right )}} + \frac {13 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{6}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}} x}{2 \, e^{5}} + \frac {3 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^5}{\sqrt {d^2-e^2\,x^2}\,{\left (d+e\,x\right )}^3} \,d x \]
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 \[ \int \frac {x^{5}}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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