Optimal. Leaf size=204 \[ \frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \]
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Rubi [A] time = 0.59, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \]
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 \sqrt {d^2-e^2 x^2}}{(d+e x)^4} \, dx &=\int \frac {x^5 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d-e x)^3 \left (-\frac {4 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)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d-e x)^2 \left (-\frac {60 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)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x) \left (-\frac {240 d^5}{e^5}+\frac {45 d^4 x}{e^4}-\frac {15 d^3 x^2}{e^3}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\int \frac {\frac {720 d^6}{e^3}-\frac {885 d^5 x}{e^2}+\frac {180 d^4 x^2}{e}}{\sqrt {d^2-e^2 x^2}} \, dx}{45 d^3 e^2}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}-\frac {\int \frac {-\frac {1620 d^6}{e}+1770 d^5 x}{\sqrt {d^2-e^2 x^2}} \, dx}{90 d^3 e^4}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\left (18 d^3\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^5}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {\left (18 d^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}\\ &=\frac {d^4 (d-e x)^4}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {8 d^3 (d-e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {10 d^2 (d-e x)^2}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {59 d^2 \sqrt {d^2-e^2 x^2}}{3 e^6}-\frac {2 d x \sqrt {d^2-e^2 x^2}}{e^5}+\frac {x^2 \sqrt {d^2-e^2 x^2}}{3 e^4}+\frac {18 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 109, normalized size = 0.53 \[ \frac {270 d^3 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (424 d^5+1002 d^4 e x+674 d^3 e^2 x^2+70 d^2 e^3 x^3-15 d e^4 x^4+5 e^5 x^5\right )}{(d+e x)^3}}{15 e^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 200, normalized size = 0.98 \[ \frac {424 \, d^{3} e^{3} x^{3} + 1272 \, d^{4} e^{2} x^{2} + 1272 \, d^{5} e x + 424 \, d^{6} - 540 \, {\left (d^{3} e^{3} x^{3} + 3 \, d^{4} e^{2} x^{2} + 3 \, d^{5} e x + d^{6}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (5 \, e^{5} x^{5} - 15 \, d e^{4} x^{4} + 70 \, d^{2} e^{3} x^{3} + 674 \, d^{3} e^{2} x^{2} + 1002 \, d^{4} e x + 424 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]
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 = 297, normalized size = 1.46 \[ \frac {20 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{\sqrt {e^{2}}\, e^{5}}-\frac {2 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{5}}-\frac {2 \sqrt {-e^{2} x^{2}+d^{2}}\, d x}{e^{5}}+\frac {20 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{2}}{e^{6}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{4}}{5 \left (x +\frac {d}{e}\right )^{4} e^{10}}-\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{3}}{5 \left (x +\frac {d}{e}\right )^{3} e^{9}}+\frac {10 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{2}}{\left (x +\frac {d}{e}\right )^{2} e^{8}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}{3 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,\sqrt {d^2-e^2\,x^2}}{{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {x^{5} \sqrt {- \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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