Optimal. Leaf size=252 \[ \frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}+\frac {65 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^6}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3} \]
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Rubi [A] time = 0.66, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \[ \frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}+\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {65 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^6} \]
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 \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx &=\int \frac {x^5 (d-e x)^4}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {(d-e x)^3 \left (-\frac {4 d^5}{e^5}+\frac {d^4 x}{e^4}-\frac {d^3 x^2}{e^3}+\frac {d^2 x^3}{e^2}-\frac {d x^4}{e}\right )}{\sqrt {d^2-e^2 x^2}} \, dx}{d}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {\frac {28 d^8}{e^3}-\frac {91 d^7 x}{e^2}+\frac {112 d^6 x^2}{e}-77 d^5 x^3+56 d^4 e x^4-55 d^3 e^2 x^5+28 d^2 e^3 x^6}{\sqrt {d^2-e^2 x^2}} \, dx}{7 d e^2}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-\frac {168 d^8}{e}+546 d^7 x-672 d^6 e x^2+462 d^5 e^2 x^3-476 d^4 e^3 x^4+330 d^3 e^4 x^5}{\sqrt {d^2-e^2 x^2}} \, dx}{42 d e^4}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {840 d^8 e-2730 d^7 e^2 x+3360 d^6 e^3 x^2-3630 d^5 e^4 x^3+2380 d^4 e^5 x^4}{\sqrt {d^2-e^2 x^2}} \, dx}{210 d e^6}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-3360 d^8 e^3+10920 d^7 e^4 x-20580 d^6 e^5 x^2+14520 d^5 e^6 x^3}{\sqrt {d^2-e^2 x^2}} \, dx}{840 d e^8}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\int \frac {10080 d^8 e^5-61800 d^7 e^6 x+61740 d^6 e^7 x^2}{\sqrt {d^2-e^2 x^2}} \, dx}{2520 d e^{10}}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}-\frac {\int \frac {-81900 d^8 e^7+123600 d^7 e^8 x}{\sqrt {d^2-e^2 x^2}} \, dx}{5040 d e^{12}}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\left (65 d^7\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{4 e^5}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {\left (65 d^7\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^5}\\ &=\frac {d^4 (d-e x)^4}{e^6 \sqrt {d^2-e^2 x^2}}+\frac {515 d^6 \sqrt {d^2-e^2 x^2}}{21 e^6}-\frac {49 d^5 x \sqrt {d^2-e^2 x^2}}{4 e^5}+\frac {121 d^4 x^2 \sqrt {d^2-e^2 x^2}}{21 e^4}-\frac {17 d^3 x^3 \sqrt {d^2-e^2 x^2}}{6 e^3}+\frac {11 d^2 x^4 \sqrt {d^2-e^2 x^2}}{7 e^2}-\frac {2 d x^5 \sqrt {d^2-e^2 x^2}}{3 e}+\frac {1}{7} x^6 \sqrt {d^2-e^2 x^2}+\frac {65 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{4 e^6}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 131, normalized size = 0.52 \[ \frac {1365 d^7 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )+\frac {\sqrt {d^2-e^2 x^2} \left (2144 d^7+779 d^6 e x-293 d^5 e^2 x^2+162 d^4 e^3 x^3-106 d^3 e^4 x^4+76 d^2 e^5 x^5-44 d e^6 x^6+12 e^7 x^7\right )}{d+e x}}{84 e^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 156, normalized size = 0.62 \[ \frac {2144 \, d^{7} e x + 2144 \, d^{8} - 2730 \, {\left (d^{7} e x + d^{8}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (12 \, e^{7} x^{7} - 44 \, d e^{6} x^{6} + 76 \, d^{2} e^{5} x^{5} - 106 \, d^{3} e^{4} x^{4} + 162 \, d^{4} e^{3} x^{3} - 293 \, d^{5} e^{2} x^{2} + 779 \, d^{6} e x + 2144 \, d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{84 \, {\left (e^{7} x + d e^{6}\right )}} \]
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 = 0.03, size = 416, normalized size = 1.65 \[ \frac {35 d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{5}}-\frac {5 d^{7} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{4 \sqrt {e^{2}}\, e^{5}}+\frac {35 \sqrt {2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}}\, d^{5} x}{2 e^{5}}-\frac {5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5} x}{4 e^{5}}+\frac {35 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {3}{2}} d^{3} x}{3 e^{5}}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{3} x}{6 e^{5}}-\frac {2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d x}{3 e^{5}}+\frac {28 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {5}{2}} d^{2}}{3 e^{6}}+\frac {\left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{4}}{\left (x +\frac {d}{e}\right )^{4} e^{10}}+\frac {8 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{3}}{\left (x +\frac {d}{e}\right )^{3} e^{9}}+\frac {22 \left (2 \left (x +\frac {d}{e}\right ) d e -\left (x +\frac {d}{e}\right )^{2} e^{2}\right )^{\frac {7}{2}} d^{2}}{3 \left (x +\frac {d}{e}\right )^{2} e^{8}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{7 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 1.06, size = 478, normalized size = 1.90 \[ -\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5}}{2 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6}}{2 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7}}{e^{7} x + d e^{6}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4}}{3 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} + \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5}}{6 \, {\left (e^{7} x + d e^{6}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3}}{2 \, {\left (e^{7} x + d e^{6}\right )}} + \frac {5 i \, d^{7} \arcsin \left (\frac {e x}{d} + 2\right )}{2 \, e^{6}} + \frac {75 \, d^{7} \arcsin \left (\frac {e x}{d}\right )}{4 \, e^{6}} - \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{5} x}{2 \, e^{5}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{5} x}{4 \, e^{5}} - \frac {5 \, \sqrt {e^{2} x^{2} + 4 \, d e x + 3 \, d^{2}} d^{6}}{e^{6}} + \frac {25 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{6}}{2 \, e^{6}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{3} x}{3 \, e^{5}} - \frac {25 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{4}}{6 \, e^{6}} - \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d x}{3 \, e^{5}} + \frac {2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{2}}{e^{6}} - \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{7 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^5\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]
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 \[ \int \frac {x^{5} \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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