Optimal. Leaf size=122 \[ \frac {x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}+\frac {8 d+15 e x}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6}-\frac {x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {819, 778, 217, 203} \[ \frac {x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d+15 e x}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 778
Rule 819
Rubi steps
\begin {align*} \int \frac {x^5 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {x^3 \left (4 d^3+5 d^2 e x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2 e^2}\\ &=\frac {x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {x \left (8 d^5+15 d^4 e x\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^4 e^4}\\ &=\frac {x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d+15 e x}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^5}\\ &=\frac {x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d+15 e x}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\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 {x^4 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^2 (4 d+5 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {8 d+15 e x}{15 e^6 \sqrt {d^2-e^2 x^2}}-\frac {\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 = 0.08, size = 130, normalized size = 1.07 \[ \frac {8 d^4+7 d^3 e x-27 d^2 e^2 x^2-15 (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )-8 d e^3 x^3+23 e^4 x^4}{15 e^6 (d-e x)^2 (d+e x) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 247, normalized size = 2.02 \[ \frac {8 \, e^{5} x^{5} - 8 \, d e^{4} x^{4} - 16 \, d^{2} e^{3} x^{3} + 16 \, d^{3} e^{2} x^{2} + 8 \, d^{4} e x - 8 \, d^{5} + 30 \, {\left (e^{5} x^{5} - d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} + 2 \, d^{3} e^{2} x^{2} + d^{4} e x - d^{5}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (23 \, e^{4} x^{4} - 8 \, d e^{3} x^{3} - 27 \, d^{2} e^{2} x^{2} + 7 \, d^{3} e x + 8 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{11} x^{5} - d e^{10} x^{4} - 2 \, d^{2} e^{9} x^{3} + 2 \, d^{3} e^{8} x^{2} + d^{4} e^{7} x - d^{5} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 97, normalized size = 0.80 \[ -\arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\relax (d) - \frac {{\left (8 \, d^{5} e^{\left (-6\right )} + {\left (15 \, d^{4} e^{\left (-5\right )} - {\left (20 \, d^{3} e^{\left (-4\right )} + {\left (35 \, d^{2} e^{\left (-3\right )} - {\left (23 \, x e^{\left (-1\right )} + 15 \, d e^{\left (-2\right )}\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{15 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 166, normalized size = 1.36 \[ \frac {x^{5}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {d \,x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}-\frac {4 d^{3} x^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}-\frac {x^{3}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}+\frac {8 d^{5}}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}+\frac {x}{\sqrt {-e^{2} x^{2}+d^{2}}\, e^{5}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.01, size = 250, normalized size = 2.05 \[ \frac {1}{15} \, e x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{3 \, e} + \frac {d x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {4 \, d^{3} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{5}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} + \frac {4 \, d^{2} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}} - \frac {7 \, x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{5}} - \frac {\arcsin \left (\frac {e x}{d}\right )}{e^{6}} \]
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 \[ \int \frac {x^5\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 73.14, size = 1739, normalized size = 14.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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