Optimal. Leaf size=174 \[ \frac {127 d^2 (d+e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {3 d \sqrt {d^2-e^2 x^2}}{e^6}-\frac {13 d^2 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2 e^6}+\frac {x \sqrt {d^2-e^2 x^2}}{2 e^5}+\frac {d^4 (d+e x)^3}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {23 d^3 (d+e x)^2}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.40, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {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 1635
Rule 1815
Rubi steps
\begin {align*} \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 = 0.24, size = 131, normalized size = 0.75 \[ \frac {(d+e x) \left (\sqrt {1-\frac {e^2 x^2}{d^2}} \left (304 d^4-717 d^3 e x+479 d^2 e^2 x^2-45 d e^3 x^3-15 e^4 x^4\right )-195 d (d-e x)^3 \sin ^{-1}\left (\frac {e x}{d}\right )\right )}{30 e^6 (d-e x)^2 \sqrt {d^2-e^2 x^2} \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 192, normalized size = 1.10 \[ \frac {304 \, d^{2} e^{3} x^{3} - 912 \, d^{3} e^{2} x^{2} + 912 \, d^{4} e x - 304 \, d^{5} + 390 \, {\left (d^{2} e^{3} x^{3} - 3 \, d^{3} e^{2} x^{2} + 3 \, d^{4} e x - d^{5}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{4} x^{4} + 45 \, d e^{3} x^{3} - 479 \, d^{2} e^{2} x^{2} + 717 \, d^{3} e x - 304 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{30 \, {\left (e^{9} x^{3} - 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x - d^{3} e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 118, normalized size = 0.68 \[ -\frac {13}{2} \, d^{2} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\relax (d) - \frac {{\left (304 \, d^{7} e^{\left (-6\right )} + {\left (195 \, d^{6} e^{\left (-5\right )} - {\left (760 \, d^{5} e^{\left (-4\right )} + {\left (455 \, d^{4} e^{\left (-3\right )} - {\left (570 \, d^{3} e^{\left (-2\right )} + {\left (299 \, d^{2} e^{\left (-1\right )} - 15 \, {\left (x e + 6 \, d\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{30 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 222, normalized size = 1.28 \[ -\frac {e \,x^{7}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d \,x^{6}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {13 d^{2} x^{5}}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {19 d^{3} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}-\frac {76 d^{5} x^{2}}{3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}-\frac {13 d^{2} x^{3}}{6 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{3}}+\frac {152 d^{7}}{15 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{6}}+\frac {13 d^{2} x}{2 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{5}}-\frac {13 d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2 \sqrt {e^{2}}\, e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 305, normalized size = 1.75 \[ -\frac {e x^{7}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {13}{30} \, d^{2} e x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {3 \, d x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {13 \, d^{2} x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{6 \, e} + \frac {19 \, d^{3} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {76 \, d^{5} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {152 \, d^{7}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} + \frac {26 \, d^{4} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}} - \frac {91 \, d^{2} x}{30 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{5}} - \frac {13 \, d^{2} \arcsin \left (\frac {e x}{d}\right )}{2 \, e^{6}} \]
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 \[ \int \frac {x^5\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5} \left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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