Optimal. Leaf size=142 \[ -\frac {6 d^2 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {24 d (d+e x)}{5 e^5 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^5}-\frac {3 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}+\frac {d^3 (d+e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.32, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1635, 641, 217, 203} \[ \frac {d^3 (d+e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 d^2 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {24 d (d+e x)}{5 e^5 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^5}-\frac {3 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5} \]
Antiderivative was successfully verified.
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Rule 203
Rule 217
Rule 641
Rule 1635
Rubi steps
\begin {align*} \int \frac {x^4 (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {d^3 (d+e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x)^2 \left (\frac {3 d^4}{e^4}+\frac {5 d^3 x}{e^3}+\frac {5 d^2 x^2}{e^2}+\frac {5 d x^3}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d}\\ &=\frac {d^3 (d+e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 d^2 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {\int \frac {(d+e x) \left (\frac {27 d^4}{e^4}+\frac {30 d^3 x}{e^3}+\frac {15 d^2 x^2}{e^2}\right )}{\left (d^2-e^2 x^2\right )^{3/2}} \, dx}{15 d^2}\\ &=\frac {d^3 (d+e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 d^2 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {24 d (d+e x)}{5 e^5 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {\frac {45 d^4}{e^4}+\frac {15 d^3 x}{e^3}}{\sqrt {d^2-e^2 x^2}} \, dx}{15 d^3}\\ &=\frac {d^3 (d+e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 d^2 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {24 d (d+e x)}{5 e^5 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^5}-\frac {(3 d) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{e^4}\\ &=\frac {d^3 (d+e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 d^2 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {24 d (d+e x)}{5 e^5 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^5}-\frac {(3 d) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4}\\ &=\frac {d^3 (d+e x)^3}{5 e^5 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {6 d^2 (d+e x)^2}{5 e^5 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {24 d (d+e x)}{5 e^5 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^5}-\frac {3 d \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^5}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 119, normalized size = 0.84 \[ \frac {(d+e x) \left (\sqrt {1-\frac {e^2 x^2}{d^2}} \left (24 d^3-57 d^2 e x+39 d e^2 x^2-5 e^3 x^3\right )-15 (d-e x)^3 \sin ^{-1}\left (\frac {e x}{d}\right )\right )}{5 e^5 (d-e x)^2 \sqrt {d^2-e^2 x^2} \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 177, normalized size = 1.25 \[ \frac {24 \, d e^{3} x^{3} - 72 \, d^{2} e^{2} x^{2} + 72 \, d^{3} e x - 24 \, d^{4} + 30 \, {\left (d e^{3} x^{3} - 3 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x - d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (5 \, e^{3} x^{3} - 39 \, d e^{2} x^{2} + 57 \, d^{2} e x - 24 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (e^{8} x^{3} - 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x - d^{3} e^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 107, normalized size = 0.75 \[ -3 \, d \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\relax (d) - \frac {{\left (24 \, d^{6} e^{\left (-5\right )} + {\left (15 \, d^{5} e^{\left (-4\right )} - {\left (60 \, d^{4} e^{\left (-3\right )} + {\left (35 \, d^{3} e^{\left (-2\right )} - {\left (45 \, d^{2} e^{\left (-1\right )} - {\left (5 \, x e - 24 \, d\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}}}{5 \, {\left (x^{2} e^{2} - d^{2}\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 262, normalized size = 1.85 \[ -\frac {e \,x^{6}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {3 d \,x^{5}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {9 d^{2} x^{4}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e}+\frac {d^{3} x^{3}}{2 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{2}}-\frac {12 d^{4} x^{2}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{3}}-\frac {3 d^{5} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{4}}-\frac {d \,x^{3}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{2}}+\frac {24 d^{6}}{5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} e^{5}}+\frac {d^{3} x}{10 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} e^{4}}+\frac {16 d x}{5 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{4}}-\frac {3 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{\sqrt {e^{2}}\, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.02, size = 324, normalized size = 2.28 \[ \frac {1}{5} \, d e^{2} x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {e x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - d x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )} + \frac {9 \, d^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} + \frac {d^{3} x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {12 \, d^{4} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} - \frac {3 \, d^{5} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {24 \, d^{6}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}} + \frac {9 \, d^{3} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}} - \frac {6 \, d x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{4}} - \frac {3 \, d \arcsin \left (\frac {e x}{d}\right )}{e^{5}} \]
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 \[ \int \frac {x^4\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {x^{4} \left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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