Optimal. Leaf size=230 \[ -\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2}+\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2} \]
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Rubi [A] time = 0.12, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {795, 671, 641, 195, 217, 203} \[ \frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 641
Rule 671
Rule 795
Rubi steps
\begin {align*} \int x (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {(3 d) \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx}{10 e}\\ &=-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (11 d^2\right ) \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx}{30 e}\\ &=-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^3\right ) \int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{80 e}\\ &=-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^4\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{80 e}\\ &=\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (11 d^6\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{32 e}\\ &=\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^8\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{128 e}\\ &=\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^{10}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{256 e}\\ &=\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {\left (33 d^{10}\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}\\ &=\frac {33 d^8 x \sqrt {d^2-e^2 x^2}}{256 e}+\frac {11 d^6 x \left (d^2-e^2 x^2\right )^{3/2}}{128 e}+\frac {11 d^4 x \left (d^2-e^2 x^2\right )^{5/2}}{160 e}-\frac {33 d^3 \left (d^2-e^2 x^2\right )^{7/2}}{560 e^2}-\frac {11 d^2 (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{240 e^2}-\frac {d (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{30 e^2}-\frac {(d+e x)^3 \left (d^2-e^2 x^2\right )^{7/2}}{10 e^2}+\frac {33 d^{10} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e^2}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 167, normalized size = 0.73 \[ \frac {\sqrt {d^2-e^2 x^2} \left (3465 d^9 \sin ^{-1}\left (\frac {e x}{d}\right )+\sqrt {1-\frac {e^2 x^2}{d^2}} \left (-6400 d^9-3465 d^8 e x+10240 d^7 e^2 x^2+24570 d^6 e^3 x^3+7680 d^5 e^4 x^4-23352 d^4 e^5 x^5-20480 d^3 e^6 x^6+3024 d^2 e^7 x^7+8960 d e^8 x^8+2688 e^9 x^9\right )\right )}{26880 e^2 \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 150, normalized size = 0.65 \[ -\frac {6930 \, d^{10} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (2688 \, e^{9} x^{9} + 8960 \, d e^{8} x^{8} + 3024 \, d^{2} e^{7} x^{7} - 20480 \, d^{3} e^{6} x^{6} - 23352 \, d^{4} e^{5} x^{5} + 7680 \, d^{5} e^{4} x^{4} + 24570 \, d^{6} e^{3} x^{3} + 10240 \, d^{7} e^{2} x^{2} - 3465 \, d^{8} e x - 6400 \, d^{9}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{26880 \, e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 128, normalized size = 0.56 \[ \frac {33}{256} \, d^{10} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-2\right )} \mathrm {sgn}\relax (d) - \frac {1}{26880} \, {\left (6400 \, d^{9} e^{\left (-2\right )} + {\left (3465 \, d^{8} e^{\left (-1\right )} - 2 \, {\left (5120 \, d^{7} + {\left (12285 \, d^{6} e + 4 \, {\left (960 \, d^{5} e^{2} - {\left (2919 \, d^{4} e^{3} + 2 \, {\left (1280 \, d^{3} e^{4} - 7 \, {\left (27 \, d^{2} e^{5} + 8 \, {\left (3 \, x e^{7} + 10 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 191, normalized size = 0.83 \[ \frac {33 d^{10} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 \sqrt {e^{2}}\, e}+\frac {33 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{8} x}{256 e}+\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{6} x}{128 e}+\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{4} x}{160 e}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e \,x^{3}}{10}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d \,x^{2}}{3}-\frac {33 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2} x}{80 e}-\frac {5 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3}}{21 e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 170, normalized size = 0.74 \[ \frac {33 \, d^{10} \arcsin \left (\frac {e x}{d}\right )}{256 \, e^{2}} + \frac {33 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{8} x}{256 \, e} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{6} x}{128 \, e} - \frac {1}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{3} + \frac {11 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{4} x}{160 \, e} - \frac {1}{3} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{2} - \frac {33 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x}{80 \, e} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3}}{21 \, e^{2}} \]
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 \[ \int x\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 40.18, size = 1554, normalized size = 6.76 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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