Optimal. Leaf size=310 \[ -\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}+\frac {35 d^{14} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2048 e^6}+\frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2} \]
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Rubi [A] time = 0.49, antiderivative size = 310, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1809, 833, 780, 195, 217, 203} \[ \frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}+\frac {35 d^{14} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2048 e^6} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 780
Rule 833
Rule 1809
Rubi steps
\begin {align*} \int x^5 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^5 \left (d^2-e^2 x^2\right )^{5/2} \left (-14 d^3 e^2-49 d^2 e^3 x-42 d e^4 x^2\right ) \, dx}{14 e^2}\\ &=-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^5 \left (434 d^3 e^4+637 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{182 e^4}\\ &=-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^4 \left (-3185 d^4 e^5-5208 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{2184 e^6}\\ &=-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^3 \left (20832 d^5 e^6+35035 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{24024 e^8}\\ &=-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^2 \left (-105105 d^6 e^7-208320 d^5 e^8 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{240240 e^{10}}\\ &=-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x \left (416640 d^7 e^8+945945 d^6 e^9 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{2162160 e^{12}}\\ &=-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (7 d^8\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{128 e^5}\\ &=\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (35 d^{10}\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{768 e^5}\\ &=\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (35 d^{12}\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{1024 e^5}\\ &=\frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (35 d^{14}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{2048 e^5}\\ &=\frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {\left (35 d^{14}\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{2048 e^5}\\ &=\frac {35 d^{12} x \sqrt {d^2-e^2 x^2}}{2048 e^5}+\frac {35 d^{10} x \left (d^2-e^2 x^2\right )^{3/2}}{3072 e^5}+\frac {7 d^8 x \left (d^2-e^2 x^2\right )^{5/2}}{768 e^5}-\frac {124 d^5 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{1287 e^4}-\frac {7 d^4 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{48 e^3}-\frac {31 d^3 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^2}-\frac {7 d^2 x^5 \left (d^2-e^2 x^2\right )^{7/2}}{24 e}-\frac {3}{13} d x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{14} e x^7 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^6 (31744 d+63063 e x) \left (d^2-e^2 x^2\right )^{7/2}}{1153152 e^6}+\frac {35 d^{14} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{2048 e^6}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 212, normalized size = 0.68 \[ \frac {\sqrt {d^2-e^2 x^2} \left (315315 d^{13} \sin ^{-1}\left (\frac {e x}{d}\right )-\sqrt {1-\frac {e^2 x^2}{d^2}} \left (507904 d^{13}+315315 d^{12} e x+253952 d^{11} e^2 x^2+210210 d^{10} e^3 x^3+190464 d^9 e^4 x^4+168168 d^8 e^5 x^5-2916352 d^7 e^6 x^6-7763184 d^6 e^7 x^7-2551808 d^5 e^8 x^8+9499776 d^4 e^9 x^9+8773632 d^3 e^{10} x^{10}-1427712 d^2 e^{11} x^{11}-4257792 d e^{12} x^{12}-1317888 e^{13} x^{13}\right )\right )}{18450432 e^6 \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 194, normalized size = 0.63 \[ -\frac {630630 \, d^{14} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (1317888 \, e^{13} x^{13} + 4257792 \, d e^{12} x^{12} + 1427712 \, d^{2} e^{11} x^{11} - 8773632 \, d^{3} e^{10} x^{10} - 9499776 \, d^{4} e^{9} x^{9} + 2551808 \, d^{5} e^{8} x^{8} + 7763184 \, d^{6} e^{7} x^{7} + 2916352 \, d^{7} e^{6} x^{6} - 168168 \, d^{8} e^{5} x^{5} - 190464 \, d^{9} e^{4} x^{4} - 210210 \, d^{10} e^{3} x^{3} - 253952 \, d^{11} e^{2} x^{2} - 315315 \, d^{12} e x - 507904 \, d^{13}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{18450432 \, e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 170, normalized size = 0.55 \[ \frac {35}{2048} \, d^{14} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-6\right )} \mathrm {sgn}\relax (d) - \frac {1}{18450432} \, {\left (507904 \, d^{13} e^{\left (-6\right )} + {\left (315315 \, d^{12} e^{\left (-5\right )} + 2 \, {\left (126976 \, d^{11} e^{\left (-4\right )} + {\left (105105 \, d^{10} e^{\left (-3\right )} + 4 \, {\left (23808 \, d^{9} e^{\left (-2\right )} + {\left (21021 \, d^{8} e^{\left (-1\right )} - 2 \, {\left (182272 \, d^{7} + {\left (485199 \, d^{6} e + 8 \, {\left (19936 \, d^{5} e^{2} - 3 \, {\left (24739 \, d^{4} e^{3} + 2 \, {\left (11424 \, d^{3} e^{4} - 11 \, {\left (169 \, d^{2} e^{5} + 12 \, {\left (13 \, x e^{7} + 42 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 291, normalized size = 0.94 \[ \frac {35 d^{14} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{2048 \sqrt {e^{2}}\, e^{5}}+\frac {35 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{12} x}{2048 e^{5}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e \,x^{7}}{14}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d \,x^{6}}{13}+\frac {35 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{10} x}{3072 e^{5}}-\frac {7 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2} x^{5}}{24 e}-\frac {31 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} x^{4}}{143 e^{2}}+\frac {7 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{8} x}{768 e^{5}}-\frac {7 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{4} x^{3}}{48 e^{3}}-\frac {124 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{5} x^{2}}{1287 e^{4}}-\frac {7 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{6} x}{128 e^{5}}-\frac {248 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{7}}{9009 e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.00, size = 270, normalized size = 0.87 \[ -\frac {1}{14} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{7} - \frac {3}{13} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{6} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{5}}{24 \, e} + \frac {35 \, d^{14} \arcsin \left (\frac {e x}{d}\right )}{2048 \, e^{6}} + \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{12} x}{2048 \, e^{5}} - \frac {31 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{4}}{143 \, e^{2}} + \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{10} x}{3072 \, e^{5}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x^{3}}{48 \, e^{3}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{8} x}{768 \, e^{5}} - \frac {124 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5} x^{2}}{1287 \, e^{4}} - \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{6} x}{128 \, e^{5}} - \frac {248 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{7}}{9009 \, e^{6}} \]
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 \[ \int x^5\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 101.43, size = 2273, normalized size = 7.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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