Optimal. Leaf size=188 \[ -\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {55 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e}+\frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2} \]
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Rubi [A] time = 0.08, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {671, 641, 195, 217, 203} \[ \frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {55 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 641
Rule 671
Rubi steps
\begin {align*} \int (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{9} (11 d) \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{8} \left (11 d^2\right ) \int (d+e x) \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{8} \left (11 d^3\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{48} \left (55 d^5\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{64} \left (55 d^7\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{128} \left (55 d^9\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {1}{128} \left (55 d^9\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )\\ &=\frac {55}{128} d^7 x \sqrt {d^2-e^2 x^2}+\frac {55}{192} d^5 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {11}{48} d^3 x \left (d^2-e^2 x^2\right )^{5/2}-\frac {11 d^2 \left (d^2-e^2 x^2\right )^{7/2}}{56 e}-\frac {11 d (d+e x) \left (d^2-e^2 x^2\right )^{7/2}}{72 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2}}{9 e}+\frac {55 d^9 \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{128 e}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 156, normalized size = 0.83 \[ \frac {\sqrt {d^2-e^2 x^2} \left (3465 d^8 \sin ^{-1}\left (\frac {e x}{d}\right )+\sqrt {1-\frac {e^2 x^2}{d^2}} \left (-3712 d^8+4599 d^7 e x+10240 d^6 e^2 x^2+3066 d^5 e^3 x^3-8448 d^4 e^4 x^4-7224 d^3 e^5 x^5+1024 d^2 e^6 x^6+3024 d e^7 x^7+896 e^8 x^8\right )\right )}{8064 e \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 139, normalized size = 0.74 \[ -\frac {6930 \, d^{9} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (896 \, e^{8} x^{8} + 3024 \, d e^{7} x^{7} + 1024 \, d^{2} e^{6} x^{6} - 7224 \, d^{3} e^{5} x^{5} - 8448 \, d^{4} e^{4} x^{4} + 3066 \, d^{5} e^{3} x^{3} + 10240 \, d^{6} e^{2} x^{2} + 4599 \, d^{7} e x - 3712 \, d^{8}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{8064 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 117, normalized size = 0.62 \[ \frac {55}{128} \, d^{9} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\relax (d) - \frac {1}{8064} \, {\left (3712 \, d^{8} e^{\left (-1\right )} - {\left (4599 \, d^{7} + 2 \, {\left (5120 \, d^{6} e + {\left (1533 \, d^{5} e^{2} - 4 \, {\left (1056 \, d^{4} e^{3} + {\left (903 \, d^{3} e^{4} - 2 \, {\left (64 \, d^{2} e^{5} + 7 \, {\left (8 \, x e^{7} + 27 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 154, normalized size = 0.82 \[ \frac {55 d^{9} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{128 \sqrt {e^{2}}}+\frac {55 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{7} x}{128}+\frac {55 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{5} x}{192}+\frac {11 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{3} x}{48}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e \,x^{2}}{9}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d x}{8}-\frac {29 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2}}{63 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 136, normalized size = 0.72 \[ \frac {55 \, d^{9} \arcsin \left (\frac {e x}{d}\right )}{128 \, e} + \frac {55}{128} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{7} x + \frac {55}{192} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{5} x + \frac {11}{48} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{3} x - \frac {1}{9} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{2} - \frac {3}{8} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x - \frac {29 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2}}{63 \, e} \]
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 \[ \int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 25.88, size = 1284, normalized size = 6.83 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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