Optimal. Leaf size=212 \[ -\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {91 d^{11} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}+\frac {91}{256} d^9 x \sqrt {d^2-e^2 x^2}+\frac {91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2} \]
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Rubi [A] time = 0.09, antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {91}{256} d^9 x \sqrt {d^2-e^2 x^2}+\frac {91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {91 d^{11} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e} \]
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 )^{7/2} \, dx &=-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {1}{11} (13 d) \int (d+e x)^2 \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {1}{10} \left (13 d^2\right ) \int (d+e x) \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=-\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {1}{10} \left (13 d^3\right ) \int \left (d^2-e^2 x^2\right )^{7/2} \, dx\\ &=\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {1}{80} \left (91 d^5\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx\\ &=\frac {91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {1}{96} \left (91 d^7\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx\\ &=\frac {91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {1}{128} \left (91 d^9\right ) \int \sqrt {d^2-e^2 x^2} \, dx\\ &=\frac {91}{256} d^9 x \sqrt {d^2-e^2 x^2}+\frac {91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {1}{256} \left (91 d^{11}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx\\ &=\frac {91}{256} d^9 x \sqrt {d^2-e^2 x^2}+\frac {91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {1}{256} \left (91 d^{11}\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 {91}{256} d^9 x \sqrt {d^2-e^2 x^2}+\frac {91}{384} d^7 x \left (d^2-e^2 x^2\right )^{3/2}+\frac {91}{480} d^5 x \left (d^2-e^2 x^2\right )^{5/2}+\frac {13}{80} d^3 x \left (d^2-e^2 x^2\right )^{7/2}-\frac {13 d^2 \left (d^2-e^2 x^2\right )^{9/2}}{90 e}-\frac {13 d (d+e x) \left (d^2-e^2 x^2\right )^{9/2}}{110 e}-\frac {(d+e x)^2 \left (d^2-e^2 x^2\right )^{9/2}}{11 e}+\frac {91 d^{11} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{256 e}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 178, normalized size = 0.84 \[ \frac {\sqrt {d^2-e^2 x^2} \left (45045 d^{10} \sin ^{-1}\left (\frac {e x}{d}\right )+\sqrt {1-\frac {e^2 x^2}{d^2}} \left (-44800 d^{10}+81675 d^9 e x+167680 d^8 e^2 x^2+12210 d^7 e^3 x^3-222720 d^6 e^4 x^4-142296 d^5 e^5 x^5+110080 d^4 e^6 x^6+131472 d^3 e^7 x^7+1280 d^2 e^8 x^8-38016 d e^9 x^9-11520 e^{10} x^{10}\right )\right )}{126720 e \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 160, normalized size = 0.75 \[ -\frac {90090 \, d^{11} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (11520 \, e^{10} x^{10} + 38016 \, d e^{9} x^{9} - 1280 \, d^{2} e^{8} x^{8} - 131472 \, d^{3} e^{7} x^{7} - 110080 \, d^{4} e^{6} x^{6} + 142296 \, d^{5} e^{5} x^{5} + 222720 \, d^{6} e^{4} x^{4} - 12210 \, d^{7} e^{3} x^{3} - 167680 \, d^{8} e^{2} x^{2} - 81675 \, d^{9} e x + 44800 \, d^{10}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{126720 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 138, normalized size = 0.65 \[ \frac {91}{256} \, d^{11} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-1\right )} \mathrm {sgn}\relax (d) - \frac {1}{126720} \, {\left (44800 \, d^{10} e^{\left (-1\right )} - {\left (81675 \, d^{9} + 2 \, {\left (83840 \, d^{8} e + {\left (6105 \, d^{7} e^{2} - 4 \, {\left (27840 \, d^{6} e^{3} + {\left (17787 \, d^{5} e^{4} - 2 \, {\left (6880 \, d^{4} e^{5} + {\left (8217 \, d^{3} e^{6} + 8 \, {\left (10 \, d^{2} e^{7} - 9 \, {\left (10 \, x e^{9} + 33 \, d e^{8}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 174, normalized size = 0.82 \[ \frac {91 d^{11} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{256 \sqrt {e^{2}}}+\frac {91 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{9} x}{256}+\frac {91 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{7} x}{384}+\frac {91 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{5} x}{480}+\frac {13 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} x}{80}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}} e \,x^{2}}{11}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}} d x}{10}-\frac {35 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {9}{2}} d^{2}}{99 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 156, normalized size = 0.74 \[ \frac {91 \, d^{11} \arcsin \left (\frac {e x}{d}\right )}{256 \, e} + \frac {91}{256} \, \sqrt {-e^{2} x^{2} + d^{2}} d^{9} x + \frac {91}{384} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{7} x + \frac {91}{480} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{5} x + \frac {13}{80} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x - \frac {1}{11} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} e x^{2} - \frac {3}{10} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d x - \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {9}{2}} d^{2}}{99 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d^2-e^2\,x^2\right )}^{7/2}\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 41.08, size = 1496, normalized size = 7.06 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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