Optimal. Leaf size=281 \[ -\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}+\frac {27 d^{13} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5}+\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.41, antiderivative size = 281, normalized size of antiderivative = 1.00, number of steps used = 11, 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} \begin {gather*} \frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}+\frac {27 d^{13} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 203
Rule 217
Rule 780
Rule 833
Rule 1809
Rubi steps
\begin {align*} \int x^4 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^4 \left (d^2-e^2 x^2\right )^{5/2} \left (-13 d^3 e^2-45 d^2 e^3 x-39 d e^4 x^2\right ) \, dx}{13 e^2}\\ &=-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^4 \left (351 d^3 e^4+540 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{156 e^4}\\ &=-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^3 \left (-2160 d^4 e^5-3861 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{1716 e^6}\\ &=-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^2 \left (11583 d^5 e^6+21600 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{17160 e^8}\\ &=-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x \left (-43200 d^6 e^7-104247 d^5 e^8 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{154440 e^{10}}\\ &=-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^7\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{320 e^4}\\ &=\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (9 d^9\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{128 e^4}\\ &=\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{11}\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{512 e^4}\\ &=\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{13}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{1024 e^4}\\ &=\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {\left (27 d^{13}\right ) \operatorname {Subst}\left (\int \frac {1}{1+e^2 x^2} \, dx,x,\frac {x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^4}\\ &=\frac {27 d^{11} x \sqrt {d^2-e^2 x^2}}{1024 e^4}+\frac {9 d^9 x \left (d^2-e^2 x^2\right )^{3/2}}{512 e^4}+\frac {9 d^7 x \left (d^2-e^2 x^2\right )^{5/2}}{640 e^4}-\frac {20 d^4 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{143 e^3}-\frac {9 d^3 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{40 e^2}-\frac {45 d^2 x^4 \left (d^2-e^2 x^2\right )^{7/2}}{143 e}-\frac {1}{4} d x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{13} e x^6 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^5 (12800 d+27027 e x) \left (d^2-e^2 x^2\right )^{7/2}}{320320 e^5}+\frac {27 d^{13} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.33, size = 200, normalized size = 0.71 \begin {gather*} \frac {\sqrt {d^2-e^2 x^2} \left (135135 d^{12} \sin ^{-1}\left (\frac {e x}{d}\right )+\sqrt {1-\frac {e^2 x^2}{d^2}} \left (-204800 d^{12}-135135 d^{11} e x-102400 d^{10} e^2 x^2-90090 d^9 e^3 x^3-76800 d^8 e^4 x^4+952952 d^7 e^5 x^5+2498560 d^6 e^6 x^6+816816 d^5 e^7 x^7-2938880 d^4 e^8 x^8-2690688 d^3 e^9 x^9+430080 d^2 e^{10} x^{10}+1281280 d e^{11} x^{11}+394240 e^{12} x^{12}\right )\right )}{5125120 e^5 \sqrt {1-\frac {e^2 x^2}{d^2}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.63, size = 202, normalized size = 0.72 \begin {gather*} \frac {27 d^{13} \sqrt {-e^2} \log \left (\sqrt {d^2-e^2 x^2}-\sqrt {-e^2} x\right )}{1024 e^6}+\frac {\sqrt {d^2-e^2 x^2} \left (-204800 d^{12}-135135 d^{11} e x-102400 d^{10} e^2 x^2-90090 d^9 e^3 x^3-76800 d^8 e^4 x^4+952952 d^7 e^5 x^5+2498560 d^6 e^6 x^6+816816 d^5 e^7 x^7-2938880 d^4 e^8 x^8-2690688 d^3 e^9 x^9+430080 d^2 e^{10} x^{10}+1281280 d e^{11} x^{11}+394240 e^{12} x^{12}\right )}{5125120 e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 183, normalized size = 0.65 \begin {gather*} -\frac {270270 \, d^{13} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (394240 \, e^{12} x^{12} + 1281280 \, d e^{11} x^{11} + 430080 \, d^{2} e^{10} x^{10} - 2690688 \, d^{3} e^{9} x^{9} - 2938880 \, d^{4} e^{8} x^{8} + 816816 \, d^{5} e^{7} x^{7} + 2498560 \, d^{6} e^{6} x^{6} + 952952 \, d^{7} e^{5} x^{5} - 76800 \, d^{8} e^{4} x^{4} - 90090 \, d^{9} e^{3} x^{3} - 102400 \, d^{10} e^{2} x^{2} - 135135 \, d^{11} e x - 204800 \, d^{12}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5125120 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.26, size = 160, normalized size = 0.57 \begin {gather*} \frac {27}{1024} \, d^{13} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-5\right )} \mathrm {sgn}\relax (d) - \frac {1}{5125120} \, {\left (204800 \, d^{12} e^{\left (-5\right )} + {\left (135135 \, d^{11} e^{\left (-4\right )} + 2 \, {\left (51200 \, d^{10} e^{\left (-3\right )} + {\left (45045 \, d^{9} e^{\left (-2\right )} + 4 \, {\left (9600 \, d^{8} e^{\left (-1\right )} - {\left (119119 \, d^{7} + 2 \, {\left (156160 \, d^{6} e + 7 \, {\left (7293 \, d^{5} e^{2} - 8 \, {\left (3280 \, d^{4} e^{3} + {\left (3003 \, d^{3} e^{4} - 10 \, {\left (48 \, d^{2} e^{5} + 11 \, {\left (4 \, x e^{7} + 13 \, 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 )} \sqrt {-x^{2} e^{2} + d^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 266, normalized size = 0.95 \begin {gather*} \frac {27 d^{13} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{1024 \sqrt {e^{2}}\, e^{4}}+\frac {27 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{11} x}{1024 e^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e \,x^{6}}{13}+\frac {9 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{9} x}{512 e^{4}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d \,x^{5}}{4}-\frac {45 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2} x^{4}}{143 e}+\frac {9 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{7} x}{640 e^{4}}-\frac {9 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} x^{3}}{40 e^{2}}-\frac {20 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{4} x^{2}}{143 e^{3}}-\frac {27 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{5} x}{320 e^{4}}-\frac {40 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{6}}{1001 e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 245, normalized size = 0.87 \begin {gather*} -\frac {1}{13} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{6} - \frac {1}{4} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{5} + \frac {27 \, d^{13} \arcsin \left (\frac {e x}{d}\right )}{1024 \, e^{5}} + \frac {27 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{11} x}{1024 \, e^{4}} - \frac {45 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{4}}{143 \, e} + \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{9} x}{512 \, e^{4}} - \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{3}}{40 \, e^{2}} + \frac {9 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{7} x}{640 \, e^{4}} - \frac {20 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x^{2}}{143 \, e^{3}} - \frac {27 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5} x}{320 \, e^{4}} - \frac {40 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{6}}{1001 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 64.64, size = 2028, normalized size = 7.22
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________