Optimal. Leaf size=252 \[ -\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}+\frac {41 d^{12} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^4}+\frac {41 d^{10} x \sqrt {d^2-e^2 x^2}}{1024 e^3}+\frac {41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {41 d^{10} x \sqrt {d^2-e^2 x^2}}{1024 e^3}+\frac {41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}+\frac {41 d^{12} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 195
Rule 203
Rule 217
Rule 780
Rule 833
Rule 1809
Rubi steps
\begin {align*} \int x^3 (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^3 \left (d^2-e^2 x^2\right )^{5/2} \left (-12 d^3 e^2-41 d^2 e^3 x-36 d e^4 x^2\right ) \, dx}{12 e^2}\\ &=-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x^3 \left (276 d^3 e^4+451 d^2 e^5 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{132 e^4}\\ &=-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {\int x^2 \left (-1353 d^4 e^5-2760 d^3 e^6 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{1320 e^6}\\ &=-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}+\frac {\int x \left (5520 d^5 e^6+12177 d^4 e^7 x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{11880 e^8}\\ &=-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac {\left (41 d^6\right ) \int \left (d^2-e^2 x^2\right )^{5/2} \, dx}{320 e^3}\\ &=\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac {\left (41 d^8\right ) \int \left (d^2-e^2 x^2\right )^{3/2} \, dx}{384 e^3}\\ &=\frac {41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac {\left (41 d^{10}\right ) \int \sqrt {d^2-e^2 x^2} \, dx}{512 e^3}\\ &=\frac {41 d^{10} x \sqrt {d^2-e^2 x^2}}{1024 e^3}+\frac {41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac {\left (41 d^{12}\right ) \int \frac {1}{\sqrt {d^2-e^2 x^2}} \, dx}{1024 e^3}\\ &=\frac {41 d^{10} x \sqrt {d^2-e^2 x^2}}{1024 e^3}+\frac {41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac {\left (41 d^{12}\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^3}\\ &=\frac {41 d^{10} x \sqrt {d^2-e^2 x^2}}{1024 e^3}+\frac {41 d^8 x \left (d^2-e^2 x^2\right )^{3/2}}{1536 e^3}+\frac {41 d^6 x \left (d^2-e^2 x^2\right )^{5/2}}{1920 e^3}-\frac {23 d^3 x^2 \left (d^2-e^2 x^2\right )^{7/2}}{99 e^2}-\frac {41 d^2 x^3 \left (d^2-e^2 x^2\right )^{7/2}}{120 e}-\frac {3}{11} d x^4 \left (d^2-e^2 x^2\right )^{7/2}-\frac {1}{12} e x^5 \left (d^2-e^2 x^2\right )^{7/2}-\frac {d^4 (14720 d+28413 e x) \left (d^2-e^2 x^2\right )^{7/2}}{221760 e^4}+\frac {41 d^{12} \tan ^{-1}\left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{1024 e^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.30, size = 189, normalized size = 0.75 \[ \frac {\sqrt {d^2-e^2 x^2} \left (142065 d^{11} \sin ^{-1}\left (\frac {e x}{d}\right )+\sqrt {1-\frac {e^2 x^2}{d^2}} \left (-235520 d^{11}-142065 d^{10} e x-117760 d^9 e^2 x^2-94710 d^8 e^3 x^3+798720 d^7 e^4 x^4+2053128 d^6 e^5 x^5+665600 d^5 e^6 x^6-2295216 d^4 e^7 x^7-2078720 d^3 e^8 x^8+325248 d^2 e^9 x^9+967680 d e^{10} x^{10}+295680 e^{11} x^{11}\right )\right )}{3548160 e^4 \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.97, size = 172, normalized size = 0.68 \[ -\frac {284130 \, d^{12} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) - {\left (295680 \, e^{11} x^{11} + 967680 \, d e^{10} x^{10} + 325248 \, d^{2} e^{9} x^{9} - 2078720 \, d^{3} e^{8} x^{8} - 2295216 \, d^{4} e^{7} x^{7} + 665600 \, d^{5} e^{6} x^{6} + 2053128 \, d^{6} e^{5} x^{5} + 798720 \, d^{7} e^{4} x^{4} - 94710 \, d^{8} e^{3} x^{3} - 117760 \, d^{9} e^{2} x^{2} - 142065 \, d^{10} e x - 235520 \, d^{11}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3548160 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 149, normalized size = 0.59 \[ \frac {41}{1024} \, d^{12} \arcsin \left (\frac {x e}{d}\right ) e^{\left (-4\right )} \mathrm {sgn}\relax (d) - \frac {1}{3548160} \, {\left (235520 \, d^{11} e^{\left (-4\right )} + {\left (142065 \, d^{10} e^{\left (-3\right )} + 2 \, {\left (58880 \, d^{9} e^{\left (-2\right )} + {\left (47355 \, d^{8} e^{\left (-1\right )} - 4 \, {\left (99840 \, d^{7} + {\left (256641 \, d^{6} e + 2 \, {\left (41600 \, d^{5} e^{2} - 7 \, {\left (20493 \, d^{4} e^{3} + 8 \, {\left (2320 \, d^{3} e^{4} - 3 \, {\left (121 \, d^{2} e^{5} + 10 \, {\left (11 \, x e^{7} + 36 \, d e^{6}\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \sqrt {-x^{2} e^{2} + d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 241, normalized size = 0.96 \[ \frac {41 d^{12} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{1024 \sqrt {e^{2}}\, e^{3}}+\frac {41 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{10} x}{1024 e^{3}}+\frac {41 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}} d^{8} x}{1536 e^{3}}-\frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} e \,x^{5}}{12}-\frac {3 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d \,x^{4}}{11}+\frac {41 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} d^{6} x}{1920 e^{3}}-\frac {41 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{2} x^{3}}{120 e}-\frac {23 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{3} x^{2}}{99 e^{2}}-\frac {41 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{4} x}{320 e^{3}}-\frac {46 \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}} d^{5}}{693 e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.99, size = 220, normalized size = 0.87 \[ -\frac {1}{12} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} e x^{5} + \frac {41 \, d^{12} \arcsin \left (\frac {e x}{d}\right )}{1024 \, e^{4}} + \frac {41 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{10} x}{1024 \, e^{3}} - \frac {3}{11} \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d x^{4} + \frac {41 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{8} x}{1536 \, e^{3}} - \frac {41 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{2} x^{3}}{120 \, e} + \frac {41 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d^{6} x}{1920 \, e^{3}} - \frac {23 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{3} x^{2}}{99 \, e^{2}} - \frac {41 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{4} x}{320 \, e^{3}} - \frac {46 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} d^{5}}{693 \, e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 59.74, size = 1919, normalized size = 7.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________