Optimal. Leaf size=139 \[ -\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}+\frac {20 F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}+\frac {16 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{3 e} \]
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Rubi [A] time = 0.14, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3120, 3149, 3118, 2653, 3126, 2661} \[ -\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{3 e}+\frac {20 F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}+\frac {16 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{3 e} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 3118
Rule 3120
Rule 3126
Rule 3149
Rubi steps
\begin {align*} \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2} \, dx &=-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{3 e}+\frac {2}{3} \int \frac {23+12 \cos (d+e x)+20 \sin (d+e x)}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx\\ &=-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{3 e}+\frac {8}{3} \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx+10 \int \frac {1}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx\\ &=-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{3 e}+\frac {8}{3} \int \sqrt {2+\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )} \, dx+10 \int \frac {1}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )}} \, dx\\ &=\frac {16 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{3 e}+\frac {20 F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{3 e}\\ \end {align*}
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Mathematica [C] time = 3.42, size = 349, normalized size = 2.51 \[ \frac {2 \left (\sqrt {\sin ^2\left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )} \left (23 \sqrt {30} \sqrt {\sqrt {34} \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+2} \sqrt {\cos ^2\left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )} \sqrt {\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+2} \sec \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right ) F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};\frac {17 \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+\sqrt {34}}{-17+\sqrt {34}},\frac {17 \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+\sqrt {34}}{17+\sqrt {34}}\right )-15 (-18 \sin (d+e x)+8 \sin (2 (d+e x))+30 \cos (d+e x)+15 \cos (2 (d+e x)))\right )-60 \sqrt {30} \sin \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right ) F_1\left (-\frac {1}{2};-\frac {1}{2},-\frac {1}{2};\frac {1}{2};\frac {17 \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+\sqrt {34}}{-17+\sqrt {34}},\frac {17 \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+\sqrt {34}}{17+\sqrt {34}}\right )\right )}{45 e \sqrt {\sin ^2\left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )} \sqrt {\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {3}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.46, size = 686, normalized size = 4.94 \[ \frac {\frac {8 \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}\, \sqrt {17}\, \sqrt {\frac {1+\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )}{-\sqrt {34}+17}}\, \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \EllipticF \left (\sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}, i \sqrt {\frac {\sqrt {34}+17}{-\sqrt {34}+17}}\right ) \sqrt {34}}{17}+8 \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}\, \sqrt {17}\, \sqrt {\frac {1+\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )}{-\sqrt {34}+17}}\, \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \EllipticF \left (\sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{\sqrt {34}+17}}, i \sqrt {\frac {\sqrt {34}+17}{-\sqrt {34}+17}}\right )+\frac {68 \left (\sin ^{3}\left (e x +d +\arctan \left (\frac {3}{5}\right )\right )\right )}{3}-\frac {68 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )}{3}+\frac {4 \sqrt {34}\, \left (\sin ^{2}\left (e x +d +\arctan \left (\frac {3}{5}\right )\right )\right )}{3}-\frac {4 \sqrt {34}}{3}-4 \sqrt {17}\, \sqrt {\frac {1+\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )}{-\sqrt {34}+17}}\, \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}\, \EllipticF \left (\sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right ) \sqrt {34}-12 \sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \sqrt {17}\, \sqrt {\frac {1+\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )}{-\sqrt {34}+17}}\, \EllipticF \left (\sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right )+\frac {80 \sqrt {17}\, \sqrt {\frac {1+\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )}{-\sqrt {34}+17}}\, \sqrt {-\frac {17 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-1\right )}{\sqrt {34}+17}}\, \sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}\, \EllipticE \left (\sqrt {-\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\sqrt {34}}{-\sqrt {34}+17}}, i \sqrt {\frac {-\sqrt {34}+17}{\sqrt {34}+17}}\right ) \sqrt {34}}{17}}{\cos \left (e x +d +\arctan \left (\frac {3}{5}\right )\right ) \sqrt {\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+2}\, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (5 \sin {\left (d + e x \right )} + 3 \cos {\left (d + e x \right )} + 2\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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