Optimal. Leaf size=187 \[ \frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}+\frac {F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{45 \sqrt {2+\sqrt {34}} e}+\frac {4 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{675 e} \]
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Rubi [A] time = 0.20, antiderivative size = 187, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {3129, 3156, 3149, 3118, 2653, 3126, 2661} \[ \frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}+\frac {F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{45 \sqrt {2+\sqrt {34}} e}+\frac {4 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{675 e} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 3118
Rule 3126
Rule 3129
Rule 3149
Rule 3156
Rubi steps
\begin {align*} \int \frac {1}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2}} \, dx &=-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {1}{45} \int \frac {-3+\frac {3}{2} \cos (d+e x)+\frac {5}{2} \sin (d+e x)}{(2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}} \, dx\\ &=-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {1}{675} \int \frac {\frac {23}{2}+6 \cos (d+e x)+10 \sin (d+e x)}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx\\ &=-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {2}{675} \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx+\frac {1}{90} \int \frac {1}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx\\ &=-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}+\frac {2}{675} \int \sqrt {2+\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )} \, dx+\frac {1}{90} \int \frac {1}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )}} \, dx\\ &=\frac {4 \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 )}{675 e}+\frac {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 )}{45 \sqrt {2+\sqrt {34}} e}-\frac {5 \cos (d+e x)-3 \sin (d+e x)}{45 e (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}+\frac {4 (5 \cos (d+e x)-3 \sin (d+e x))}{675 e \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}\\ \end {align*}
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Mathematica [C] time = 3.18, size = 430, normalized size = 2.30 \[ \frac {23 \sqrt {\frac {10}{3}} \sqrt {\sqrt {34} \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+2} \sqrt {\cos ^2\left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right ) F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};\frac {17 \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+\sqrt {34}}{-17+\sqrt {34}},\frac {17 \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+\sqrt {34}}{17+\sqrt {34}}\right )-\frac {20 \sqrt {30} \sqrt {\sin ^2\left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )} \csc \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right ) F_1\left (-\frac {1}{2};-\frac {1}{2},-\frac {1}{2};\frac {1}{2};\frac {17 \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+\sqrt {34}}{-17+\sqrt {34}},\frac {17 \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+\sqrt {34}}{17+\sqrt {34}}\right )}{\sqrt {\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+2}}+\frac {100 (17 \sin (d+e x)+5)}{(5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}-\frac {10 (136 \sin (d+e x)+115)}{3 \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}+\frac {272}{3} \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}-24 \sqrt {\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+2}+\frac {20 \sin \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )}{\sqrt {\frac {\cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )}{\sqrt {34}}+\frac {1}{17}}}}{6750 e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}}{198 \, \cos \left (e x + d\right )^{3} + 96 \, \cos \left (e x + d\right )^{2} - 5 \, {\left (2 \, \cos \left (e x + d\right )^{2} + 36 \, \cos \left (e x + d\right ) + 37\right )} \sin \left (e x + d\right ) - 261 \, \cos \left (e x + d\right ) - 158}, x\right ) \]
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 \[ \int \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.61, size = 542, normalized size = 2.90 \[ \frac {\sqrt {-\left (-\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-2\right ) \left (\cos ^{2}\left (e x +d +\arctan \left (\frac {3}{5}\right )\right )\right )}\, \left (-\frac {\sqrt {34}\, \sqrt {-\left (-\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-2\right ) \left (\cos ^{2}\left (e x +d +\arctan \left (\frac {3}{5}\right )\right )\right )}}{1530 \left (\sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+\frac {\sqrt {34}}{17}\right )^{2}}+\frac {68 \sqrt {34}\, \left (\cos ^{2}\left (e x +d +\arctan \left (\frac {3}{5}\right )\right )\right )}{675 \sqrt {-\left (-289 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-17 \sqrt {34}\right ) \sqrt {34}\, \left (\cos ^{2}\left (e x +d +\arctan \left (\frac {3}{5}\right )\right )\right )}}+\frac {23 \left (-1+\frac {\sqrt {34}}{17}\right ) \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{\sqrt {34}+17}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{-\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 )}{675 \sqrt {-\left (-\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-2\right ) \left (\cos ^{2}\left (e x +d +\arctan \left (\frac {3}{5}\right )\right )\right )}}+\frac {4 \sqrt {34}\, \left (-1+\frac {\sqrt {34}}{17}\right ) \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-\sqrt {34}}{-\sqrt {34}+17}}\, \sqrt {\frac {-17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{\sqrt {34}+17}}\, \sqrt {\frac {17 \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )+17}{-\sqrt {34}+17}}\, \left (\left (-\frac {\sqrt {34}}{17}-1\right ) \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 )+\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 )\right )}{675 \sqrt {-\left (-\sqrt {34}\, \sin \left (e x +d +\arctan \left (\frac {3}{5}\right )\right )-2\right ) \left (\cos ^{2}\left (e x +d +\arctan \left (\frac {3}{5}\right )\right )\right )}}\right )}{\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 \[ \int \frac {1}{{\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (5 \sin {\left (d + e x \right )} + 3 \cos {\left (d + e x \right )} + 2\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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