Optimal. Leaf size=185 \[ -\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}-\frac {32 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{15 e}+\frac {64 F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}+\frac {796 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{15 e} \]
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Rubi [A] time = 0.27, antiderivative size = 185, 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 = {3120, 3146, 3149, 3118, 2653, 3126, 2661} \[ -\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (5 \sin (d+e x)+3 \cos (d+e x)+2)^{3/2}}{5 e}-\frac {32 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2}}{15 e}+\frac {64 F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{\sqrt {2+\sqrt {34}} e}+\frac {796 \sqrt {2+\sqrt {34}} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )|\frac {2}{15} \left (17-\sqrt {34}\right )\right )}{15 e} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 3118
Rule 3120
Rule 3126
Rule 3146
Rule 3149
Rubi steps
\begin {align*} \int (2+3 \cos (d+e x)+5 \sin (d+e x))^{5/2} \, dx &=-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}{5 e}+\frac {2}{5} \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} (61+24 \cos (d+e x)+40 \sin (d+e x)) \, dx\\ &=-\frac {32 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{15 e}-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}{5 e}+\frac {2}{15} \int \frac {638+597 \cos (d+e x)+995 \sin (d+e x)}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx\\ &=-\frac {32 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{15 e}-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}{5 e}+\frac {398}{15} \int \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)} \, dx+32 \int \frac {1}{\sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}} \, dx\\ &=-\frac {32 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{15 e}-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}{5 e}+\frac {398}{15} \int \sqrt {2+\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )} \, dx+32 \int \frac {1}{\sqrt {2+\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )}} \, dx\\ &=\frac {796 \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 )}{15 e}+\frac {64 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 {32 (5 \cos (d+e x)-3 \sin (d+e x)) \sqrt {2+3 \cos (d+e x)+5 \sin (d+e x)}}{15 e}-\frac {2 (5 \cos (d+e x)-3 \sin (d+e x)) (2+3 \cos (d+e x)+5 \sin (d+e x))^{3/2}}{5 e}\\ \end {align*}
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Mathematica [C] time = 5.91, size = 399, normalized size = 2.16 \[ \frac {1276 \sqrt {\frac {10}{3}} \sqrt {\sqrt {34} \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+2} \sqrt {\cos ^2\left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )} \sec \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right ) F_1\left (\frac {1}{2};\frac {1}{2},\frac {1}{2};\frac {3}{2};\frac {17 \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+\sqrt {34}}{-17+\sqrt {34}},\frac {17 \sin \left (d+e x+\tan ^{-1}\left (\frac {3}{5}\right )\right )+\sqrt {34}}{17+\sqrt {34}}\right )-\frac {1990 \sqrt {30} \sqrt {\sin ^2\left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )} \csc \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right ) F_1\left (-\frac {1}{2};-\frac {1}{2},-\frac {1}{2};\frac {1}{2};\frac {17 \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+\sqrt {34}}{-17+\sqrt {34}},\frac {17 \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+\sqrt {34}}{17+\sqrt {34}}\right )}{\sqrt {\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+2}}-2 \sqrt {5 \sin (d+e x)+3 \cos (d+e x)+2} (550 \cos (d+e x)+3 (-110 \sin (d+e x)+40 \sin (2 (d+e x))+75 \cos (2 (d+e x))-398))-2388 \sqrt {\sqrt {34} \cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )+2}+\frac {1990 \sin \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )}{\sqrt {\frac {\cos \left (d+e x-\tan ^{-1}\left (\frac {5}{3}\right )\right )}{\sqrt {34}}+\frac {1}{17}}}}{75 e} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (16 \, \cos \left (e x + d\right )^{2} - 10 \, {\left (3 \, \cos \left (e x + d\right ) + 2\right )} \sin \left (e x + d\right ) - 12 \, \cos \left (e x + d\right ) - 29\right )} \sqrt {3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2}, x\right ) \]
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 \[ \int {\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.66, size = 701, normalized size = 3.79 \[ \text {result too large to display} \]
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 \[ \int {\left (3 \, \cos \left (e x + d\right ) + 5 \, \sin \left (e x + d\right ) + 2\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (3\,\cos \left (d+e\,x\right )+5\,\sin \left (d+e\,x\right )+2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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