Optimal. Leaf size=139 \[ -\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}-\frac{16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}{3 e}-\frac{320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}} \]
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Rubi [A] time = 0.0648625, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3113, 3112} \[ -\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}-\frac{16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}}{3 e}-\frac{320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt{3 \sin (d+e x)+4 \cos (d+e x)+5}} \]
Antiderivative was successfully verified.
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Rule 3113
Rule 3112
Rubi steps
\begin{align*} \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx &=-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{5 e}+8 \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx\\ &=-\frac{16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e}-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{5 e}+\frac{160}{3} \int \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx\\ &=-\frac{320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)}}-\frac{16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt{5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e}-\frac{2 (3 \cos (d+e x)-4 \sin (d+e x)) (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{5 e}\\ \end{align*}
Mathematica [A] time = 0.627387, size = 130, normalized size = 0.94 \[ -\frac{(3 \sin (d+e x)+4 \cos (d+e x)+5)^{5/2} \left (3750 \cos \left (\frac{1}{2} (d+e x)\right )+1625 \cos \left (\frac{3}{2} (d+e x)\right )+3 \left (-3750 \sin \left (\frac{1}{2} (d+e x)\right )-375 \sin \left (\frac{3}{2} (d+e x)\right )+3 \sin \left (\frac{5}{2} (d+e x)\right )+79 \cos \left (\frac{5}{2} (d+e x)\right )\right )\right )}{30 e \left (\sin \left (\frac{1}{2} (d+e x)\right )+3 \cos \left (\frac{1}{2} (d+e x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.355, size = 74, normalized size = 0.5 \begin{align*}{\frac{ \left ( 50+50\,\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) \right ) \left ( \sin \left ( ex+d+\arctan \left ({\frac{4}{3}} \right ) \right ) -1 \right ) \left ( 3\, \left ( \sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) \right ) ^{2}+14\,\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) +43 \right ) }{3\,\cos \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) e}{\frac{1}{\sqrt{5+5\,\sin \left ( ex+d+\arctan \left ( 4/3 \right ) \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82639, size = 293, normalized size = 2.11 \begin{align*} -\frac{2 \,{\left (237 \, \cos \left (e x + d\right )^{3} + 931 \, \cos \left (e x + d\right )^{2} + 9 \,{\left (\cos \left (e x + d\right )^{2} - 62 \, \cos \left (e x + d\right ) - 344\right )} \sin \left (e x + d\right ) + 1166 \, \cos \left (e x + d\right ) + 472\right )} \sqrt{4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}{15 \,{\left (3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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