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 = 0.07, antiderivative size = 139, normalized size of antiderivative = 1.00, 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 3112
Rule 3113
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*}
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Mathematica [A] time = 0.49, size = 127, normalized size = 0.91 \[ \frac {(3 \sin (d+e x)+4 \cos (d+e x)-5)^{5/2} \left (3750 \sin \left (\frac {1}{2} (d+e x)\right )-1625 \sin \left (\frac {3}{2} (d+e x)\right )+237 \sin \left (\frac {5}{2} (d+e x)\right )+11250 \cos \left (\frac {1}{2} (d+e x)\right )-1125 \cos \left (\frac {3}{2} (d+e x)\right )-9 \cos \left (\frac {5}{2} (d+e x)\right )\right )}{30 e \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 101, normalized size = 0.73 \[ -\frac {2 \, {\left (9 \, \cos \left (e x + d\right )^{3} + 567 \, \cos \left (e x + d\right )^{2} - {\left (237 \, \cos \left (e x + d\right )^{2} - 694 \, \cos \left (e x + d\right ) + 472\right )} \sin \left (e x + d\right ) - 2538 \, \cos \left (e x + d\right ) - 3096\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{15 \, {\left (e \cos \left (e x + d\right ) - 3 \, e \sin \left (e x + d\right ) + e\right )}} \]
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 \[ \int {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 74, normalized size = 0.53 \[ \frac {50 \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \left (3 \left (\sin ^{2}\left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right )-14 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+43\right )}{3 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e} \]
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 \[ \int {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )-5\right )}^{5/2} \,d x \]
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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