Optimal. Leaf size=93 \[ -\frac {2 \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5} (3 \cos (d+e x)-4 \sin (d+e x))}{3 e}-\frac {40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}} \]
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Rubi [A] time = 0.04, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3113, 3112} \[ -\frac {2 \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5} (3 \cos (d+e x)-4 \sin (d+e x))}{3 e}-\frac {40 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}} \]
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))^{3/2} \, dx &=-\frac {2 (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 {20}{3} \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx\\ &=-\frac {40 (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 {2 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 104, normalized size = 1.12 \[ \frac {(3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2} \left (9 \left (15 \sin \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {3}{2} (d+e x)\right )\right )-45 \cos \left (\frac {1}{2} (d+e x)\right )-13 \cos \left (\frac {3}{2} (d+e x)\right )\right )}{3 e \left (\sin \left (\frac {1}{2} (d+e x)\right )+3 \cos \left (\frac {1}{2} (d+e x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 81, normalized size = 0.87 \[ -\frac {2 \, {\left (13 \, \cos \left (e x + d\right )^{2} - 9 \, {\left (\cos \left (e x + d\right ) + 8\right )} \sin \left (e x + d\right ) + 29 \, \cos \left (e x + d\right ) + 16\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}{3 \, {\left (3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e\right )}} \]
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 \[ \int {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 60, normalized size = 0.65 \[ \frac {50 \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+5\right )}{3 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e} \]
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 \[ \int {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )+5\right )}^{3/2} \,d x \]
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 \[ \int \left (3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} + 5\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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