Optimal. Leaf size=100 \[ \frac {2 a e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {2 b (e \sin (c+d x))^{5/2}}{5 d e} \]
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Rubi [A]
time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2748, 2715,
2721, 2720} \begin {gather*} \frac {2 a e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {2 b (e \sin (c+d x))^{5/2}}{5 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rubi steps
\begin {align*} \int (a+b \cos (c+d x)) (e \sin (c+d x))^{3/2} \, dx &=\frac {2 b (e \sin (c+d x))^{5/2}}{5 d e}+a \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {2 b (e \sin (c+d x))^{5/2}}{5 d e}+\frac {1}{3} \left (a e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {2 b (e \sin (c+d x))^{5/2}}{5 d e}+\frac {\left (a e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 \sqrt {e \sin (c+d x)}}\\ &=\frac {2 a e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {2 b (e \sin (c+d x))^{5/2}}{5 d e}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 80, normalized size = 0.80 \begin {gather*} \frac {2 (e \sin (c+d x))^{3/2} \left (-5 a F\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )+\sqrt {\sin (c+d x)} \left (-5 a \cos (c+d x)+3 b \sin ^2(c+d x)\right )\right )}{15 d \sin ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 116, normalized size = 1.16
method | result | size |
default | \(\frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{5 e}-\frac {e^{2} a \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\sin ^{3}\left (d x +c \right )\right )+2 \sin \left (d x +c \right )\right )}{3 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(116\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.10, size = 104, normalized size = 1.04 \begin {gather*} \frac {5 \, \sqrt {2} \sqrt {-i} a e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} \sqrt {i} a e^{\frac {3}{2}} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 2 \, {\left (3 \, b \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} + 5 \, a \cos \left (d x + c\right ) e^{\frac {3}{2}} - 3 \, b e^{\frac {3}{2}}\right )} \sqrt {\sin \left (d x + c\right )}}{15 \, d} \end {gather*}
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 \begin {gather*} \int \left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (a + b \cos {\left (c + d x \right )}\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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