Optimal. Leaf size=154 \[ \frac {2 \left (7 a^2+2 b^2\right ) e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 d \sqrt {e \sin (c+d x)}}-\frac {2 \left (7 a^2+2 b^2\right ) e \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e} \]
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Rubi [A]
time = 0.11, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2771, 2748,
2715, 2721, 2720} \begin {gather*} \frac {2 e^2 \left (7 a^2+2 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 d \sqrt {e \sin (c+d x)}}-\frac {2 e \left (7 a^2+2 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{7 d e} \end {gather*}
Antiderivative was successfully verified.
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Rule 2715
Rule 2720
Rule 2721
Rule 2748
Rule 2771
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{3/2} \, dx &=\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}+\frac {2}{7} \int \left (\frac {7 a^2}{2}+b^2+\frac {9}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2} \, dx\\ &=\frac {18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}+\frac {1}{7} \left (7 a^2+2 b^2\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {2 \left (7 a^2+2 b^2\right ) e \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}+\frac {1}{21} \left (\left (7 a^2+2 b^2\right ) e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=-\frac {2 \left (7 a^2+2 b^2\right ) e \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}+\frac {\left (\left (7 a^2+2 b^2\right ) e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 \sqrt {e \sin (c+d x)}}\\ &=\frac {2 \left (7 a^2+2 b^2\right ) e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{21 d \sqrt {e \sin (c+d x)}}-\frac {2 \left (7 a^2+2 b^2\right ) e \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {18 a b (e \sin (c+d x))^{5/2}}{35 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{7 d e}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 117, normalized size = 0.76 \begin {gather*} \frac {\left (-\frac {1}{2} \left (5 \left (28 a^2+5 b^2\right ) \cos (c+d x)+3 b (-28 a+28 a \cos (2 (c+d x))+5 b \cos (3 (c+d x)))\right ) \csc (c+d x)-\frac {10 \left (7 a^2+2 b^2\right ) F\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right )}{\sin ^{\frac {3}{2}}(c+d x)}\right ) (e \sin (c+d x))^{3/2}}{105 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 229, normalized size = 1.49
method | result | size |
default | \(-\frac {e^{2} \left (30 b^{2} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )+35 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+10 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}+84 a b \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+70 a^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-10 b^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-84 a b \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{105 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(229\) |
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 = 0.12, size = 151, normalized size = 0.98 \begin {gather*} \frac {5 \, \sqrt {2} \sqrt {-i} {\left (7 \, a^{2} + 2 \, b^{2}\right )} 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} {\left (7 \, a^{2} + 2 \, b^{2}\right )} 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 (15 \, b^{2} \cos \left (d x + c\right )^{3} e^{\frac {3}{2}} + 42 \, a b \cos \left (d x + c\right )^{2} e^{\frac {3}{2}} - 42 \, a b e^{\frac {3}{2}} + 5 \, {\left (7 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right ) e^{\frac {3}{2}}\right )} \sqrt {\sin \left (d x + c\right )}}{105 \, 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 )^{2}\, 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 )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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