Optimal. Leaf size=202 \[ \frac {2 a e^2 \left (7 a^2+6 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 d \sqrt {e \sin (c+d x)}}+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{315 d e}-\frac {2 a e \left (7 a^2+6 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{63 d e} \]
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Rubi [A] time = 0.30, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2692, 2862, 2669, 2635, 2642, 2641} \[ \frac {2 a e^2 \left (7 a^2+6 b^2\right ) \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{21 d \sqrt {e \sin (c+d x)}}+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{315 d e}-\frac {2 a e \left (7 a^2+6 b^2\right ) \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {2 b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))^2}{9 d e}+\frac {26 a b (e \sin (c+d x))^{5/2} (a+b \cos (c+d x))}{63 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rule 2692
Rule 2862
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 (e \sin (c+d x))^{3/2} \, dx &=\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}}{9 d e}+\frac {2}{9} \int (a+b \cos (c+d x)) \left (\frac {9 a^2}{2}+2 b^2+\frac {13}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2} \, dx\\ &=\frac {26 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}}{9 d e}+\frac {4}{63} \int \left (\frac {9}{4} a \left (7 a^2+6 b^2\right )+\frac {1}{4} b \left (89 a^2+28 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{3/2} \, dx\\ &=\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{315 d e}+\frac {26 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}}{9 d e}+\frac {1}{7} \left (a \left (7 a^2+6 b^2\right )\right ) \int (e \sin (c+d x))^{3/2} \, dx\\ &=-\frac {2 a \left (7 a^2+6 b^2\right ) e \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{315 d e}+\frac {26 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}}{9 d e}+\frac {1}{21} \left (a \left (7 a^2+6 b^2\right ) e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=-\frac {2 a \left (7 a^2+6 b^2\right ) e \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{315 d e}+\frac {26 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}}{9 d e}+\frac {\left (a \left (7 a^2+6 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 a \left (7 a^2+6 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 a \left (7 a^2+6 b^2\right ) e \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 d}+\frac {2 b \left (89 a^2+28 b^2\right ) (e \sin (c+d x))^{5/2}}{315 d e}+\frac {26 a b (a+b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2}}{9 d e}\\ \end {align*}
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Mathematica [A] time = 1.20, size = 147, normalized size = 0.73 \[ \frac {(e \sin (c+d x))^{3/2} \left (-20 a \left (28 a^2+15 b^2\right ) \cot (c+d x)-\frac {2}{3} b \csc (c+d x) \left (28 \left (27 a^2+4 b^2\right ) \cos (2 (c+d x))-756 a^2+270 a b \cos (3 (c+d x))+35 b^2 \cos (4 (c+d x))-147 b^2\right )-\frac {80 a \left (7 a^2+6 b^2\right ) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )}{\sin ^{\frac {3}{2}}(c+d x)}\right )}{840 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} e \cos \left (d x + c\right )^{3} + 3 \, a b^{2} e \cos \left (d x + c\right )^{2} + 3 \, a^{2} b e \cos \left (d x + c\right ) + a^{3} e\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right ), x\right ) \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 226, normalized size = 1.12 \[ \frac {\frac {2 b \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}} \left (5 \left (\cos ^{2}\left (d x +c \right )\right ) b^{2}+27 a^{2}+4 b^{2}\right )}{45 e}-\frac {e^{2} a \left (18 b^{2} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+\left (14 a^{2}-6 b^{2}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+7 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+6 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}\right )}{21 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3 \,d x \]
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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