Optimal. Leaf size=154 \[ \frac {2 e^2 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 d \sqrt {\sin (c+d x)}}-\frac {2 e \left (9 a^2+2 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 d}+\frac {22 a b (e \sin (c+d x))^{7/2}}{63 d e}+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e} \]
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Rubi [A] time = 0.16, 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 = {2692, 2669, 2635, 2640, 2639} \[ \frac {2 e^2 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 d \sqrt {\sin (c+d x)}}-\frac {2 e \left (9 a^2+2 b^2\right ) \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 d}+\frac {22 a b (e \sin (c+d x))^{7/2}}{63 d e}+\frac {2 b (e \sin (c+d x))^{7/2} (a+b \cos (c+d x))}{9 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2640
Rule 2669
Rule 2692
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^2 (e \sin (c+d x))^{5/2} \, dx &=\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{9 d e}+\frac {2}{9} \int \left (\frac {9 a^2}{2}+b^2+\frac {11}{2} a b \cos (c+d x)\right ) (e \sin (c+d x))^{5/2} \, dx\\ &=\frac {22 a b (e \sin (c+d x))^{7/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{9 d e}+\frac {1}{9} \left (9 a^2+2 b^2\right ) \int (e \sin (c+d x))^{5/2} \, dx\\ &=-\frac {2 \left (9 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 d}+\frac {22 a b (e \sin (c+d x))^{7/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{9 d e}+\frac {1}{15} \left (\left (9 a^2+2 b^2\right ) e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx\\ &=-\frac {2 \left (9 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 d}+\frac {22 a b (e \sin (c+d x))^{7/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{9 d e}+\frac {\left (\left (9 a^2+2 b^2\right ) e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 \sqrt {\sin (c+d x)}}\\ &=\frac {2 \left (9 a^2+2 b^2\right ) e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 d \sqrt {\sin (c+d x)}}-\frac {2 \left (9 a^2+2 b^2\right ) e \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 d}+\frac {22 a b (e \sin (c+d x))^{7/2}}{63 d e}+\frac {2 b (a+b \cos (c+d x)) (e \sin (c+d x))^{7/2}}{9 d e}\\ \end {align*}
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Mathematica [A] time = 0.80, size = 116, normalized size = 0.75 \[ -\frac {(e \sin (c+d x))^{5/2} \left (84 \left (9 a^2+2 b^2\right ) E\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )+\sin ^{\frac {3}{2}}(c+d x) \left (21 \left (12 a^2+b^2\right ) \cos (c+d x)+5 b (36 a \cos (2 (c+d x))-36 a+7 b \cos (3 (c+d x)))\right )\right )}{630 d \sin ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} e^{2} \cos \left (d x + c\right )^{4} + 2 \, a b e^{2} \cos \left (d x + c\right )^{3} - 2 \, a b e^{2} \cos \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} e^{2} \cos \left (d x + c\right )^{2} - a^{2} e^{2}\right )} \sqrt {e \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 332, normalized size = 2.16 \[ \frac {\frac {4 a b \left (e \sin \left (d x +c \right )\right )^{\frac {7}{2}}}{7 e}-\frac {e^{3} \left (10 b^{2} \left (\sin ^{6}\left (d x +c \right )\right )+54 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}+12 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticE \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}-27 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{2}-6 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) b^{2}-18 a^{2} \left (\sin ^{4}\left (d x +c \right )\right )-14 \left (\sin ^{4}\left (d x +c \right )\right ) b^{2}+18 \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}+4 \left (\sin ^{2}\left (d x +c \right )\right ) b^{2}\right )}{45 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d} \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2 \,d x \]
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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