Optimal. Leaf size=149 \[ \frac {2 e^2 \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 e \left (7 a^2+2 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e} \]
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Rubi [A] time = 0.16, antiderivative size = 149, 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, 2642, 2641} \[ \frac {2 e^2 \left (7 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 e \left (7 a^2+2 b^2\right ) \sin (c+d x) \sqrt {e \cos (c+d x)}}{21 d}-\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rule 2692
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2 \, dx &=-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}+\frac {2}{7} \int (e \cos (c+d x))^{3/2} \left (\frac {7 a^2}{2}+b^2+\frac {9}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}+\frac {1}{7} \left (7 a^2+2 b^2\right ) \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}+\frac {2 \left (7 a^2+2 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}+\frac {1}{21} \left (\left (7 a^2+2 b^2\right ) e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}+\frac {2 \left (7 a^2+2 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}+\frac {\left (\left (7 a^2+2 b^2\right ) e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {e \cos (c+d x)}}\\ &=-\frac {18 a b (e \cos (c+d x))^{5/2}}{35 d e}+\frac {2 \left (7 a^2+2 b^2\right ) e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {e \cos (c+d x)}}+\frac {2 \left (7 a^2+2 b^2\right ) e \sqrt {e \cos (c+d x)} \sin (c+d x)}{21 d}-\frac {2 b (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))}{7 d e}\\ \end {align*}
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Mathematica [A] time = 1.18, size = 115, normalized size = 0.77 \[ \frac {(e \cos (c+d x))^{3/2} \left (20 \left (7 a^2+2 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sqrt {\cos (c+d x)} \left (5 \left (28 a^2+5 b^2\right ) \sin (c+d x)-3 b (28 a+5 b \sin (3 (c+d x)))-84 a b \cos (2 (c+d x))\right )\right )}{210 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b^{2} e \cos \left (d x + c\right )^{3} - 2 \, a b e \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{2} + b^{2}\right )} e \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.72, size = 343, normalized size = 2.30 \[ -\frac {2 e^{2} \left (-240 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-336 a b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+360 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+140 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 a b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-140 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{2}+10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) b^{2}-70 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-252 a b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+42 a b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}^{2}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^2 \,d x \]
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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