Optimal. Leaf size=95 \[ \frac {2 a e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 a e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}-\frac {2 b (e \cos (c+d x))^{5/2}}{5 d e} \]
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Rubi [A] time = 0.07, antiderivative size = 95, 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 = {2669, 2635, 2642, 2641} \[ \frac {2 a e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 a e \sin (c+d x) \sqrt {e \cos (c+d x)}}{3 d}-\frac {2 b (e \cos (c+d x))^{5/2}}{5 d e} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rule 2642
Rule 2669
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{3/2} (a+b \sin (c+d x)) \, dx &=-\frac {2 b (e \cos (c+d x))^{5/2}}{5 d e}+a \int (e \cos (c+d x))^{3/2} \, dx\\ &=-\frac {2 b (e \cos (c+d x))^{5/2}}{5 d e}+\frac {2 a e \sqrt {e \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{3} \left (a e^2\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 b (e \cos (c+d x))^{5/2}}{5 d e}+\frac {2 a e \sqrt {e \cos (c+d x)} \sin (c+d x)}{3 d}+\frac {\left (a e^2 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 \sqrt {e \cos (c+d x)}}\\ &=-\frac {2 b (e \cos (c+d x))^{5/2}}{5 d e}+\frac {2 a e^2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {e \cos (c+d x)}}+\frac {2 a e \sqrt {e \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 79, normalized size = 0.83 \[ \frac {(e \cos (c+d x))^{3/2} \left (\sqrt {\cos (c+d x)} (10 a \sin (c+d x)-3 b \cos (2 (c+d x))-3 b)+10 a F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{15 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b e \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a e \cos \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.29, size = 185, normalized size = 1.95 \[ -\frac {2 e^{2} \left (-24 b \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+36 b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a -10 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-18 b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (a+b\,\sin \left (c+d\,x\right )\right ) \,d x \]
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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