Optimal. Leaf size=140 \[ -\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {22 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{3/2}}{35 d e}+\frac {22 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e} \]
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Rubi [A] time = 0.15, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2678, 2669, 2640, 2639} \[ -\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {22 \left (a^3 \sin (c+d x)+a^3\right ) (e \cos (c+d x))^{3/2}}{35 d e}+\frac {22 a^3 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (a \sin (c+d x)+a)^2 (e \cos (c+d x))^{3/2}}{7 d e} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2669
Rule 2678
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}+\frac {1}{7} (11 a) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e}+\frac {1}{5} \left (11 a^2\right ) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x)) \, dx\\ &=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e}+\frac {1}{5} \left (11 a^3\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e}+\frac {\left (11 a^3 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {22 a^3 (e \cos (c+d x))^{3/2}}{15 d e}+\frac {22 a^3 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^2}{7 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^3+a^3 \sin (c+d x)\right )}{35 d e}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 66, normalized size = 0.47 \[ -\frac {16\ 2^{3/4} a^3 (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac {11}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (\sin (c+d x)+1)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (d x + c\right )^{2} - 4 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.85, size = 214, normalized size = 1.53 \[ \frac {2 a^{3} e \left (240 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-480 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-200 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-126 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+440 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-125 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
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 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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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