Optimal. Leaf size=243 \[ -\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a \sin (c+d x)+a}}-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}+\frac {5 a \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {5 a \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)} \]
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Rubi [A] time = 0.36, antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2678, 2684, 2775, 203, 2833, 63, 215} \[ -\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a \sin (c+d x)+a}}-\frac {a \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}{2 d e}+\frac {5 a \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {5 a \sqrt {e} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 215
Rule 2678
Rule 2684
Rule 2775
Rule 2833
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2} \, dx &=-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {1}{4} (5 a) \int \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {1}{8} \left (5 a^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {\left (5 a^2 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{8 (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (5 a^2 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{8 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {\left (5 a^2 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (5 a^2 e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+e x^2} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {5 a^2 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (5 a^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{e}}} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {5 a^2 (e \cos (c+d x))^{3/2}}{4 d e \sqrt {a+a \sin (c+d x)}}-\frac {a (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}{2 d e}+\frac {5 a^2 \sqrt {e} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {5 a^2 \sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 77, normalized size = 0.32 \[ -\frac {8 \sqrt [4]{2} (a (\sin (c+d x)+1))^{3/2} (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac {5}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (\sin (c+d x)+1)^{9/4}} \]
Antiderivative was successfully verified.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 262, normalized size = 1.08 \[ \frac {\left (5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )+5 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )+4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-4 \left (\cos ^{3}\left (d x +c \right )\right )+10 \cos \left (d x +c \right ) \sin \left (d x +c \right )+14 \left (\cos ^{2}\left (d x +c \right )\right )-10 \cos \left (d x +c \right )\right ) \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}} \sqrt {e \cos \left (d x +c \right )}}{8 d \left (\cos \left (d x +c \right ) \sin \left (d x +c \right )+\cos ^{2}\left (d x +c \right )-2 \sin \left (d x +c \right )+\cos \left (d x +c \right )-2\right ) \cos \left (d x +c \right )} \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {e \cos {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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