Optimal. Leaf size=138 \[ -\frac {64 a^3 (7 c+5 d) \cos (e+f x)}{105 f \sqrt {a \sin (e+f x)+a}}-\frac {16 a^2 (7 c+5 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{105 f}-\frac {2 a (7 c+5 d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac {2 d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f} \]
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Rubi [A] time = 0.11, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2751, 2647, 2646} \[ -\frac {64 a^3 (7 c+5 d) \cos (e+f x)}{105 f \sqrt {a \sin (e+f x)+a}}-\frac {16 a^2 (7 c+5 d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{105 f}-\frac {2 a (7 c+5 d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac {2 d \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} (c+d \sin (e+f x)) \, dx &=-\frac {2 d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac {1}{7} (7 c+5 d) \int (a+a \sin (e+f x))^{5/2} \, dx\\ &=-\frac {2 a (7 c+5 d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac {1}{35} (8 a (7 c+5 d)) \int (a+a \sin (e+f x))^{3/2} \, dx\\ &=-\frac {16 a^2 (7 c+5 d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {2 a (7 c+5 d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac {1}{105} \left (32 a^2 (7 c+5 d)\right ) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {64 a^3 (7 c+5 d) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 (7 c+5 d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {2 a (7 c+5 d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 d \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}\\ \end {align*}
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Mathematica [A] time = 1.50, size = 119, normalized size = 0.86 \[ -\frac {a^2 \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) ((392 c+505 d) \sin (e+f x)-6 (7 c+20 d) \cos (2 (e+f x))+1246 c-15 d \sin (3 (e+f x))+1040 d)}{210 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 206, normalized size = 1.49 \[ \frac {2 \, {\left (15 \, a^{2} d \cos \left (f x + e\right )^{4} + 3 \, {\left (7 \, a^{2} c + 20 \, a^{2} d\right )} \cos \left (f x + e\right )^{3} - 224 \, a^{2} c - 160 \, a^{2} d - {\left (77 \, a^{2} c + 85 \, a^{2} d\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (161 \, a^{2} c + 145 \, a^{2} d\right )} \cos \left (f x + e\right ) + {\left (15 \, a^{2} d \cos \left (f x + e\right )^{3} + 224 \, a^{2} c + 160 \, a^{2} d - 3 \, {\left (7 \, a^{2} c + 15 \, a^{2} d\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (49 \, a^{2} c + 65 \, a^{2} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{105 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.78, size = 99, normalized size = 0.72 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (-15 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) d +\left (98 c +130 d \right ) \sin \left (f x +e \right )+\left (-21 c -60 d \right ) \left (\cos ^{2}\left (f x +e \right )\right )+322 c +290 d \right )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (c + d \sin {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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