Integrand size = 25, antiderivative size = 138 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx=-\frac {64 a^3 (7 A+5 B) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 (7 A+5 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {2 a (7 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f} \]
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Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2830, 2726, 2725} \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx=-\frac {64 a^3 (7 A+5 B) \cos (e+f x)}{105 f \sqrt {a \sin (e+f x)+a}}-\frac {16 a^2 (7 A+5 B) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{105 f}-\frac {2 a (7 A+5 B) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 f}-\frac {2 B \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{7 f} \]
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Rule 2725
Rule 2726
Rule 2830
Rubi steps \begin{align*} \text {integral}& = -\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac {1}{7} (7 A+5 B) \int (a+a \sin (e+f x))^{5/2} \, dx \\ & = -\frac {2 a (7 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac {1}{35} (8 a (7 A+5 B)) \int (a+a \sin (e+f x))^{3/2} \, dx \\ & = -\frac {16 a^2 (7 A+5 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {2 a (7 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f}+\frac {1}{105} \left (32 a^2 (7 A+5 B)\right ) \int \sqrt {a+a \sin (e+f x)} \, dx \\ & = -\frac {64 a^3 (7 A+5 B) \cos (e+f x)}{105 f \sqrt {a+a \sin (e+f x)}}-\frac {16 a^2 (7 A+5 B) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{105 f}-\frac {2 a (7 A+5 B) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 f}-\frac {2 B \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{7 f} \\ \end{align*}
Time = 2.56 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.86 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx=-\frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} (1246 A+1040 B-6 (7 A+20 B) \cos (2 (e+f x))+(392 A+505 B) \sin (e+f x)-15 B \sin (3 (e+f x)))}{210 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
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Time = 2.00 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.72
method | result | size |
default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (-15 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-21 A -60 B \right ) \left (\cos ^{2}\left (f x +e \right )\right )+\left (98 A +130 B \right ) \sin \left (f x +e \right )+322 A +290 B \right )}{105 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(99\) |
parts | \(\frac {2 A \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (3 \left (\sin ^{2}\left (f x +e \right )\right )+14 \sin \left (f x +e \right )+43\right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}+\frac {2 B \left (1+\sin \left (f x +e \right )\right ) a^{3} \left (\sin \left (f x +e \right )-1\right ) \left (3 \left (\sin ^{3}\left (f x +e \right )\right )+12 \left (\sin ^{2}\left (f x +e \right )\right )+23 \sin \left (f x +e \right )+46\right )}{21 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(142\) |
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Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.38 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx=\frac {2 \, {\left (15 \, B a^{2} \cos \left (f x + e\right )^{4} + 3 \, {\left (7 \, A + 20 \, B\right )} a^{2} \cos \left (f x + e\right )^{3} - {\left (77 \, A + 85 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (161 \, A + 145 \, B\right )} a^{2} \cos \left (f x + e\right ) - 32 \, {\left (7 \, A + 5 \, B\right )} a^{2} + {\left (15 \, B a^{2} \cos \left (f x + e\right )^{3} - 3 \, {\left (7 \, A + 15 \, B\right )} a^{2} \cos \left (f x + e\right )^{2} - 2 \, {\left (49 \, A + 65 \, B\right )} a^{2} \cos \left (f x + e\right ) + 32 \, {\left (7 \, A + 5 \, B\right )} a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{105 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
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\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx=\int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \]
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\[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.46 \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx=\frac {\sqrt {2} {\left (15 \, B a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 525 \, {\left (4 \, A a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, B a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 35 \, {\left (10 \, A a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 11 \, B a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 21 \, {\left (2 \, A a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 5 \, B a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right )\right )} \sqrt {a}}{420 \, f} \]
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Timed out. \[ \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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