Optimal. Leaf size=378 \[ \frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{315 d^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f} \]
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Rubi [A] time = 0.67, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2763, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {4 a^2 \left (-45 c^2 d+5 c^3-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (c^2-d^2\right ) \left (-45 c^2 d+5 c^3-141 c d^2-75 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{315 d^2 f \sqrt {c+d \sin (e+f x)}}-\frac {4 a^2 \left (-381 c^2 d^2-45 c^3 d+5 c^4-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2763
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^{5/2} \, dx &=-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {2 \int \left (8 a^2 d-a^2 (c-9 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2} \, dx}{9 d}\\ &=\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {4 \int (c+d \sin (e+f x))^{3/2} \left (\frac {3}{2} a^2 d (17 c+15 d)-\frac {1}{2} a^2 \left (5 c (c-9 d)-56 d^2\right ) \sin (e+f x)\right ) \, dx}{63 d}\\ &=\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {8 \int \sqrt {c+d \sin (e+f x)} \left (6 a^2 d \left (10 c^2+15 c d+7 d^2\right )-\frac {3}{4} a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \sin (e+f x)\right ) \, dx}{315 d}\\ &=\frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {16 \int \frac {\frac {3}{8} a^2 d \left (235 c^3+405 c^2 d+309 c d^2+75 d^3\right )-\frac {3}{8} a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{945 d}\\ &=\frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}+\frac {\left (2 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{315 d^2}-\frac {\left (2 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{315 d^2}\\ &=\frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {\left (2 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{315 d^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (2 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{315 d^2 \sqrt {c+d \sin (e+f x)}}\\ &=\frac {4 a^2 \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{315 d f}+\frac {4 a^2 \left (5 c (c-9 d)-56 d^2\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{315 d f}+\frac {4 a^2 (c-9 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{63 d f}-\frac {2 a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{9 d f}-\frac {4 a^2 \left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{315 d^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {4 a^2 \left (c^2-d^2\right ) \left (5 c^3-45 c^2 d-141 c d^2-75 d^3\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{315 d^2 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 1.98, size = 322, normalized size = 0.85 \[ \frac {a^2 (\sin (e+f x)+1)^2 \left (16 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \left (\left (5 c^4-45 c^3 d-381 c^2 d^2-435 c d^3-168 d^4\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )-d^2 \left (235 c^3+405 c^2 d+309 c d^2+75 d^3\right ) F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )-d (c+d \sin (e+f x)) \left (2 d \left (\sin (2 (e+f x)) \left (150 c^2+540 c d-35 d^2 \cos (2 (e+f x))+259 d^2\right )-5 d (19 c+18 d) \cos (3 (e+f x))\right )+2 \left (20 c^3+1080 c^2 d+1671 c d^2+690 d^3\right ) \cos (e+f x)\right )\right )}{1260 d^2 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^4 \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} d^{2} \cos \left (f x + e\right )^{4} + 2 \, a^{2} c^{2} + 4 \, a^{2} c d + 2 \, a^{2} d^{2} - {\left (a^{2} c^{2} + 4 \, a^{2} c d + 3 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (a^{2} c^{2} + 2 \, a^{2} c d + a^{2} d^{2} - {\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.59, size = 1614, normalized size = 4.27 \[ \text {result too large to display} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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 \[ a^{2} \left (\int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx + \int 2 c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int c^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 2 d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{4}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}\, dx + \int 4 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{2}{\left (e + f x \right )}\, dx + \int 2 c d \sqrt {c + d \sin {\left (e + f x \right )}} \sin ^{3}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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