Optimal. Leaf size=642 \[ \frac {2 b \left (-297 a^2 d^2+66 a b c d-\left (b^2 \left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac {2 \left (693 a^3 d^3+1485 a^2 b c d^2-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3465 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f} \]
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Rubi [A] time = 1.40, antiderivative size = 642, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac {2 b \left (-297 a^2 d^2+66 a b c d+b^2 \left (-\left (8 c^2+81 d^2\right )\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}-\frac {2 \left (495 a^2 b d^2 \left (3 c^2+5 d^2\right )+1848 a^3 c d^3-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (57 c^2 d^2+8 c^4+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (c^2-d^2\right ) \left (495 a^2 b d^2 \left (3 c^2+5 d^2\right )+1848 a^3 c d^3-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (57 c^2 d^2+8 c^4+135 d^4\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3465 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 \left (495 a^2 b c d^2 \left (3 c^2+29 d^2\right )+231 a^3 d^3 \left (23 c^2+9 d^2\right )-33 a b^2 d \left (-279 c^2 d^2+10 c^4-147 d^4\right )+5 b^3 \left (51 c^3 d^2+8 c^5+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2753
Rule 2793
Rule 3023
Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^3 (c+d \sin (e+f x))^{5/2} \, dx &=-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \int (c+d \sin (e+f x))^{5/2} \left (\frac {1}{2} \left (2 b^3 c+11 a^3 d+7 a b^2 d\right )-\frac {1}{2} b \left (2 a b c-33 a^2 d-9 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-6 a d) \sin ^2(e+f x)\right ) \, dx}{11 d}\\ &=\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {4 \int (c+d \sin (e+f x))^{5/2} \left (-\frac {1}{4} d \left (10 b^3 c-99 a^3 d-231 a b^2 d\right )-\frac {1}{4} b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{99 d^2}\\ &=\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {8 \int (c+d \sin (e+f x))^{3/2} \left (\frac {3}{8} d \left (231 a^3 c d+429 a b^2 c d+495 a^2 b d^2-5 b^3 \left (2 c^2-27 d^2\right )\right )+\frac {1}{8} \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \sin (e+f x)\right ) \, dx}{693 d^2}\\ &=-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {16 \int \sqrt {c+d \sin (e+f x)} \left (\frac {3}{16} d \left (3960 a^2 b c d^2+231 a^3 d \left (5 c^2+3 d^2\right )+33 a b^2 d \left (55 c^2+49 d^2\right )-10 b^3 \left (c^3-101 c d^2\right )\right )+\frac {3}{16} \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \sin (e+f x)\right ) \, dx}{3465 d^2}\\ &=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {32 \int \frac {\frac {3}{32} d \left (495 a^2 b d^2 \left (27 c^2+5 d^2\right )+231 a^3 c d \left (15 c^2+17 d^2\right )+33 a b^2 d \left (155 c^3+261 c d^2\right )+5 b^3 \left (2 c^4+663 c^2 d^2+135 d^4\right )\right )+\frac {3}{32} \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{10395 d^2}\\ &=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}-\frac {\left (\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3465 d^3}+\frac {\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{3465 d^3}\\ &=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {\left (\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3465 d^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {\left (\left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\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}{3465 d^3 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3465 d^2 f}-\frac {2 \left (1485 a^2 b c d^2+693 a^3 d^3-33 a b^2 d \left (10 c^2-49 d^2\right )+5 b^3 \left (8 c^3+67 c d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{3465 d^2 f}+\frac {2 b \left (66 a b c d-297 a^2 d^2-b^2 \left (8 c^2+81 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{693 d^2 f}+\frac {8 b^2 (b c-6 a d) \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{99 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{7/2}}{11 d f}+\frac {2 \left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )-33 a b^2 d \left (10 c^4-279 c^2 d^2-147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\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)}}{3465 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (c^2-d^2\right ) \left (1848 a^3 c d^3+495 a^2 b d^2 \left (3 c^2+5 d^2\right )-66 a b^2 d \left (5 c^3-57 c d^2\right )+5 b^3 \left (8 c^4+57 c^2 d^2+135 d^4\right )\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}}}{3465 d^3 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.61, size = 545, normalized size = 0.85 \[ \frac {d (c+d \sin (e+f x)) \left (5 b d^2 \left (1188 a^2 d^2+2508 a b c d+b^2 \left (452 c^2+513 d^2\right )\right ) \cos (3 (e+f x))-4 d \left (1386 a^3 d^3+8910 a^2 b c d^2+33 a b^2 d \left (150 c^2+133 d^2\right )+5 b^3 \left (6 c^3+619 c d^2\right )\right ) \sin (2 (e+f x))+2 \left (-20328 a^3 c d^3-990 a^2 b d^2 \left (36 c^2+23 d^2\right )-66 a b^2 d \left (20 c^3+747 c d^2\right )+5 b^3 \left (32 c^4-1866 c^2 d^2-1305 d^4\right )\right ) \cos (e+f x)+70 b^2 d^3 (33 a d+23 b c) \sin (4 (e+f x))-315 b^3 d^4 \cos (5 (e+f x))\right )-16 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \left (d^2 \left (231 a^3 c d \left (15 c^2+17 d^2\right )+495 a^2 b d^2 \left (27 c^2+5 d^2\right )+33 a b^2 d \left (155 c^3+261 c d^2\right )+5 b^3 \left (2 c^4+663 c^2 d^2+135 d^4\right )\right ) F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )+\left (231 a^3 d^3 \left (23 c^2+9 d^2\right )+495 a^2 b c d^2 \left (3 c^2+29 d^2\right )+33 a b^2 d \left (-10 c^4+279 c^2 d^2+147 d^4\right )+5 b^3 \left (8 c^5+51 c^3 d^2+741 c d^4\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )\right )}{27720 d^3 f \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left ({\left (2 \, b^{3} c d + 3 \, a b^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} + {\left (a^{3} + 3 \, a b^{2}\right )} c^{2} + 2 \, {\left (3 \, a^{2} b + b^{3}\right )} c d + {\left (a^{3} + 3 \, a b^{2}\right )} d^{2} - {\left (3 \, a b^{2} c^{2} + 2 \, {\left (3 \, a^{2} b + 2 \, b^{3}\right )} c d + {\left (a^{3} + 6 \, a b^{2}\right )} d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (b^{3} d^{2} \cos \left (f x + e\right )^{4} + {\left (3 \, a^{2} b + b^{3}\right )} c^{2} + 2 \, {\left (a^{3} + 3 \, a b^{2}\right )} c d + {\left (3 \, a^{2} b + b^{3}\right )} d^{2} - {\left (b^{3} c^{2} + 6 \, a b^{2} c d + {\left (3 \, a^{2} b + 2 \, b^{3}\right )} d^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {d \sin \left (f x + e\right ) + c}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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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maple [B] time = 9.40, size = 2728, normalized size = 4.25 \[ \text {Expression too large to display} \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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