Optimal. Leaf size=1295 \[ -\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {\left (-\left (\left (15 c^2-64 d^2\right ) b^2\right )+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {(a+b)^{3/2} \left (-\left (\left (45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right ) b^4\right )+2 a d \left (165 c^3+917 d c^2+2392 d^2 c+516 d^3\right ) b^3+30 a^2 d^2 \left (64 c^2+23 d c+22 d^2\right ) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt {c+d} f}-\frac {\left (-\left (\left (45 c^3-516 c d^2\right ) b^3\right )+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a d \left (45 c^3+791 d^2 c\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4\right )+8 a d \left (45 c^3+791 d^2 c\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{1920 b^2 d^2 (b c-a d) f}-\frac {\sqrt {c+d} (b c+a d) \left (-\left (\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4\right )+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt {a+b} d^3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 7.84, antiderivative size = 1295, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2793, 3049, 3061, 3053, 2811, 2998, 2818, 2996} \[ -\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {\left (-\left (15 c^2-64 d^2\right ) b^2+110 a c d b+93 a^2 d^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {(a+b)^{3/2} \left (-\left (45 c^4-30 d c^3-1692 d^2 c^2-1544 d^3 c-1024 d^4\right ) b^4+2 a d \left (165 c^3+917 d c^2+2392 d^2 c+516 d^3\right ) b^3+30 a^2 d^2 \left (64 c^2+23 d c+22 d^2\right ) b^2-30 a^3 d^3 (11 c+3 d) b+45 a^4 d^4\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt {c+d} f}-\frac {\left (-\left (45 c^3-516 c d^2\right ) b^3+a d \left (345 c^2+772 d^2\right ) b^2+917 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (-\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4+8 a d \left (45 c^3+791 d^2 c\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (-\left (45 c^4-1692 d^2 c^2-1024 d^4\right ) b^4+8 a d \left (45 c^3+791 d^2 c\right ) b^3+2 a^2 d^2 \left (1877 c^2+846 d^2\right ) b^2+360 a^3 c d^3 b-45 a^4 d^4\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{1920 b^2 d^2 (b c-a d) f}-\frac {\sqrt {c+d} (b c+a d) \left (-\left (3 c^4+40 d^2 c^2+240 d^4\right ) b^4+28 a c d \left (c^2-20 d^2\right ) b^3-2 a^2 d^2 \left (89 c^2+20 d^2\right ) b^2+28 a^3 c d^3 b-3 a^4 d^4\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt {a+b} d^3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2793
Rule 2811
Rule 2818
Rule 2996
Rule 2998
Rule 3049
Rule 3053
Rule 3061
Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {(c+d \sin (e+f x))^{5/2} \left (\frac {1}{2} \left (b^3 c+10 a^3 d+7 a b^2 d\right )-b \left (a b c-15 a^2 d-4 b^2 d\right ) \sin (e+f x)-\frac {3}{2} b^2 (b c-7 a d) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{5 d}\\ &=\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {(c+d \sin (e+f x))^{3/2} \left (\frac {1}{4} b \left (5 b^3 c^2+80 a^3 c d+62 a b^2 c d+105 a^2 b d^2\right )+\frac {1}{2} b \left (99 a^2 b c d+27 b^3 c d+40 a^3 d^2-a b^2 \left (5 c^2-91 d^2\right )\right ) \sin (e+f x)+\frac {1}{4} b^2 \left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{20 b d}\\ &=-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {\sqrt {c+d \sin (e+f x)} \left (\frac {1}{8} b^2 \left (1053 a^2 b c d^2+3 a^3 d \left (160 c^2+93 d^2\right )+a b^2 d \left (437 c^2+192 d^2\right )+b^3 \left (15 c^3+64 c d^2\right )\right )+\frac {1}{4} b^2 \left (387 a^3 c d^2-3 a b^2 c \left (5 c^2-296 d^2\right )+b^3 d \left (147 c^2+128 d^2\right )+a^2 b d \left (484 c^2+501 d^2\right )\right ) \sin (e+f x)+\frac {1}{8} b^2 \left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{60 b^2 d}\\ &=-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {\frac {1}{16} b^2 \left (15 a^4 d^4+128 a^3 b c d \left (15 c^2+16 d^2\right )+2 a^2 b^2 d^2 \left (2737 c^2+386 d^2\right )+8 a b^3 d \left (256 c^3+257 c d^2\right )+b^4 \left (15 c^4+772 c^2 d^2\right )\right )-\frac {1}{8} b^2 (b c+a d) \left (15 a b^2 c^3-1606 a^2 b c^2 d-573 b^3 c^2 d+15 a^3 c d^2-3682 a b^2 c d^2-573 a^2 b d^3-1156 b^3 d^3\right ) \sin (e+f x)+\frac {1}{16} b^2 \left (360 a^3 b c d^3-45 a^4 d^4+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )+8 a b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{120 b^3 d}\\ &=-\frac {\left (360 a^3 b c d^3-45 a^4 d^4+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )+8 a b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {-\frac {1}{16} b^2 \left (15 a^5 d^5-54 a^2 b^3 c d^2 \left (13 c^2+162 d^2\right )-2 a^3 b^2 d^3 \left (7171 c^2+1618 d^2\right )-a^4 b c d^2 \left (3840 c^2+4501 d^2\right )+a b^4 d \left (375 c^4+3092 c^2 d^2-1024 d^4\right )-b^5 \left (45 c^5-1692 c^3 d^2-1024 c d^4\right )\right )+\frac {1}{8} b^2 \left (15 a^5 c d^4+b^5 c^2 d \left (15 c^2+772 d^2\right )+a^4 b d^3 \left (2822 c^2+1161 d^2\right )+2 a^3 b^2 c d^2 \left (674 c^2+4433 d^2\right )-6 a^2 b^3 d \left (65 c^4-1276 c^2 d^2-514 d^4\right )+a b^4 c \left (45 c^4+1502 c^2 d^2+3344 d^4\right )\right ) \sin (e+f x)+\frac {15}{16} b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{240 b^3 d^2}\\ &=-\frac {\left (360 a^3 b c d^3-45 a^4 d^4+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )+8 a b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}+\frac {\int \frac {-\frac {15}{16} a^2 b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right )-\frac {1}{16} b^4 \left (15 a^5 d^5-54 a^2 b^3 c d^2 \left (13 c^2+162 d^2\right )-2 a^3 b^2 d^3 \left (7171 c^2+1618 d^2\right )-a^4 b c d^2 \left (3840 c^2+4501 d^2\right )+a b^4 d \left (375 c^4+3092 c^2 d^2-1024 d^4\right )-b^5 \left (45 c^5-1692 c^3 d^2-1024 c d^4\right )\right )+b \left (-\frac {15}{8} a b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right )+\frac {1}{8} b^3 \left (15 a^5 c d^4+b^5 c^2 d \left (15 c^2+772 d^2\right )+a^4 b d^3 \left (2822 c^2+1161 d^2\right )+2 a^3 b^2 c d^2 \left (674 c^2+4433 d^2\right )-6 a^2 b^3 d \left (65 c^4-1276 c^2 d^2-514 d^4\right )+a b^4 c \left (45 c^4+1502 c^2 d^2+3344 d^4\right )\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{240 b^5 d^2}-\frac {\left ((b c+a d) \left (28 a^3 b c d^3-3 a^4 d^4+28 a b^3 c d \left (c^2-20 d^2\right )-2 a^2 b^2 d^2 \left (89 c^2+20 d^2\right )-b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{256 b^3 d^2}\\ &=-\frac {\sqrt {c+d} (b c+a d) \left (28 a^3 b c d^3-3 a^4 d^4+28 a b^3 c d \left (c^2-20 d^2\right )-2 a^2 b^2 d^2 \left (89 c^2+20 d^2\right )-b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt {a+b} d^3 f}-\frac {\left (360 a^3 b c d^3-45 a^4 d^4+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )+8 a b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}-\frac {\left ((a+b) \left (45 a^5 d^5-15 a^4 b d^4 (25 c+6 d)+30 a^3 b^2 d^3 \left (75 c^2+26 c d+22 d^2\right )-2 a^2 b^3 d^2 \left (795 c^3-572 c^2 d-2062 c d^2-516 d^3\right )+b^5 c \left (45 c^4-30 c^3 d-1692 c^2 d^2-1544 c d^3-1024 d^4\right )-a b^4 d \left (375 c^4+1804 c^3 d+3092 c^2 d^2-512 c d^3-1024 d^4\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{3840 b^3 d^2}+-\frac {\left (-a b \left (-\frac {15}{8} a b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right )+\frac {1}{8} b^3 \left (15 a^5 c d^4+b^5 c^2 d \left (15 c^2+772 d^2\right )+a^4 b d^3 \left (2822 c^2+1161 d^2\right )+2 a^3 b^2 c d^2 \left (674 c^2+4433 d^2\right )-6 a^2 b^3 d \left (65 c^4-1276 c^2 d^2-514 d^4\right )+a b^4 c \left (45 c^4+1502 c^2 d^2+3344 d^4\right )\right )\right )+b \left (-\frac {15}{16} a^2 b^2 (b c+a d) \left (3 b^4 c^4-28 a b^3 c^3 d+178 a^2 b^2 c^2 d^2+40 b^4 c^2 d^2-28 a^3 b c d^3+560 a b^3 c d^3+3 a^4 d^4+40 a^2 b^2 d^4+240 b^4 d^4\right )-\frac {1}{16} b^4 \left (15 a^5 d^5-54 a^2 b^3 c d^2 \left (13 c^2+162 d^2\right )-2 a^3 b^2 d^3 \left (7171 c^2+1618 d^2\right )-a^4 b c d^2 \left (3840 c^2+4501 d^2\right )+a b^4 d \left (375 c^4+3092 c^2 d^2-1024 d^4\right )-b^5 \left (45 c^5-1692 c^3 d^2-1024 c d^4\right )\right )\right )\right ) \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{240 (a-b) b^5 d^2}\\ &=\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (360 a^3 b c d^3-45 a^4 d^4+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )+8 a b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{1920 b^2 d^2 (b c-a d) f}-\frac {\sqrt {c+d} (b c+a d) \left (28 a^3 b c d^3-3 a^4 d^4+28 a b^3 c d \left (c^2-20 d^2\right )-2 a^2 b^2 d^2 \left (89 c^2+20 d^2\right )-b^4 \left (3 c^4+40 c^2 d^2+240 d^4\right )\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{128 b^3 \sqrt {a+b} d^3 f}-\frac {\left (360 a^3 b c d^3-45 a^4 d^4+2 a^2 b^2 d^2 \left (1877 c^2+846 d^2\right )+8 a b^3 d \left (45 c^3+791 c d^2\right )-b^4 \left (45 c^4-1692 c^2 d^2-1024 d^4\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{1920 b d^2 f \sqrt {a+b \sin (e+f x)}}-\frac {\left (917 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (345 c^2+772 d^2\right )-b^3 \left (45 c^3-516 c d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{960 b d f}+\frac {(a+b)^{3/2} \left (45 a^4 d^4-30 a^3 b d^3 (11 c+3 d)+30 a^2 b^2 d^2 \left (64 c^2+23 c d+22 d^2\right )+2 a b^3 d \left (165 c^3+917 c^2 d+2392 c d^2+516 d^3\right )-b^4 \left (45 c^4-30 c^3 d-1692 c^2 d^2-1544 c d^3-1024 d^4\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{1920 b^3 d^2 \sqrt {c+d} f}-\frac {\left (110 a b c d+93 a^2 d^2-b^2 \left (15 c^2-64 d^2\right )\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{240 d f}+\frac {3 b (b c-7 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{40 d f}-\frac {b^2 \cos (e+f x) \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{7/2}}{5 d f}\\ \end {align*}
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Mathematica [A] time = 8.47, size = 2276, normalized size = 1.76 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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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(-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 = 79.80, size = 755109, normalized size = 583.10 \[ \text {output too large to display} \]
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 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^{5/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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