Optimal. Leaf size=1071 \[ \text{result too large to display} \]
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Rubi [A] time = 4.99506, antiderivative size = 1071, normalized size of antiderivative = 1., number of steps used = 9, 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))^{5/2}}{4 d f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{(a+b)^{3/2} \left (\left (9 c^3-6 d c^2-156 d^2 c-72 d^3\right ) b^3-a d \left (51 c^2+172 d c+212 d^2\right ) b^2-15 a^2 d^2 (11 c+2 d) b+15 a^3 d^3\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))}{192 b^2 d^2 \sqrt{c+d} f}-\frac{\left (-9 \left (c^2-4 d^2\right ) b^2+54 a c d b+59 a^2 d^2\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}-\frac{\left (-\left (9 c^3-156 c d^2\right ) b^3+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (-\left (9 c^3-156 c d^2\right ) b^3+a d \left (57 c^2+284 d^2\right ) b^2+337 a^2 c d^2 b+15 a^3 d^3\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))}{192 b d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left (3 \left (c^2+4 d^2\right )^2 b^4-20 a c d \left (c^2-12 d^2\right ) b^3+30 a^2 d^2 \left (3 c^2+4 d^2\right ) b^2+60 a^3 c d^3 b-5 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))}{64 b^2 \sqrt{a+b} d^3 f} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3049
Rule 3061
Rule 3053
Rule 2811
Rule 2998
Rule 2818
Rule 2996
Rubi steps
\begin{align*} \int (a+b \sin (e+f x))^{5/2} (c+d \sin (e+f x))^{3/2} \, dx &=-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{(c+d \sin (e+f x))^{3/2} \left (\frac{1}{2} \left (b^3 c+8 a^3 d+5 a b^2 d\right )-b \left (a b c-12 a^2 d-3 b^2 d\right ) \sin (e+f x)-\frac{1}{2} b^2 (3 b c-17 a d) \sin ^2(e+f x)\right )}{\sqrt{a+b \sin (e+f x)}} \, dx}{4 d}\\ &=\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{1}{4} b \left (3 b^3 c^2+48 a^3 c d+38 a b^2 c d+51 a^2 b d^2\right )+\frac{1}{2} b \left (55 a^2 b c d+15 b^3 c d+24 a^3 d^2-a b^2 \left (3 c^2-49 d^2\right )\right ) \sin (e+f x)+\frac{1}{4} b^2 \left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \sin ^2(e+f x)\right )}{\sqrt{a+b \sin (e+f x)}} \, dx}{12 b d}\\ &=-\frac{\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{\frac{1}{8} b^2 \left (317 a^2 b c d^2+a b^2 d \left (197 c^2+36 d^2\right )+a^3 d \left (192 c^2+59 d^2\right )+3 b^3 \left (c^3+12 c d^2\right )\right )+\frac{1}{4} b^2 \left (133 a^3 c d^2-a b^2 c \left (3 c^2-290 d^2\right )+3 b^3 d \left (19 c^2+12 d^2\right )+a^2 b d \left (166 c^2+161 d^2\right )\right ) \sin (e+f x)+\frac{1}{8} b^2 \left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \sin ^2(e+f x)}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx}{24 b^2 d}\\ &=-\frac{\left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 d^2 f \sqrt{a+b \sin (e+f x)}}-\frac{\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{\frac{1}{8} b^2 \left (956 a^3 b c d^3+a^4 d^2 \left (384 c^2+133 d^2\right )+2 a^2 b^2 d^2 \left (57 c^2+178 d^2\right )-4 a b^3 d \left (15 c^3+14 c d^2\right )+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right )+\frac{1}{4} b^2 \left (251 a^4 c d^3-a^2 b^2 c d \left (63 c^2-613 d^2\right )+3 b^4 c d \left (c^2+12 d^2\right )+a^3 b d^2 \left (187 c^2+381 d^2\right )+a b^3 \left (9 c^4+155 c^2 d^2+108 d^4\right )\right ) \sin (e+f x)+\frac{3}{8} b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx}{48 b^2 d^2}\\ &=-\frac{\left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 d^2 f \sqrt{a+b \sin (e+f x)}}-\frac{\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{\int \frac{-\frac{3}{8} a^2 b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac{1}{8} b^4 \left (956 a^3 b c d^3+a^4 d^2 \left (384 c^2+133 d^2\right )+2 a^2 b^2 d^2 \left (57 c^2+178 d^2\right )-4 a b^3 d \left (15 c^3+14 c d^2\right )+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right )+b \left (-\frac{3}{4} a b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac{1}{4} b^3 \left (251 a^4 c d^3-a^2 b^2 c d \left (63 c^2-613 d^2\right )+3 b^4 c d \left (c^2+12 d^2\right )+a^3 b d^2 \left (187 c^2+381 d^2\right )+a b^3 \left (9 c^4+155 c^2 d^2+108 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}{48 b^4 d^2}+\frac{\left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right ) \int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx}{128 b^2 d^2}\\ &=\frac{\sqrt{c+d} \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\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))}{64 b^2 \sqrt{a+b} d^3 f}-\frac{\left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 d^2 f \sqrt{a+b \sin (e+f x)}}-\frac{\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}+\frac{\left (-\frac{3}{8} a^2 b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac{1}{8} b^4 \left (956 a^3 b c d^3+a^4 d^2 \left (384 c^2+133 d^2\right )+2 a^2 b^2 d^2 \left (57 c^2+178 d^2\right )-4 a b^3 d \left (15 c^3+14 c d^2\right )+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right )-b \left (-\frac{3}{4} a b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac{1}{4} b^3 \left (251 a^4 c d^3-a^2 b^2 c d \left (63 c^2-613 d^2\right )+3 b^4 c d \left (c^2+12 d^2\right )+a^3 b d^2 \left (187 c^2+381 d^2\right )+a b^3 \left (9 c^4+155 c^2 d^2+108 d^4\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx}{48 (a-b) b^4 d^2}-\frac{\left (b \left (-\frac{3}{8} a^2 b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac{1}{8} b^4 \left (956 a^3 b c d^3+a^4 d^2 \left (384 c^2+133 d^2\right )+2 a^2 b^2 d^2 \left (57 c^2+178 d^2\right )-4 a b^3 d \left (15 c^3+14 c d^2\right )+3 b^4 \left (3 c^4-52 c^2 d^2\right )\right )\right )-a b \left (-\frac{3}{4} a b^2 \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\right )\right )+\frac{1}{4} b^3 \left (251 a^4 c d^3-a^2 b^2 c d \left (63 c^2-613 d^2\right )+3 b^4 c d \left (c^2+12 d^2\right )+a^3 b d^2 \left (187 c^2+381 d^2\right )+a b^3 \left (9 c^4+155 c^2 d^2+108 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}{48 (a-b) b^4 d^2}\\ &=\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\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))}{192 b d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left (60 a^3 b c d^3-5 a^4 d^4-20 a b^3 c d \left (c^2-12 d^2\right )+3 b^4 \left (c^2+4 d^2\right )^2+30 a^2 b^2 d^2 \left (3 c^2+4 d^2\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))}{64 b^2 \sqrt{a+b} d^3 f}-\frac{\left (337 a^2 b c d^2+15 a^3 d^3+a b^2 d \left (57 c^2+284 d^2\right )-b^3 \left (9 c^3-156 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 d^2 f \sqrt{a+b \sin (e+f x)}}-\frac{\left (54 a b c d+59 a^2 d^2-9 b^2 \left (c^2-4 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 d f}-\frac{(a+b)^{3/2} \left (15 a^3 d^3-15 a^2 b d^2 (11 c+2 d)-a b^2 d \left (51 c^2+172 c d+212 d^2\right )+b^3 \left (9 c^3-6 c^2 d-156 c d^2-72 d^3\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))}{192 b^2 d^2 \sqrt{c+d} f}+\frac{b (3 b c-17 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{24 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2}}{4 d f}\\ \end{align*}
Mathematica [A] time = 7.39697, size = 2061, normalized size = 1.92 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 17.068, size = 577718, normalized size = 539.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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