Optimal. Leaf size=1080 \[ \text{result too large to display} \]
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Rubi [A] time = 5.23808, antiderivative size = 1080, 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{d^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}}{4 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}}{24 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 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^2 d (b c-a d) f}-\frac{\sqrt{c+d} \left (\left (5 c^4-120 d^2 c^2-48 d^4\right ) b^4-60 a c d \left (c^2+4 d^2\right ) b^3-6 a^2 d^2 \left (15 c^2+4 d^2\right ) b^2+20 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))}{64 b^3 \sqrt{a+b} d^2 f}-\frac{\left (\left (59 c^2+36 d^2\right ) b^2+54 a c d b-9 a^2 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)} \sqrt{a+b \sin (e+f x)}}{96 b f}+\frac{(a+b)^{3/2} \left (\left (15 c^3+118 d c^2+284 d^2 c+72 d^3\right ) b^3+3 a d \left (73 c^2+36 d c+28 d^2\right ) b^2-3 a^2 d^2 (17 c+6 d) b+9 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^3 d \sqrt{c+d} f}-\frac{\left (\left (15 c^3+284 d^2 c\right ) b^3+a d \left (337 c^2+156 d^2\right ) b^2+57 a^2 c d^2 b-9 a^3 d^3\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 b d f \sqrt{a+b \sin (e+f x)}} \]
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))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}}{4 b f}+\frac{\int \frac{(a+b \sin (e+f x))^{3/2} \left (\frac{1}{2} \left (8 b c^3+5 b c d^2+a d^3\right )-d \left (a c d-3 b \left (4 c^2+d^2\right )\right ) \sin (e+f x)+\frac{1}{2} d^2 (17 b c-3 a d) \sin ^2(e+f x)\right )}{\sqrt{c+d \sin (e+f x)}} \, dx}{4 b}\\ &=-\frac{d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{24 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}}{4 b f}+\frac{\int \frac{\sqrt{a+b \sin (e+f x)} \left (\frac{1}{4} d \left (51 b^2 c^2 d+3 a^2 d^3+a b \left (48 c^3+38 c d^2\right )\right )-\frac{1}{2} d \left (3 a^2 c d^2-5 a b d \left (11 c^2+3 d^2\right )-b^2 \left (24 c^3+49 c d^2\right )\right ) \sin (e+f x)+\frac{1}{4} d^2 \left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \sin ^2(e+f x)\right )}{\sqrt{c+d \sin (e+f x)}} \, dx}{12 b d}\\ &=-\frac{\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{24 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}}{4 b f}+\frac{\int \frac{\frac{1}{8} d^2 \left (3 a^3 d^3+b^3 c \left (59 c^2+36 d^2\right )+a b^2 d \left (317 c^2+36 d^2\right )+a^2 b c \left (192 c^2+197 d^2\right )\right )-\frac{1}{4} d^2 \left (3 a^3 c d^2-b^3 d \left (161 c^2+36 d^2\right )-a^2 b d \left (166 c^2+57 d^2\right )-a b^2 c \left (133 c^2+290 d^2\right )\right ) \sin (e+f x)+\frac{1}{8} d^2 \left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 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 d^2}\\ &=-\frac{\left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 b d f \sqrt{a+b \sin (e+f x)}}-\frac{\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{24 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}}{4 b f}+\frac{\int \frac{-\frac{1}{8} d^2 \left (3 a^4 d^4-2 a^2 b^2 d^2 \left (457 c^2+114 d^2\right )-4 a^3 b c d \left (96 c^2+115 d^2\right )+4 a b^3 d \left (51 c^3-50 c d^2\right )+b^4 \left (15 c^4+284 c^2 d^2\right )\right )+\frac{1}{4} d^2 \left (3 a^4 c d^3+b^4 c d \left (59 c^2+36 d^2\right )+a^3 b d^2 \left (275 c^2+117 d^2\right )+a^2 b^2 c d \left (121 c^2+621 d^2\right )-a b^3 \left (15 c^4-355 c^2 d^2-108 d^4\right )\right ) \sin (e+f x)-\frac{3}{8} d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\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 d^3}\\ &=-\frac{\left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 b d f \sqrt{a+b \sin (e+f x)}}-\frac{\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{24 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}}{4 b f}+\frac{\int \frac{-\frac{1}{8} b^2 d^2 \left (3 a^4 d^4-2 a^2 b^2 d^2 \left (457 c^2+114 d^2\right )-4 a^3 b c d \left (96 c^2+115 d^2\right )+4 a b^3 d \left (51 c^3-50 c d^2\right )+b^4 \left (15 c^4+284 c^2 d^2\right )\right )+\frac{3}{8} a^2 d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right )+b \left (\frac{1}{4} b d^2 \left (3 a^4 c d^3+b^4 c d \left (59 c^2+36 d^2\right )+a^3 b d^2 \left (275 c^2+117 d^2\right )+a^2 b^2 c d \left (121 c^2+621 d^2\right )-a b^3 \left (15 c^4-355 c^2 d^2-108 d^4\right )\right )+\frac{3}{4} a d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 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^3 d^3}-\frac{\left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right ) \int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx}{128 b^3 d}\\ &=-\frac{\sqrt{c+d} \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 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))}{64 b^3 \sqrt{a+b} d^2 f}-\frac{\left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 b d f \sqrt{a+b \sin (e+f x)}}-\frac{\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{24 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}}{4 b f}+\frac{\left (-\frac{1}{8} b^2 d^2 \left (3 a^4 d^4-2 a^2 b^2 d^2 \left (457 c^2+114 d^2\right )-4 a^3 b c d \left (96 c^2+115 d^2\right )+4 a b^3 d \left (51 c^3-50 c d^2\right )+b^4 \left (15 c^4+284 c^2 d^2\right )\right )+\frac{3}{8} a^2 d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right )-b \left (\frac{1}{4} b d^2 \left (3 a^4 c d^3+b^4 c d \left (59 c^2+36 d^2\right )+a^3 b d^2 \left (275 c^2+117 d^2\right )+a^2 b^2 c d \left (121 c^2+621 d^2\right )-a b^3 \left (15 c^4-355 c^2 d^2-108 d^4\right )\right )+\frac{3}{4} a d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 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^3 d^3}-\frac{\left (-a b \left (\frac{1}{4} b d^2 \left (3 a^4 c d^3+b^4 c d \left (59 c^2+36 d^2\right )+a^3 b d^2 \left (275 c^2+117 d^2\right )+a^2 b^2 c d \left (121 c^2+621 d^2\right )-a b^3 \left (15 c^4-355 c^2 d^2-108 d^4\right )\right )+\frac{3}{4} a d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 d^4\right )\right )\right )+b \left (-\frac{1}{8} b^2 d^2 \left (3 a^4 d^4-2 a^2 b^2 d^2 \left (457 c^2+114 d^2\right )-4 a^3 b c d \left (96 c^2+115 d^2\right )+4 a b^3 d \left (51 c^3-50 c d^2\right )+b^4 \left (15 c^4+284 c^2 d^2\right )\right )+\frac{3}{8} a^2 d^2 \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 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^3 d^3}\\ &=\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 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^2 d (b c-a d) f}-\frac{\sqrt{c+d} \left (20 a^3 b c d^3-3 a^4 d^4-60 a b^3 c d \left (c^2+4 d^2\right )-6 a^2 b^2 d^2 \left (15 c^2+4 d^2\right )+b^4 \left (5 c^4-120 c^2 d^2-48 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))}{64 b^3 \sqrt{a+b} d^2 f}-\frac{\left (57 a^2 b c d^2-9 a^3 d^3+a b^2 d \left (337 c^2+156 d^2\right )+b^3 \left (15 c^3+284 c d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{192 b d f \sqrt{a+b \sin (e+f x)}}-\frac{\left (54 a b c d-9 a^2 d^2+b^2 \left (59 c^2+36 d^2\right )\right ) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{96 b f}-\frac{d (17 b c-3 a d) \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{24 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{5/2} \sqrt{c+d \sin (e+f x)}}{4 b f}+\frac{(a+b)^{3/2} \left (9 a^3 d^3-3 a^2 b d^2 (17 c+6 d)+3 a b^2 d \left (73 c^2+36 c d+28 d^2\right )+b^3 \left (15 c^3+118 c^2 d+284 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^3 d \sqrt{c+d} f}\\ \end{align*}
Mathematica [A] time = 7.47473, size = 2061, normalized size = 1.91 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 20.865, size = 577146, normalized size = 534.4 \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{3}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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