Optimal. Leaf size=888 \[ -\frac{\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} d^2}{3 b f}-\frac{(13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} d}{12 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}+\frac{(a+b)^{3/2} \left (\left (33 c^2+26 d c+16 d^2\right ) b^2-6 a d (2 c+d) b+3 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt{c+d} f}-\frac{\left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{c+d} \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt{a+b} f d} \]
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Rubi [A] time = 3.38603, antiderivative size = 888, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)} d^2}{3 b f}-\frac{(13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} d}{12 b f}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}+\frac{(a+b)^{3/2} \left (\left (33 c^2+26 d c+16 d^2\right ) b^2-6 a d (2 c+d) b+3 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt{c+d} f}-\frac{\left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{\sqrt{c+d} \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt{a+b} f d} \]
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 \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{3 b f}+\frac{\int \frac{\sqrt{a+b \sin (e+f x)} \left (\frac{1}{2} \left (a d^3+3 b c \left (2 c^2+d^2\right )\right )+d \left (9 b c^2-a c d+2 b d^2\right ) \sin (e+f x)+\frac{1}{2} d^2 (13 b c-3 a d) \sin ^2(e+f x)\right )}{\sqrt{c+d \sin (e+f x)}} \, dx}{3 b}\\ &=-\frac{d (13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{3 b f}+\frac{\int \frac{\frac{1}{4} d \left (13 b^2 c^2 d+a^2 d^3+a b \left (24 c^3+22 c d^2\right )\right )-\frac{1}{2} d \left (a^2 c d^2-a b d \left (23 c^2+7 d^2\right )-b^2 \left (12 c^3+19 c d^2\right )\right ) \sin (e+f x)+\frac{1}{4} d^2 \left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \sin ^2(e+f x)}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx}{6 b d}\\ &=-\frac{\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{d (13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{3 b f}+\frac{\int \frac{-\frac{1}{4} d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac{1}{2} d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right ) \sin (e+f x)-\frac{3}{4} d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c 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}{12 b d^2}\\ &=-\frac{\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{d (13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{3 b f}+\frac{\int \frac{-\frac{1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac{3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )+b \left (\frac{1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac{3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}} \, dx}{12 b^3 d^2}-\frac{\left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \int \frac{\sqrt{a+b \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx}{16 b^3}\\ &=-\frac{\sqrt{c+d} \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c 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))}{8 b^3 \sqrt{a+b} d f}-\frac{\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{d (13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{3 b f}+\frac{\left (-\frac{1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac{3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )-b \left (\frac{1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac{3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}} \, dx}{12 (a-b) b^3 d^2}-\frac{\left (-a b \left (\frac{1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac{3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right )+b \left (-\frac{1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac{3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\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}{12 (a-b) b^3 d^2}\\ &=\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 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))}{24 b^2 (b c-a d) f}-\frac{\sqrt{c+d} \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c 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))}{8 b^3 \sqrt{a+b} d f}-\frac{\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 b f \sqrt{a+b \sin (e+f x)}}-\frac{d (13 b c-3 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 b f}-\frac{d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt{c+d \sin (e+f x)}}{3 b f}+\frac{(a+b)^{3/2} \left (3 a^2 d^2-6 a b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\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))}{24 b^3 \sqrt{c+d} f}\\ \end{align*}
Mathematica [B] time = 7.12279, size = 1948, normalized size = 2.19 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 9.008, size = 404333, normalized size = 455.3 \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 \sqrt{b \sin \left (f x + e\right ) + a}{\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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