Optimal. Leaf size=894 \[ -\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} b^2}{3 d f}+\frac{(3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} b}{12 d f}-\frac{\left (-\left (3 c^2-16 d^2\right ) b^2+14 a c d b+33 a^2 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)} b}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (-\left (3 c^2-16 d^2\right ) b^2+14 a c d b+33 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left (\left (c^3+4 d^2 c\right ) b^3-5 a d \left (c^2-4 d^2\right ) b^2+15 a^2 c d^2 b+5 a^3 d^3\right ) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 \sqrt{a+b} d^3 f b}+\frac{(a+b)^{3/2} \left (-\left (3 c^2-2 d c-16 d^2\right ) b^2+6 a d (2 c+3 d) b+15 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 d^2 \sqrt{c+d} f b} \]
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Rubi [A] time = 3.2685, antiderivative size = 894, 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} \[ -\frac{\cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2} b^2}{3 d f}+\frac{(3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)} b}{12 d f}-\frac{\left (-\left (3 c^2-16 d^2\right ) b^2+14 a c d b+33 a^2 d^2\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)} b}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{\sqrt{a+b} (c-d) \sqrt{c+d} \left (-\left (3 c^2-16 d^2\right ) b^2+14 a c d b+33 a^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left (\left (c^3+4 d^2 c\right ) b^3-5 a d \left (c^2-4 d^2\right ) b^2+15 a^2 c d^2 b+5 a^3 d^3\right ) \Pi \left (\frac{b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt{-\frac{(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt{\frac{(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 \sqrt{a+b} d^3 f b}+\frac{(a+b)^{3/2} \left (-\left (3 c^2-2 d c-16 d^2\right ) b^2+6 a d (2 c+3 d) b+15 a^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sin (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sin (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt{\frac{(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt{-\frac{(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 d^2 \sqrt{c+d} f b} \]
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} \sqrt{c+d \sin (e+f x)} \, dx &=-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac{\int \frac{\sqrt{c+d \sin (e+f x)} \left (\frac{1}{2} \left (b^3 c+6 a^3 d+3 a b^2 d\right )-b \left (a b c-9 a^2 d-2 b^2 d\right ) \sin (e+f x)-\frac{1}{2} b^2 (3 b c-13 a d) \sin ^2(e+f x)\right )}{\sqrt{a+b \sin (e+f x)}} \, dx}{3 d}\\ &=\frac{b (3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac{\int \frac{\frac{1}{4} b \left (b^3 c^2+24 a^3 c d+22 a b^2 c d+13 a^2 b d^2\right )+\frac{1}{2} b \left (23 a^2 b c d+7 b^3 c d+12 a^3 d^2-a b^2 \left (c^2-19 d^2\right )\right ) \sin (e+f x)+\frac{1}{4} b^2 \left (14 a b c d+33 a^2 d^2-b^2 \left (3 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{b \left (14 a b c d+33 a^2 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{b (3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac{\int \frac{\frac{1}{4} b \left (48 a^4 c d^2+25 a^2 b^2 c d^2+59 a^3 b d^3+b^4 \left (3 c^3-16 c d^2\right )-a b^3 \left (15 c^2 d-16 d^3\right )\right )+\frac{1}{2} b \left (b^4 c^2 d+37 a^3 b c d^2+24 a^4 d^3-a^2 b^2 d \left (16 c^2-51 d^2\right )+a b^3 c \left (3 c^2+20 d^2\right )\right ) \sin (e+f x)+\frac{3}{4} b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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{b \left (14 a b c d+33 a^2 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{b (3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac{\int \frac{-\frac{3}{4} a^2 b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right )+\frac{1}{4} b^3 \left (48 a^4 c d^2+25 a^2 b^2 c d^2+59 a^3 b d^3+b^4 \left (3 c^3-16 c d^2\right )-a b^3 \left (15 c^2 d-16 d^3\right )\right )+b \left (\frac{1}{2} b^2 \left (b^4 c^2 d+37 a^3 b c d^2+24 a^4 d^3-a^2 b^2 d \left (16 c^2-51 d^2\right )+a b^3 c \left (3 c^2+20 d^2\right )\right )-\frac{3}{2} a b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 d^2}\\ &=\frac{\sqrt{c+d} \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 \sqrt{a+b} d^3 f}-\frac{b \left (14 a b c d+33 a^2 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{b (3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 d f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac{\left (-\frac{3}{4} a^2 b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right )+\frac{1}{4} b^3 \left (48 a^4 c d^2+25 a^2 b^2 c d^2+59 a^3 b d^3+b^4 \left (3 c^3-16 c d^2\right )-a b^3 \left (15 c^2 d-16 d^3\right )\right )-b \left (\frac{1}{2} b^2 \left (b^4 c^2 d+37 a^3 b c d^2+24 a^4 d^3-a^2 b^2 d \left (16 c^2-51 d^2\right )+a b^3 c \left (3 c^2+20 d^2\right )\right )-\frac{3}{2} a b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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^2 \left (b^4 c^2 d+37 a^3 b c d^2+24 a^4 d^3-a^2 b^2 d \left (16 c^2-51 d^2\right )+a b^3 c \left (3 c^2+20 d^2\right )\right )-\frac{3}{2} a b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right )\right )+b \left (-\frac{3}{4} a^2 b^2 \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+b^3 \left (c^3+4 c d^2\right )\right )+\frac{1}{4} b^3 \left (48 a^4 c d^2+25 a^2 b^2 c d^2+59 a^3 b d^3+b^4 \left (3 c^3-16 c d^2\right )-a b^3 \left (15 c^2 d-16 d^3\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+33 a^2 d^2-b^2 \left (3 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 d^2 (b c-a d) f}+\frac{\sqrt{c+d} \left (15 a^2 b c d^2+5 a^3 d^3-5 a b^2 d \left (c^2-4 d^2\right )+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 \sqrt{a+b} d^3 f}-\frac{b \left (14 a b c d+33 a^2 d^2-b^2 \left (3 c^2-16 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{24 d^2 f \sqrt{a+b \sin (e+f x)}}+\frac{b (3 b c-13 a d) \cos (e+f x) \sqrt{a+b \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}{12 d f}+\frac{(a+b)^{3/2} \left (15 a^2 d^2+6 a b d (2 c+3 d)-b^2 \left (3 c^2-2 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 d^2 \sqrt{c+d} f}-\frac{b^2 \cos (e+f x) \sqrt{a+b \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}\\ \end{align*}
Mathematica [B] time = 7.04505, size = 1949, normalized size = 2.18 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 10.796, size = 409146, normalized size = 457.7 \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}} \sqrt{d \sin \left (f x + e\right ) + c}\,{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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