Optimal. Leaf size=375 \[ \frac{2 b \left (-105 a^2 d^2+42 a b c d+b^2 \left (-\left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{2 b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d+b^2 \left (-\left (8 c^2+25 d^2\right )\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f} \]
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Rubi [A] time = 0.677772, antiderivative size = 375, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2793, 3023, 2753, 2752, 2663, 2661, 2655, 2653} \[ \frac{2 b \left (-105 a^2 d^2+42 a b c d+b^2 \left (-\left (8 c^2+25 d^2\right )\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{2 b \left (c^2-d^2\right ) \left (-105 a^2 d^2+42 a b c d+b^2 \left (-\left (8 c^2+25 d^2\right )\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \sqrt{c+d \sin (e+f x)}}+\frac{2 \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2753
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int (a+b \sin (e+f x))^3 \sqrt{c+d \sin (e+f x)} \, dx &=-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac{2 \int \sqrt{c+d \sin (e+f x)} \left (\frac{1}{2} \left (2 b^3 c+7 a^3 d+3 a b^2 d\right )-\frac{1}{2} b \left (2 a b c-21 a^2 d-5 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-4 a d) \sin ^2(e+f x)\right ) \, dx}{7 d}\\ &=\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac{4 \int \sqrt{c+d \sin (e+f x)} \left (-\frac{1}{4} d \left (2 b^3 c-35 a^3 d-63 a b^2 d\right )-\frac{1}{4} b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{35 d^2}\\ &=\frac{2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac{8 \int \frac{\frac{1}{8} d \left (105 a^3 c d+147 a b^2 c d+105 a^2 b d^2+b^3 \left (2 c^2+25 d^2\right )\right )+\frac{1}{8} \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d^2}\\ &=\frac{2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac{\left (b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{105 d^3}+\frac{\left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{105 d^3}\\ &=\frac{2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac{\left (\left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \sqrt{c+d \sin (e+f x)}\right ) \int \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}} \, dx}{105 d^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left (b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{\sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{105 d^3 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{2 b \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{105 d^2 f}+\frac{8 b^2 (b c-4 a d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{35 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) (c+d \sin (e+f x))^{3/2}}{7 d f}+\frac{2 \left (105 a^2 b c d^2+105 a^3 d^3-21 a b^2 d \left (2 c^2-9 d^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) E\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{c+d \sin (e+f x)}}{105 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{2 b \left (c^2-d^2\right ) \left (42 a b c d-105 a^2 d^2-b^2 \left (8 c^2+25 d^2\right )\right ) F\left (\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}{105 d^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.41046, size = 306, normalized size = 0.82 \[ \frac{b d (c+d \sin (e+f x)) \left (\left (-420 a^2 d^2-84 a b c d+b^2 \left (16 c^2-115 d^2\right )\right ) \cos (e+f x)+3 b d (5 b d \cos (3 (e+f x))-2 (21 a d+b c) \sin (2 (e+f x)))\right )-4 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (d^2 \left (105 a^2 b d^2+105 a^3 c d+147 a b^2 c d+b^3 \left (2 c^2+25 d^2\right )\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )+\left (105 a^2 b c d^2+105 a^3 d^3+21 a b^2 d \left (9 d^2-2 c^2\right )+b^3 \left (8 c^3+19 c d^2\right )\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )\right )}{210 d^3 f \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.738, size = 1561, normalized size = 4.2 \begin{align*} \text{result 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 )}^{3} \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{d \sin \left (f x + e\right ) + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (e + f x \right )}\right )^{3} \sqrt{c + d \sin{\left (e + f x \right )}}\, dx \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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