Optimal. Leaf size=302 \[ -\frac{2 \left (45 a^2 b c d^2-15 a^3 d^3-15 a b^2 d \left (2 c^2+d^2\right )+b^3 \left (8 c^3+7 c d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 b \left (-45 a^2 d^2+30 a b c d+b^2 \left (-\left (8 c^2+9 d^2\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f} \]
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Rubi [A] time = 0.479075, antiderivative size = 302, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2793, 3023, 2752, 2663, 2661, 2655, 2653} \[ -\frac{2 \left (45 a^2 b c d^2-15 a^3 d^3-15 a b^2 d \left (2 c^2+d^2\right )+b^3 \left (8 c^3+7 c d^2\right )\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \sqrt{c+d \sin (e+f x)}}-\frac{2 b \left (-45 a^2 d^2+30 a b c d+b^2 \left (-\left (8 c^2+9 d^2\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f} \]
Antiderivative was successfully verified.
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Rule 2793
Rule 3023
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{(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)) \sqrt{c+d \sin (e+f x)}}{5 d f}+\frac{2 \int \frac{\frac{1}{2} \left (2 b^3 c+5 a^3 d+a b^2 d\right )-\frac{1}{2} b \left (2 a b c-15 a^2 d-3 b^2 d\right ) \sin (e+f x)-2 b^2 (b c-3 a d) \sin ^2(e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{5 d}\\ &=\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f}+\frac{4 \int \frac{\frac{1}{4} d \left (2 b^3 c+15 a^3 d+15 a b^2 d\right )-\frac{1}{4} b \left (30 a b c d-45 a^2 d^2-b^2 \left (8 c^2+9 d^2\right )\right ) \sin (e+f x)}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 d^2}\\ &=\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f}-\frac{\left (b \left (30 a b c d-45 a^2 d^2-b^2 \left (8 c^2+9 d^2\right )\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{15 d^3}-\frac{\left (45 a^2 b c d^2-15 a^3 d^3-15 a b^2 d \left (2 c^2+d^2\right )+b^3 \left (8 c^3+7 c d^2\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{15 d^3}\\ &=\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f}-\frac{\left (b \left (30 a b c d-45 a^2 d^2-b^2 \left (8 c^2+9 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}{15 d^3 \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{\left (\left (45 a^2 b c d^2-15 a^3 d^3-15 a b^2 d \left (2 c^2+d^2\right )+b^3 \left (8 c^3+7 c 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}{15 d^3 \sqrt{c+d \sin (e+f x)}}\\ &=\frac{8 b^2 (b c-3 a d) \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{15 d^2 f}-\frac{2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}}{5 d f}-\frac{2 b \left (30 a b c d-45 a^2 d^2-b^2 \left (8 c^2+9 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)}}{15 d^3 f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}-\frac{2 \left (45 a^2 b c d^2-15 a^3 d^3-15 a b^2 d \left (2 c^2+d^2\right )+b^3 \left (8 c^3+7 c 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}}}{15 d^3 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.18337, size = 219, normalized size = 0.73 \[ \frac{-2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \left (b \left (45 a^2 d^2-30 a b c d+b^2 \left (8 c^2+9 d^2\right )\right ) \left ((c+d) E\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )-c F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )+d^2 \left (15 a^3 d+15 a b^2 d+2 b^3 c\right ) F\left (\frac{1}{4} (-2 e-2 f x+\pi )|\frac{2 d}{c+d}\right )\right )-2 b^2 d \cos (e+f x) (c+d \sin (e+f x)) (15 a d-4 b c+3 b d \sin (e+f x))}{15 d^3 f \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.21, size = 1085, normalized size = 3.6 \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 \frac{{\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 (-\frac{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 )}{\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 \frac{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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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