Optimal. Leaf size=302 \[ -\frac {2 b \left (-45 a^2 d^2+30 a b c d-\left (b^2 \left (8 c^2+9 d^2\right )\right )\right ) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d^3 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 \left (-15 a^3 d^3+45 a^2 b c d^2-15 a b^2 d \left (2 c^2+d^2\right )+b^3 \left (8 c^3+7 c d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{15 d^3 f \sqrt {c+d \sin (e+f x)}}+\frac {8 b^2 (b c-3 a d) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{15 d^2 f}-\frac {2 b^2 \cos (e+f x) (a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}{5 d f} \]
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Rubi [A] time = 0.48, antiderivative size = 302, normalized size of antiderivative = 1.00, 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 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2793
Rule 3023
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*}
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Mathematica [A] time = 1.20, size = 219, normalized size = 0.73 \[ \frac {-2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \left (d^2 \left (15 a^3 d+15 a b^2 d+2 b^3 c\right ) F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )+b \left (45 a^2 d^2-30 a b c d+b^2 \left (8 c^2+9 d^2\right )\right ) \left ((c+d) E\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )-c F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 d}{c+d}\right )\right )\right )-2 b^2 d \cos (e+f x) (c+d \sin (e+f x)) (15 a d-4 b c+3 b d \sin (e+f x))}{15 d^3 f \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {3 \, a b^{2} \cos \left (f x + e\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (f x + e\right )}{\sqrt {d \sin \left (f x + e\right ) + c}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{\sqrt {d \sin \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 4.18, size = 1085, normalized size = 3.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{3}}{\sqrt {d \sin \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^3}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sin {\left (e + f x \right )}\right )^{3}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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