Optimal. Leaf size=390 \[ \frac{\left (3 a^2 d^2+2 a b c d+b^2 \left (-\left (c^2+4 d^2\right )\right )\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b^3 f \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}-\frac{\left (-3 a^2 d^2+2 a b c d+b^2 \left (-\left (c^2-2 d^2\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b^2 f \left (a^2-b^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac{\left (3 a^2 d+2 a b c-5 b^2 d\right ) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b^3 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}} \]
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Rubi [A] time = 1.26106, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2792, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{\left (3 a^2 d^2+2 a b c d+b^2 \left (-\left (c^2+4 d^2\right )\right )\right ) (b c-a d) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} F\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b^3 f \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}-\frac{\left (-3 a^2 d^2+2 a b c d+b^2 \left (-\left (c^2-2 d^2\right )\right )\right ) \sqrt{c+d \sin (e+f x)} E\left (\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b^2 f \left (a^2-b^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac{\left (3 a^2 d+2 a b c-5 b^2 d\right ) (b c-a d)^2 \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{b^3 f (a-b) (a+b)^2 \sqrt{c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{(c+d \sin (e+f x))^{5/2}}{(a+b \sin (e+f x))^2} \, dx &=\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{\int \frac{\frac{1}{2} \left (5 b^2 c^2 d+a^2 d^3-2 a b c \left (c^2+2 d^2\right )\right )-d \left (a^2 c d-3 b^2 c d+a b \left (c^2+d^2\right )\right ) \sin (e+f x)+\frac{1}{2} d \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{\int \frac{\frac{1}{2} d (b c-a d) \left (a b c^2+3 a^2 c d-5 b^2 c d+a b d^2\right )-\frac{1}{2} d (b c-a d) \left (b^2 c^2-2 a b c d-3 a^2 d^2+4 b^2 d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{b^2 \left (a^2-b^2\right ) d}-\frac{\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \int \sqrt{c+d \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac{\left ((b c-a d)^2 \left (2 a b c+3 a^2 d-5 b^2 d\right )\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{c+d \sin (e+f x)}} \, dx}{2 b^3 \left (a^2-b^2\right )}+\frac{\left ((b c-a d) \left (2 a b c d+3 a^2 d^2-b^2 \left (c^2+4 d^2\right )\right )\right ) \int \frac{1}{\sqrt{c+d \sin (e+f x)}} \, dx}{2 b^3 \left (a^2-b^2\right )}-\frac{\left (\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 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}{2 b^2 \left (a^2-b^2\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}\\ &=\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 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)}}{b^2 \left (a^2-b^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{\left ((b c-a d)^2 \left (2 a b c+3 a^2 d-5 b^2 d\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}}\right ) \int \frac{1}{(a+b \sin (e+f x)) \sqrt{\frac{c}{c+d}+\frac{d \sin (e+f x)}{c+d}}} \, dx}{2 b^3 \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}+\frac{\left ((b c-a d) \left (2 a b c d+3 a^2 d^2-b^2 \left (c^2+4 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}{2 b^3 \left (a^2-b^2\right ) \sqrt{c+d \sin (e+f x)}}\\ &=\frac{(b c-a d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac{\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 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)}}{b^2 \left (a^2-b^2\right ) f \sqrt{\frac{c+d \sin (e+f x)}{c+d}}}+\frac{(b c-a d) \left (2 a b c d+3 a^2 d^2-b^2 \left (c^2+4 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}}}{b^3 \left (a^2-b^2\right ) f \sqrt{c+d \sin (e+f x)}}+\frac{(b c-a d)^2 \left (2 a b c+3 a^2 d-5 b^2 d\right ) \Pi \left (\frac{2 b}{a+b};\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}}}{(a-b) b^3 (a+b)^2 f \sqrt{c+d \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 7.99406, size = 986, normalized size = 2.53 \[ \frac{\sqrt{c+d \sin (e+f x)} \left (-b^2 \cos (e+f x) c^2+2 a b d \cos (e+f x) c-a^2 d^2 \cos (e+f x)\right )}{b \left (b^2-a^2\right ) f (a+b \sin (e+f x))}+\frac{-\frac{2 \left (4 a b c^3-9 b^2 d c^2+6 a b d^2 c+a^2 d^3-2 b^2 d^3\right ) \sqrt{\frac{c+d \sin (e+f x)}{c+d}} \Pi \left (\frac{2 b}{a+b};\frac{1}{2} \left (-e-f x+\frac{\pi }{2}\right )|\frac{2 d}{c+d}\right )}{(a+b) \sqrt{c+d \sin (e+f x)}}-\frac{2 i \left (4 a b d^3+4 a^2 c d^2-12 b^2 c d^2+4 a b c^2 d\right ) \cos (e+f x) \left ((b c-a d) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+a d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}-\frac{2 i \left (-3 a^2 d^3+2 b^2 d^3+2 a b c d^2-b^2 c^2 d\right ) \cos (e+f x) \cos (2 (e+f x)) \left (2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )+d \left (\left (2 a^2-b^2\right ) d \Pi \left (\frac{b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )-2 (a+b) (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{c+d}} \sqrt{c+d \sin (e+f x)}\right )|\frac{c+d}{c-d}\right )\right )\right ) \sqrt{\frac{d-d \sin (e+f x)}{c+d}} \sqrt{-\frac{\sin (e+f x) d+d}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt{-\frac{1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt{1-\sin ^2(e+f x)} \left (-2 c^2+4 (c+d \sin (e+f x)) c+d^2-2 (c+d \sin (e+f x))^2\right ) \sqrt{-\frac{c^2-2 (c+d \sin (e+f x)) c-d^2+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) b (a+b) f} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 5.217, size = 1363, normalized size = 3.5 \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 (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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