Optimal. Leaf size=193 \[ \frac{8 a^2 \sqrt{d \sin (e+f x)}}{5 d f g^3 \sqrt{g \cos (e+f x)}}+\frac{8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt{g \cos (e+f x)}}-\frac{8 a b E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 d f g^4 \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}} \]
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Rubi [A] time = 0.473507, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {2888, 2838, 2563, 2571, 2572, 2639} \[ \frac{8 a^2 \sqrt{d \sin (e+f x)}}{5 d f g^3 \sqrt{g \cos (e+f x)}}+\frac{8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt{g \cos (e+f x)}}-\frac{8 a b E\left (\left .e+f x-\frac{\pi }{4}\right |2\right ) \sqrt{d \sin (e+f x)} \sqrt{g \cos (e+f x)}}{5 d f g^4 \sqrt{\sin (2 e+2 f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2888
Rule 2838
Rule 2563
Rule 2571
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+b \sin (e+f x))^2}{(g \cos (e+f x))^{7/2} \sqrt{d \sin (e+f x)}} \, dx &=\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}+\frac{(4 a) \int \frac{a+b \sin (e+f x)}{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}} \, dx}{5 g^2}\\ &=\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}+\frac{\left (4 a^2\right ) \int \frac{1}{(g \cos (e+f x))^{3/2} \sqrt{d \sin (e+f x)}} \, dx}{5 g^2}+\frac{(4 a b) \int \frac{\sqrt{d \sin (e+f x)}}{(g \cos (e+f x))^{3/2}} \, dx}{5 d g^2}\\ &=\frac{8 a^2 \sqrt{d \sin (e+f x)}}{5 d f g^3 \sqrt{g \cos (e+f x)}}+\frac{8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac{(8 a b) \int \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)} \, dx}{5 d g^4}\\ &=\frac{8 a^2 \sqrt{d \sin (e+f x)}}{5 d f g^3 \sqrt{g \cos (e+f x)}}+\frac{8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac{\left (8 a b \sqrt{g \cos (e+f x)} \sqrt{d \sin (e+f x)}\right ) \int \sqrt{\sin (2 e+2 f x)} \, dx}{5 d g^4 \sqrt{\sin (2 e+2 f x)}}\\ &=\frac{8 a^2 \sqrt{d \sin (e+f x)}}{5 d f g^3 \sqrt{g \cos (e+f x)}}+\frac{8 a b (d \sin (e+f x))^{3/2}}{5 d^2 f g^3 \sqrt{g \cos (e+f x)}}+\frac{2 \sqrt{d \sin (e+f x)} (a+b \sin (e+f x))^2}{5 d f g (g \cos (e+f x))^{5/2}}-\frac{8 a b \sqrt{g \cos (e+f x)} E\left (\left .e-\frac{\pi }{4}+f x\right |2\right ) \sqrt{d \sin (e+f x)}}{5 d f g^4 \sqrt{\sin (2 e+2 f x)}}\\ \end{align*}
Mathematica [C] time = 0.652753, size = 105, normalized size = 0.54 \[ \frac{2 \tan (e+f x) \left (3 \left (b^2-4 a^2\right ) \sin ^2(e+f x)+15 a^2+10 a b \sin (e+f x) \cos ^2(e+f x)^{5/4} \, _2F_1\left (\frac{3}{4},\frac{9}{4};\frac{7}{4};\sin ^2(e+f x)\right )\right )}{15 f g^2 \sqrt{d \sin (e+f x)} (g \cos (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.398, size = 616, normalized size = 3.2 \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 \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac{7}{2}} \sqrt{d \sin \left (f x + e\right )}}\,{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{{\left (b^{2} \cos \left (f x + e\right )^{2} - 2 \, a b \sin \left (f x + e\right ) - a^{2} - b^{2}\right )} \sqrt{g \cos \left (f x + e\right )} \sqrt{d \sin \left (f x + e\right )}}{d g^{4} \cos \left (f x + e\right )^{4} \sin \left (f x + e\right )}, x\right ) \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 (b \sin \left (f x + e\right ) + a\right )}^{2}}{\left (g \cos \left (f x + e\right )\right )^{\frac{7}{2}} \sqrt{d \sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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