Optimal. Leaf size=285 \[ \frac{5 a^{3/2} (c-15 d) (c+d)^3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{64 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.56667, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2763, 21, 2770, 2775, 205} \[ \frac{5 a^{3/2} (c-15 d) (c+d)^3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}}\right )}{64 d^{3/2} f}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a \sin (e+f x)+a}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a \sin (e+f x)+a}}+\frac{5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2763
Rule 21
Rule 2770
Rule 2775
Rule 205
Rubi steps
\begin{align*} \int (a+a \sin (e+f x))^{3/2} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a+a \sin (e+f x)}}+\frac{\int \frac{\left (-\frac{1}{2} a^2 (c-15 d)-\frac{1}{2} a^2 (c-15 d) \sin (e+f x)\right ) (c+d \sin (e+f x))^{5/2}}{\sqrt{a+a \sin (e+f x)}} \, dx}{4 d}\\ &=-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a+a \sin (e+f x)}}-\frac{(a (c-15 d)) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx}{8 d}\\ &=\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a+a \sin (e+f x)}}-\frac{(5 a (c-15 d) (c+d)) \int \sqrt{a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2} \, dx}{48 d}\\ &=\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a+a \sin (e+f x)}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a+a \sin (e+f x)}}-\frac{\left (5 a (c-15 d) (c+d)^2\right ) \int \sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)} \, dx}{64 d}\\ &=\frac{5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d f \sqrt{a+a \sin (e+f x)}}+\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a+a \sin (e+f x)}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a+a \sin (e+f x)}}-\frac{\left (5 a (c-15 d) (c+d)^3\right ) \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{c+d \sin (e+f x)}} \, dx}{128 d}\\ &=\frac{5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d f \sqrt{a+a \sin (e+f x)}}+\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a+a \sin (e+f x)}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a+a \sin (e+f x)}}+\frac{\left (5 a^2 (c-15 d) (c+d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+d x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{64 d f}\\ &=\frac{5 a^{3/2} (c-15 d) (c+d)^3 \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{d} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)} \sqrt{c+d \sin (e+f x)}}\right )}{64 d^{3/2} f}+\frac{5 a^2 (c-15 d) (c+d)^2 \cos (e+f x) \sqrt{c+d \sin (e+f x)}}{64 d f \sqrt{a+a \sin (e+f x)}}+\frac{5 a^2 (c-15 d) (c+d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{96 d f \sqrt{a+a \sin (e+f x)}}+\frac{a^2 (c-15 d) \cos (e+f x) (c+d \sin (e+f x))^{5/2}}{24 d f \sqrt{a+a \sin (e+f x)}}-\frac{a^2 \cos (e+f x) (c+d \sin (e+f x))^{7/2}}{4 d f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.3732, size = 318, normalized size = 1.12 \[ \frac{(a (\sin (e+f x)+1))^{3/2} \left (-\frac{2 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{c+d \sin (e+f x)} \left (2 d \left (59 c^2+190 c d+93 d^2\right ) \sin (e+f x)+455 c^2 d+15 c^3-4 d^2 (17 c+15 d) \cos (2 (e+f x))+653 c d^2-12 d^3 \sin (3 (e+f x))+285 d^3\right )}{3 d}-\frac{5 (c-15 d) (c+d)^3 \left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \sin \left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{\sqrt{c+d \sin (e+f x)}}\right )-\log \left (\sqrt{c+d \sin (e+f x)}+\sqrt{2} \sqrt{d} \cos \left (\frac{1}{4} (2 e+2 f x-\pi )\right )\right )+\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{d} \cos \left (\frac{1}{4} (2 e+2 f x-\pi )\right )}{\sqrt{c+d \sin (e+f x)}}\right )\right )}{d^{3/2}}\right )}{128 f \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sin \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}}\, dx \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 (a \sin \left (f x + e\right ) + a\right )}^{\frac{3}{2}}{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 14.0633, size = 3841, normalized size = 13.48 \begin{align*} \text{result too large to display} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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