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 = 0.57, antiderivative size = 285, normalized size of antiderivative = 1.00, 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 21
Rule 205
Rule 2763
Rule 2770
Rule 2775
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*}
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Mathematica [A] time = 1.37, size = 318, normalized size = 1.12 \[ \frac {(a (\sin (e+f x)+1))^{3/2} \left (-\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} \left (15 c^3+2 d \left (59 c^2+190 c d+93 d^2\right ) \sin (e+f x)+455 c^2 d-4 d^2 (17 c+15 d) \cos (2 (e+f x))+653 c d^2-12 d^3 \sin (3 (e+f x))+285 d^3\right )}{3 d}-\frac {5 (c-15 d) (c+d)^3 \left (2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\log \left (\sqrt {c+d \sin (e+f x)}+\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )+\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{\sqrt {c+d \sin (e+f x)}}\right )\right )}{d^{3/2}}\right )}{128 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.59, size = 1579, normalized size = 5.54 \[ \text {result too large to display} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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