Optimal. Leaf size=30 \[ \text {Int}\left ((c+d \sin (e+f x))^{5/2} (a+b \sin (e+f x))^m,x\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx &=\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx\\ \end {align*}
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Mathematica [A] time = 34.71, size = 0, normalized size = 0.00 \[ \int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}\right )} \sqrt {d \sin \left (f x + e\right ) + c} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (a +b \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]
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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