Optimal. Leaf size=76 \[ -\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{n+1} \, _2F_1\left (-\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) \sin ^2(a+b x)^{3/4}} \]
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Rubi [A] time = 0.0534167, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {2576} \[ -\frac{c (c \sin (a+b x))^{3/2} (d \cos (a+b x))^{n+1} \, _2F_1\left (-\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )}{b d (n+1) \sin ^2(a+b x)^{3/4}} \]
Antiderivative was successfully verified.
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Rule 2576
Rubi steps
\begin{align*} \int (d \cos (a+b x))^n (c \sin (a+b x))^{5/2} \, dx &=-\frac{c (d \cos (a+b x))^{1+n} \, _2F_1\left (-\frac{3}{4},\frac{1+n}{2};\frac{3+n}{2};\cos ^2(a+b x)\right ) (c \sin (a+b x))^{3/2}}{b d (1+n) \sin ^2(a+b x)^{3/4}}\\ \end{align*}
Mathematica [B] time = 0.415062, size = 158, normalized size = 2.08 \[ \frac{\cot (a+b x) (c \sin (a+b x))^{5/2} (d \cos (a+b x))^n \left ((n+1) \cos ^2(a+b x) \, _2F_1\left (\frac{1}{4},\frac{n+3}{2};\frac{n+5}{2};\cos ^2(a+b x)\right )-(n+3) \, _2F_1\left (-\frac{3}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )-(n+3) \, _2F_1\left (\frac{1}{4},\frac{n+1}{2};\frac{n+3}{2};\cos ^2(a+b x)\right )\right )}{2 b (n+1) (n+3) \sin ^2(a+b x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int \left ( d\cos \left ( bx+a \right ) \right ) ^{n} \left ( c\sin \left ( bx+a \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 (c \sin \left (b x + a\right )\right )^{\frac{5}{2}} \left (d \cos \left (b x + a\right )\right )^{n}\,{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 (-{\left (c^{2} \cos \left (b x + a\right )^{2} - c^{2}\right )} \sqrt{c \sin \left (b x + a\right )} \left (d \cos \left (b x + a\right )\right )^{n}, 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 \left (c \sin \left (b x + a\right )\right )^{\frac{5}{2}} \left (d \cos \left (b x + a\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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