3.810 \(\int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx\)

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\sqrt{c+d \sin (e+f x)} (a+b \sin (e+f x))^m,x\right ) \]

[Out]

Unintegrable[(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]], x]

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Rubi [A]  time = 0.0690777, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]],x]

[Out]

Defer[Int][(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]], x]

Rubi steps

\begin{align*} \int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx &=\int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx\\ \end{align*}

Mathematica [A]  time = 0.49436, size = 0, normalized size = 0. \[ \int (a+b \sin (e+f x))^m \sqrt{c+d \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]],x]

[Out]

Integrate[(a + b*Sin[e + f*x])^m*Sqrt[c + d*Sin[e + f*x]], x]

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Maple [A]  time = 0.157, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sin \left ( fx+e \right ) \right ) ^{m}\sqrt{c+d\sin \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x)

[Out]

int((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \sin \left (f x + e\right ) + c}{\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d \sin \left (f x + e\right ) + c}{\left (b \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \sin{\left (e + f x \right )}\right )^{m} \sqrt{c + d \sin{\left (e + f x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))**m*(c+d*sin(f*x+e))**(1/2),x)

[Out]

Integral((a + b*sin(e + f*x))**m*sqrt(c + d*sin(e + f*x)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \sin \left (f x + e\right ) + c}{\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m, x)