3.9.10 \(\int (a+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx\) [810]

Optimal. Leaf size=30 \[ \text {Int}\left ((a+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.04, 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 = {} \begin {gather*} \int (a+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx \end {gather*}

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 = 0.06, size = 0, normalized size = 0.00 \[\int \left (a +b \sin \left (f x +e \right )\right )^{m} \sqrt {c +d \sin \left (f x +e \right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin {\left (e + f x \right )}\right )^{m} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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