3.9.11 \(\int \frac {(a+b \sin (e+f x))^m}{\sqrt {c+d \sin (e+f x)}} \, dx\) [811]

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

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

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Mathematica [A]
time = 3.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(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 \[\int \frac {\left (a +b \sin \left (f x +e \right )\right )^{m}}{\sqrt {c +d \sin \left (f x +e \right )}}\, dx\]

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

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\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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