3.812 \(\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ \text{Unintegrable}\left (\frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}},x\right ) \]

[Out]

Unintegrable[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2), x]

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Rubi [A]  time = 0.0762709, 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 = {} \[ \int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2),x]

[Out]

Defer[Int][(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2), x]

Rubi steps

\begin{align*} \int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx &=\int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 4.57487, size = 0, normalized size = 0. \[ \int \frac{(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2),x]

[Out]

Integrate[(a + b*Sin[e + f*x])^m/(c + d*Sin[e + f*x])^(3/2), x]

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Maple [A]  time = 0.155, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\sin \left ( fx+e \right ) \right ) ^{m} \left ( c+d\sin \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}\, 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))^(3/2),x)

[Out]

int((a+b*sin(f*x+e))^m/(c+d*sin(f*x+e))^(3/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{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))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^m/(d*sin(f*x + e) + c)^(3/2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{d \sin \left (f x + e\right ) + c}{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, 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))^(3/2),x, algorithm="fricas")

[Out]

integral(-sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2
), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \sin{\left (e + f x \right )}\right )^{m}}{\left (c + d \sin{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, 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))**(3/2),x)

[Out]

Integral((a + b*sin(e + f*x))**m/(c + d*sin(e + f*x))**(3/2), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{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))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^m/(d*sin(f*x + e) + c)^(3/2), x)