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

Optimal. Leaf size=30 \[ \text {Int}\left (\frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx \]

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*}

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Mathematica [A]  time = 4.60, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{3/2}} \, dx \]

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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fricas [A]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {d \sin \left (f x + e\right ) + c} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]

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)

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maple [A]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sin \left (f x +e \right )\right )^{m}}{\left (c +d \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]

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))^(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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

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))^(3/2),x)

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \sin {\left (e + f x \right )}\right )^{m}}{\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

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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