3.808 \(\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx\)

Optimal. Leaf size=30 \[ \text {Int}\left ((c+d \sin (e+f x))^{5/2} (a+b \sin (e+f x))^m,x\right ) \]

[Out]

Unintegrable((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(5/2),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 = {} \[ \int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2),x]

[Out]

Defer[Int][(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2), x]

Rubi steps

\begin {align*} \int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx &=\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx\\ \end {align*}

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Mathematica [A]  time = 34.71, size = 0, normalized size = 0.00 \[ \int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2),x]

[Out]

Integrate[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2), x]

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fricas [A]  time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}\right )} \sqrt {d \sin \left (f x + e\right ) + c} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]

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))^(5/2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

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))^(5/2),x, algorithm="giac")

[Out]

integrate((d*sin(f*x + e) + c)^(5/2)*(b*sin(f*x + e) + a)^m, x)

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maple [A]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \left (a +b \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}\, dx \]

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))^(5/2),x)

[Out]

int((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(5/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

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))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*sin(f*x + e) + c)^(5/2)*(b*sin(f*x + e) + a)^m, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

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))^(5/2),x)

[Out]

int((a + b*sin(e + f*x))^m*(c + d*sin(e + f*x))^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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))**(5/2),x)

[Out]

Timed out

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