\(\int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx\) [808]

Optimal result

Integrand size = 27, antiderivative size = 27 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx=\text {Int}\left ((3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx=\int (a+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 35.91 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx=\int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx \]

[In]

Integrate[(3 + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2),x]

[Out]

Integrate[(3 + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^(5/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.64 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

\[\int \left (a +b \sin \left (f x +e \right )\right )^{m} \left (c +d \sin \left (f x +e \right )\right )^{\frac {5}{2}}d x\]

[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)

Fricas [N/A]

Not integrable

Time = 0.57 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 3.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx=\int { {\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 21.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^m (c+d \sin (e+f x))^{5/2} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \]

[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)