\(\int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx\) [810]

Optimal result

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

[Out]

Unintegrable((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx=\int (a+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx=\int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.14 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[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)

Fricas [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx=\int { \sqrt {d \sin \left (f x + e\right ) + c} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 2.70 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx=\int \left (a + b \sin {\left (e + f x \right )}\right )^{m} \sqrt {c + d \sin {\left (e + f x \right )}}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx=\int { \sqrt {d \sin \left (f x + e\right ) + c} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.51 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx=\int { \sqrt {d \sin \left (f x + e\right ) + c} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+b*sin(f*x+e))^m*(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 10.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int (3+b \sin (e+f x))^m \sqrt {c+d \sin (e+f x)} \, dx=\int {\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,\sqrt {c+d\,\sin \left (e+f\,x\right )} \,d x \]

[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)