\(\int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx\) [813]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {(a+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx \]

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

Mathematica [N/A]

Not integrable

Time = 11.99 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx \]

[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 = 0.05 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93

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

[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 = 0.60 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

integral(-sqrt(d*sin(f*x + e) + c)*(b*sin(f*x + e) + a)^m/(3*c*d^2*cos(f*x + e)^2 - c^3 - 3*c*d^2 + (d^3*cos(f
*x + e)^2 - 3*c^2*d - d^3)*sin(f*x + e)), x)

Sympy [N/A]

Not integrable

Time = 131.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.96 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {\left (a + b \sin {\left (e + f x \right )}\right )^{m}}{\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.64 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx=\int { \frac {{\left (b \sin \left (f x + e\right ) + a\right )}^{m}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 20.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {(3+b \sin (e+f x))^m}{(c+d \sin (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]

[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)