\(\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx\) [782]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx=\text {Int}\left ((d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2},x\right ) \]

[Out]

Unintegrable((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 25, 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 = {} \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx=\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx \]

[In]

Int[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2),x]

[Out]

Defer[Int][(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x]

Rubi steps \begin{align*} \text {integral}& = \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 88.89 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx=\int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx \]

[In]

Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2),x]

[Out]

Integrate[(d*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^(3/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.68 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

\[\int \left (d \sec \left (f x +e \right )\right )^{n} \left (a +b \sec \left (f x +e \right )\right )^{\frac {3}{2}}d x\]

[In]

int((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x)

[Out]

int((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

integral((b*sec(f*x + e) + a)^(3/2)*(d*sec(f*x + e))^n, x)

Sympy [N/A]

Not integrable

Time = 26.36 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx=\int \left (d \sec {\left (e + f x \right )}\right )^{n} \left (a + b \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((d*sec(f*x+e))**n*(a+b*sec(f*x+e))**(3/2),x)

[Out]

Integral((d*sec(e + f*x))**n*(a + b*sec(e + f*x))**(3/2), x)

Maxima [N/A]

Not integrable

Time = 1.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sec(f*x + e) + a)^(3/2)*(d*sec(f*x + e))^n, x)

Giac [N/A]

Not integrable

Time = 0.98 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx=\int { {\left (b \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \left (d \sec \left (f x + e\right )\right )^{n} \,d x } \]

[In]

integrate((d*sec(f*x+e))^n*(a+b*sec(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sec(f*x + e) + a)^(3/2)*(d*sec(f*x + e))^n, x)

Mupad [N/A]

Not integrable

Time = 15.77 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.16 \[ \int (d \sec (e+f x))^n (a+b \sec (e+f x))^{3/2} \, dx=\int {\left (a+\frac {b}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^n \,d x \]

[In]

int((a + b/cos(e + f*x))^(3/2)*(d/cos(e + f*x))^n,x)

[Out]

int((a + b/cos(e + f*x))^(3/2)*(d/cos(e + f*x))^n, x)