\(\int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx\) [349]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx=\text {Int}\left (\sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m,x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,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 = {} \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \]

[In]

Int[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m,x]

[Out]

Defer[Int][Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]

Rubi steps \begin{align*} \text {integral}& = \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.89 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx \]

[In]

Integrate[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m,x]

[Out]

Integrate[Sqrt[a + b*Sec[c + d*x]]*(e*Tan[c + d*x])^m, x]

Maple [N/A] (verified)

Not integrable

Time = 1.10 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92

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

[In]

int((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x)

[Out]

int((a+b*sec(d*x+c))^(1/2)*(e*tan(d*x+c))^m,x)

Fricas [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int { \sqrt {b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]

[In]

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

[Out]

integral(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)

Sympy [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int \left (e \tan {\left (c + d x \right )}\right )^{m} \sqrt {a + b \sec {\left (c + d x \right )}}\, dx \]

[In]

integrate((a+b*sec(d*x+c))**(1/2)*(e*tan(d*x+c))**m,x)

[Out]

Integral((e*tan(c + d*x))**m*sqrt(a + b*sec(c + d*x)), x)

Maxima [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int { \sqrt {b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)

Giac [N/A]

Not integrable

Time = 1.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int { \sqrt {b \sec \left (d x + c\right ) + a} \left (e \tan \left (d x + c\right )\right )^{m} \,d x } \]

[In]

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

[Out]

integrate(sqrt(b*sec(d*x + c) + a)*(e*tan(d*x + c))^m, x)

Mupad [N/A]

Not integrable

Time = 14.98 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt {a+b \sec (c+d x)} (e \tan (c+d x))^m \, dx=\int {\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^m\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}} \,d x \]

[In]

int((e*tan(c + d*x))^m*(a + b/cos(c + d*x))^(1/2),x)

[Out]

int((e*tan(c + d*x))^m*(a + b/cos(c + d*x))^(1/2), x)