\(\int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx\) [726]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx=\text {Int}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)},x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(7/3),x)

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx=\int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx \]

[In]

Int[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3),x]

[Out]

Defer[Int][Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 162.74 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx=\int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx \]

[In]

Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3),x]

[Out]

Integrate[Sqrt[a + b*Sec[c + d*x]]/Sec[c + d*x]^(7/3), x]

Maple [N/A] (verified)

Not integrable

Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

\[\int \frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{\frac {7}{3}}}d x\]

[In]

int((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(7/3),x)

[Out]

int((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(7/3),x)

Fricas [N/A]

Not integrable

Time = 1.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {7}{3}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(7/3),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(d*x + c) + a)/sec(d*x + c)^(7/3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))**(1/2)/sec(d*x+c)**(7/3),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.81 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {7}{3}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(7/3),x, algorithm="maxima")

[Out]

integrate(sqrt(b*sec(d*x + c) + a)/sec(d*x + c)^(7/3), x)

Giac [N/A]

Not integrable

Time = 1.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx=\int { \frac {\sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {7}{3}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(7/3),x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(d*x + c) + a)/sec(d*x + c)^(7/3), x)

Mupad [N/A]

Not integrable

Time = 16.62 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {7}{3}}(c+d x)} \, dx=\int \frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/3}} \,d x \]

[In]

int((a + b/cos(c + d*x))^(1/2)/(1/cos(c + d*x))^(7/3),x)

[Out]

int((a + b/cos(c + d*x))^(1/2)/(1/cos(c + d*x))^(7/3), x)