\(\int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx\) [742]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx=\text {Int}\left (\frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}},x\right ) \]

[Out]

Unintegrable((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(1/3),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 = {} \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx=\int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx \]

[In]

Int[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3),x]

[Out]

Defer[Int][(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx \\ \end{align*}

Mathematica [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx=\text {\$Aborted} \]

[In]

Integrate[(a + b*Sec[c + d*x])^(5/2)/Sec[c + d*x]^(1/3),x]

[Out]

$Aborted

Maple [N/A] (verified)

Not integrable

Time = 0.43 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

\[\int \frac {\left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{2}}}{\sec \left (d x +c \right )^{\frac {1}{3}}}d x\]

[In]

int((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(1/3),x)

[Out]

int((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(1/3),x)

Fricas [N/A]

Not integrable

Time = 1.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.96 \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(1/3),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))**(5/2)/sec(d*x+c)**(1/3),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 2.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx=\int { \frac {{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\sec \left (d x + c\right )^{\frac {1}{3}}} \,d x } \]

[In]

integrate((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(1/3),x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^(5/2)/sec(d*x + c)^(1/3), x)

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sec(d*x+c))^(5/2)/sec(d*x+c)^(1/3),x, algorithm="giac")

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 17.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]

[In]

int((a + b/cos(c + d*x))^(5/2)/(1/cos(c + d*x))^(1/3),x)

[Out]

int((a + b/cos(c + d*x))^(5/2)/(1/cos(c + d*x))^(1/3), x)