\(\int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx\) [772]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx=\text {Int}\left (\frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 148.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 1.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.04 \[ \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integral(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(2/3)/(b^3*sec(d*x + c)^4 + 3*a*b^2*sec(d*x + c)^3 + 3*a^2*b*se
c(d*x + c)^2 + a^3*sec(d*x + c)), x)

Sympy [N/A]

Not integrable

Time = 78.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {1}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}} \sqrt [3]{\sec {\left (c + d x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 17.81 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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