\(\int \sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx\) [731]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 94.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx=\int \sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.68 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {1}{3}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {1}{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.67 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {1}{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 15.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sqrt [3]{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \, dx=\int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{1/3} \,d x \]

[In]

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

[Out]

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