\(\int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx\) [759]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\text {Int}\left (\frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(3/2),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 = {} \[ \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx \]

[In]

Int[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2),x]

[Out]

Defer[Int][Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 125.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx \]

[In]

Integrate[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2),x]

[Out]

Integrate[Sec[c + d*x]^(4/3)/(a + b*Sec[c + d*x])^(3/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.45 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

\[\int \frac {\sec \left (d x +c \right )^{\frac {4}{3}}}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.04 \[ \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {4}{3}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 137.51 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {\sec ^{\frac {4}{3}}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(sec(d*x+c)**(4/3)/(a+b*sec(d*x+c))**(3/2),x)

[Out]

Integral(sec(c + d*x)**(4/3)/(a + b*sec(c + d*x))**(3/2), x)

Maxima [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {4}{3}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 16.49 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^{\frac {4}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{4/3}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int((1/cos(c + d*x))^(4/3)/(a + b/cos(c + d*x))^(3/2),x)

[Out]

int((1/cos(c + d*x))^(4/3)/(a + b/cos(c + d*x))^(3/2), x)