\(\int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx\) [767]

Optimal result

Integrand size = 25, antiderivative size = 25 \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Int}\left (\frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)^(7/3)/(a+b*sec(d*x+c))^(5/2),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 = {} \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx \]

[In]

Int[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2),x]

[Out]

Defer[Int][Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 129.79 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx \]

[In]

Integrate[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2),x]

[Out]

Integrate[Sec[c + d*x]^(7/3)/(a + b*Sec[c + d*x])^(5/2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.48 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

\[\int \frac {\sec \left (d x +c \right )^{\frac {7}{3}}}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{2}}}d x\]

[In]

int(sec(d*x+c)^(7/3)/(a+b*sec(d*x+c))^(5/2),x)

[Out]

int(sec(d*x+c)^(7/3)/(a+b*sec(d*x+c))^(5/2),x)

Fricas [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {7}{3}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(sec(d*x+c)^(7/3)/(a+b*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(7/3)/(b^3*sec(d*x + c)^3 + 3*a*b^2*sec(d*x + c)^2 + 3*a^2*b*se
c(d*x + c) + a^3), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)**(7/3)/(a+b*sec(d*x+c))**(5/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{\frac {7}{3}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(sec(d*x+c)^(7/3)/(a+b*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(sec(d*x + c)^(7/3)/(b*sec(d*x + c) + a)^(5/2), x)

Giac [F(-1)]

Timed out. \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\text {Timed out} \]

[In]

integrate(sec(d*x+c)^(7/3)/(a+b*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

Timed out

Mupad [N/A]

Not integrable

Time = 17.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx=\int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/3}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]

[In]

int((1/cos(c + d*x))^(7/3)/(a + b/cos(c + d*x))^(5/2),x)

[Out]

int((1/cos(c + d*x))^(7/3)/(a + b/cos(c + d*x))^(5/2), x)