\(\int \sec ^{\frac {4}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx\) [719]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \sec ^{\frac {4}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\int \sec ^{\frac {4}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 105.48 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sec ^{\frac {4}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\int \sec ^{\frac {4}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sec ^{\frac {4}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\int { \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate(sec(d*x+c)^(4/3)*(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate(sec(d*x+c)**(4/3)*(a+b*sec(d*x+c))**(1/2),x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \sec ^{\frac {4}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\int { \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {4}{3}} \,d x } \]

[In]

integrate(sec(d*x+c)^(4/3)*(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

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

[In]

integrate(sec(d*x+c)^(4/3)*(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 14.86 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \sec ^{\frac {4}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx=\int \sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{4/3} \,d x \]

[In]

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

[Out]

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