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

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0582848, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 37.6154, size = 0, normalized size = 0. \[ \int \frac{\sec ^{\frac{2}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.193, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sec \left ( dx+c \right ) \right ) ^{{\frac{2}{3}}} \left ( a+b\sec \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{\frac{2}{3}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac{2}{3}}}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{\frac{2}{3}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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