3.774 \(\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right ) \]

[Out]

Unintegrable[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)), x]

________________________________________________________________________________________

Rubi [A]  time = 0.0598054, 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 = {} \[ \int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)),x]

[Out]

Defer[Int][1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)), x]

Rubi steps

\begin{align*} \int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx &=\int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx\\ \end{align*}

Mathematica [A]  time = 50.09, size = 0, normalized size = 0. \[ \int \frac{1}{\sec ^{\frac{4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)),x]

[Out]

Integrate[1/(Sec[c + d*x]^(4/3)*(a + b*Sec[c + d*x])^(5/2)), x]

________________________________________________________________________________________

Maple [A]  time = 0.195, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \sec \left ( dx+c \right ) \right ) ^{-{\frac{4}{3}}} \left ( a+b\sec \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(5/2),x)

[Out]

int(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(5/2),x)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/((b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^(4/3)), 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^{3} \sec \left (d x + c\right )^{5} + 3 \, a b^{2} \sec \left (d x + c\right )^{4} + 3 \, a^{2} b \sec \left (d x + c\right )^{3} + a^{3} \sec \left (d x + c\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(2/3)/(b^3*sec(d*x + c)^5 + 3*a*b^2*sec(d*x + c)^4 + 3*a^2*b*se
c(d*x + c)^3 + a^3*sec(d*x + c)^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(1/sec(d*x+c)**(4/3)/(a+b*sec(d*x+c))**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{4}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(d*x+c)^(4/3)/(a+b*sec(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate(1/((b*sec(d*x + c) + a)^(5/2)*sec(d*x + c)^(4/3)), x)