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

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 49.97, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec ^{\frac {2}{3}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 2.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {1}{3}}}{b^{3} \sec \left (d x + c\right )^{4} + 3 \, a b^{2} \sec \left (d x + c\right )^{3} + 3 \, a^{2} b \sec \left (d x + c\right )^{2} + a^{3} \sec \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(d*x+c)^(2/3)/(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)^(2/3)), x)

________________________________________________________________________________________

maple [A]  time = 1.58, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sec \left (d x +c \right )^{\frac {2}{3}} \left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sec(d*x+c)^(2/3)/(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)^(2/3)), x)

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________