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

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

[Out]

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

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

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 50.14, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sec (c+d x))^{5/2}}{\sqrt [3]{\sec (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 2.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}\right )} \sqrt {b \sec \left (d x + c\right ) + a}}{\sec \left (d x + c\right )^{\frac {1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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maple [A]  time = 1.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{2}}}{\sec \left (d x +c \right )^{\frac {1}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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