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

Optimal. Leaf size=28 \[ \text {Int}\left (\sec ^{\frac {4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)^(4/3)*(a+b*sec(d*x+c))^(5/2),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 = {} \[ \int \sec ^{\frac {4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 42.23, size = 0, normalized size = 0.00 \[ \int \sec ^{\frac {4}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \sec \left (d x + c\right )^{3} + 2 \, a b \sec \left (d x + c\right )^{2} + a^{2} \sec \left (d x + c\right )\right )} \sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {1}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {4}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.37, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{\frac {4}{3}}\left (d x +c \right )\right ) \left (a +b \sec \left (d x +c \right )\right )^{\frac {5}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {4}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{4/3} \,d x \]

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))^(4/3),x)

[Out]

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

[Out]

Timed out

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