3.737 \(\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx\)

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2},x\right ) \]

[Out]

Unintegrable[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2), x]

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Rubi [A]  time = 0.0560519, 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 = {} \[ \int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2),x]

[Out]

Defer[Int][Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2), x]

Rubi steps

\begin{align*} \int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx &=\int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx\\ \end{align*}

Mathematica [A]  time = 37.8797, size = 0, normalized size = 0. \[ \int \sec ^{\frac{7}{3}}(c+d x) (a+b \sec (c+d x))^{5/2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2),x]

[Out]

Integrate[Sec[c + d*x]^(7/3)*(a + b*Sec[c + d*x])^(5/2), x]

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Maple [A]  time = 0.205, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( dx+c \right ) \right ) ^{{\frac{7}{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(sec(d*x+c)^(7/3)*(a+b*sec(d*x+c))^(5/2),x)

[Out]

int(sec(d*x+c)^(7/3)*(a+b*sec(d*x+c))^(5/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{7}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/3)*(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)^(7/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/3)*(a+b*sec(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

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

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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)**(7/3)*(a+b*sec(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{7}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/3)*(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)^(7/3), x)