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

Optimal. Leaf size=28 \[ \text {Int}\left (\frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}},x\right ) \]

[Out]

Unintegrable(sec(d*x+c)^(7/3)/(a+b*sec(d*x+c))^(3/2),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 = {} \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 34.97, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{\frac {7}{3}}(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {7}{3}}}{b^{2} \sec \left (d x + c\right )^{2} + 2 \, a b \sec \left (d x + c\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(7/3)/(b^2*sec(d*x + c)^2 + 2*a*b*sec(d*x + c) + a^2), 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(sec(d*x+c)^(7/3)/(a+b*sec(d*x+c))^(3/2),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 1.48, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{\frac {7}{3}}\left (d x +c \right )}{\left (a +b \sec \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (d x + c\right )^{\frac {7}{3}}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/3}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cos(c + d*x))^(7/3)/(a + b/cos(c + d*x))^(3/2),x)

[Out]

int((1/cos(c + d*x))^(7/3)/(a + b/cos(c + d*x))^(3/2), 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)**(7/3)/(a+b*sec(d*x+c))**(3/2),x)

[Out]

Timed out

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