3.717 \(\int \sec ^{\frac {7}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.05, 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 = {} \[ \int \sec ^{\frac {7}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 37.73, size = 0, normalized size = 0.00 \[ \int \sec ^{\frac {7}{3}}(c+d x) \sqrt {a+b \sec (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )^{\frac {7}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/3)*(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(d*x + c) + a)*sec(d*x + c)^(7/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/3)*(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 1.66, size = 0, normalized size = 0.00 \[ \int \left (\sec ^{\frac {7}{3}}\left (d x +c \right )\right ) \sqrt {a +b \sec \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^(7/3)*(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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