3.8.24 \(\int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {4}{3}}(c+d x)} \, dx\) [724]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 = {} \begin {gather*} \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {4}{3}}(c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 34.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b \sec (c+d x)}}{\sec ^{\frac {4}{3}}(c+d x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {a +b \sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{\frac {4}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(4/3),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(4/3),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b \sec {\left (c + d x \right )}}}{\sec ^{\frac {4}{3}}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sec(c + d*x))/sec(c + d*x)**(4/3), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(d*x+c))^(1/2)/sec(d*x+c)^(4/3),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{4/3}} \,d x \end {gather*}

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))^(4/3),x)

[Out]

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

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