3.69 \(\int \frac{1}{x^{3/2} (a+b \sec (c+d \sqrt{x}))^2} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2},x\right ) \]

[Out]

Unintegrable[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]

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Rubi [A]  time = 0.0231875, 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 = {} \[ \int \frac{1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2),x]

[Out]

Defer[Int][1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]

Rubi steps

\begin{align*} \int \frac{1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2} \, dx &=\int \frac{1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 24.8994, size = 0, normalized size = 0. \[ \int \frac{1}{x^{3/2} \left (a+b \sec \left (c+d \sqrt{x}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2),x]

[Out]

Integrate[1/(x^(3/2)*(a + b*Sec[c + d*Sqrt[x]])^2), x]

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Maple [A]  time = 0.1, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-{\frac{3}{2}}} \left ( a+b\sec \left ( c+d\sqrt{x} \right ) \right ) ^{-2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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