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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 23.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \sqrt {x} \sec \left (d \sqrt {x} + c\right )^{2} + 2 \, a b \sqrt {x} \sec \left (d \sqrt {x} + c\right ) + a^{2} \sqrt {x}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}}{x^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {4 \, b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + 4 \, {\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} \int \frac {b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x} + d \int \frac {b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x} \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \int \frac {b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x} + d \int \frac {b^{2} \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) + {\left (a b d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) \cos \left (d \sqrt {x} + c\right ) + a b d \sin \left (2 \, d \sqrt {x} + 2 \, c\right ) \sin \left (d \sqrt {x} + c\right ) + a b d \cos \left (d \sqrt {x} + c\right )\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x^{2}}\,{d x}\right )} x - 2 \, {\left (a^{2} d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + a^{2} d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, a^{2} d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + a^{2} d\right )} \sqrt {x}}{{\left (d \cos \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + d \sin \left (2 \, d \sqrt {x} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d \sqrt {x} + 2 \, c\right ) + d\right )} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*b^2*sin(2*d*sqrt(x) + 2*c) + (d*cos(2*d*sqrt(x) + 2*c)^2*integrate(4*(b^2*sin(2*d*sqrt(x) + 2*c) + (a*b*d*c
os(2*d*sqrt(x) + 2*c)*cos(d*sqrt(x) + c) + a*b*d*sin(2*d*sqrt(x) + 2*c)*sin(d*sqrt(x) + c) + a*b*d*cos(d*sqrt(
x) + c))*sqrt(x))/((d*cos(2*d*sqrt(x) + 2*c)^2 + d*sin(2*d*sqrt(x) + 2*c)^2 + 2*d*cos(2*d*sqrt(x) + 2*c) + d)*
x^2), x) + d*integrate(4*(b^2*sin(2*d*sqrt(x) + 2*c) + (a*b*d*cos(2*d*sqrt(x) + 2*c)*cos(d*sqrt(x) + c) + a*b*
d*sin(2*d*sqrt(x) + 2*c)*sin(d*sqrt(x) + c) + a*b*d*cos(d*sqrt(x) + c))*sqrt(x))/((d*cos(2*d*sqrt(x) + 2*c)^2
+ d*sin(2*d*sqrt(x) + 2*c)^2 + 2*d*cos(2*d*sqrt(x) + 2*c) + d)*x^2), x)*sin(2*d*sqrt(x) + 2*c)^2 + 2*d*cos(2*d
*sqrt(x) + 2*c)*integrate(4*(b^2*sin(2*d*sqrt(x) + 2*c) + (a*b*d*cos(2*d*sqrt(x) + 2*c)*cos(d*sqrt(x) + c) + a
*b*d*sin(2*d*sqrt(x) + 2*c)*sin(d*sqrt(x) + c) + a*b*d*cos(d*sqrt(x) + c))*sqrt(x))/((d*cos(2*d*sqrt(x) + 2*c)
^2 + d*sin(2*d*sqrt(x) + 2*c)^2 + 2*d*cos(2*d*sqrt(x) + 2*c) + d)*x^2), x) + d*integrate(4*(b^2*sin(2*d*sqrt(x
) + 2*c) + (a*b*d*cos(2*d*sqrt(x) + 2*c)*cos(d*sqrt(x) + c) + a*b*d*sin(2*d*sqrt(x) + 2*c)*sin(d*sqrt(x) + c)
+ a*b*d*cos(d*sqrt(x) + c))*sqrt(x))/((d*cos(2*d*sqrt(x) + 2*c)^2 + d*sin(2*d*sqrt(x) + 2*c)^2 + 2*d*cos(2*d*s
qrt(x) + 2*c) + d)*x^2), x))*x - 2*(a^2*d*cos(2*d*sqrt(x) + 2*c)^2 + a^2*d*sin(2*d*sqrt(x) + 2*c)^2 + 2*a^2*d*
cos(2*d*sqrt(x) + 2*c) + a^2*d)*sqrt(x))/((d*cos(2*d*sqrt(x) + 2*c)^2 + d*sin(2*d*sqrt(x) + 2*c)^2 + 2*d*cos(2
*d*sqrt(x) + 2*c) + d)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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