\(\int \frac {a+b \sec (c+d \sqrt {x})}{x^{3/2}} \, dx\) [54]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx=-\frac {2 a}{\sqrt {x}}+b \text {Int}\left (\frac {\sec \left (c+d \sqrt {x}\right )}{x^{3/2}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx=\int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 22.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx=\int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.53 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80

\[\int \frac {a +b \sec \left (c +d \sqrt {x}\right )}{x^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx=\int { \frac {b \sec \left (d \sqrt {x} + c\right ) + a}{x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.88 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx=\int \frac {a + b \sec {\left (c + d \sqrt {x} \right )}}{x^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.74 (sec) , antiderivative size = 112, normalized size of antiderivative = 5.60 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx=\int { \frac {b \sec \left (d \sqrt {x} + c\right ) + a}{x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx=\int { \frac {b \sec \left (d \sqrt {x} + c\right ) + a}{x^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 13.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \sec \left (c+d \sqrt {x}\right )}{x^{3/2}} \, dx=\int \frac {a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}}{x^{3/2}} \,d x \]

[In]

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

[Out]

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