\(\int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx\) [1455]

Optimal result

Integrand size = 31, antiderivative size = 102 \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx=-\frac {4 \left (3 b-3 a x+2 c^2 x\right ) \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{15 b x^2}-\frac {4 c \sqrt {-b x+a x^2} \sqrt {-x \left (-c x+\sqrt {-b x+a x^2}\right )}}{15 b x^2} \]

[Out]

-4/15*(2*c^2*x-3*a*x+3*b)*(-x*(-c*x+(a*x^2-b*x)^(1/2)))^(1/2)/b/x^2-4/15*c*(a*x^2-b*x)^(1/2)*(-x*(-c*x+(a*x^2-
b*x)^(1/2)))^(1/2)/b/x^2

Rubi [F]

\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx=\int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 4.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx=\frac {4 \sqrt {x \left (c x-\sqrt {x (-b+a x)}\right )} \left (-3 b+3 a x-c \left (2 c x+\sqrt {x (-b+a x)}\right )\right )}{15 b x^2} \]

[In]

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

[Out]

(4*Sqrt[x*(c*x - Sqrt[x*(-b + a*x)])]*(-3*b + 3*a*x - c*(2*c*x + Sqrt[x*(-b + a*x)])))/(15*b*x^2)

Maple [F]

\[\int \frac {\sqrt {c \,x^{2}-x \sqrt {a \,x^{2}-b x}}}{x^{3}}d x\]

[In]

int((c*x^2-x*(a*x^2-b*x)^(1/2))^(1/2)/x^3,x)

[Out]

int((c*x^2-x*(a*x^2-b*x)^(1/2))^(1/2)/x^3,x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx=-\frac {4 \, \sqrt {c x^{2} - \sqrt {a x^{2} - b x} x} {\left ({\left (2 \, c^{2} - 3 \, a\right )} x + \sqrt {a x^{2} - b x} c + 3 \, b\right )}}{15 \, b x^{2}} \]

[In]

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

[Out]

-4/15*sqrt(c*x^2 - sqrt(a*x^2 - b*x)*x)*((2*c^2 - 3*a)*x + sqrt(a*x^2 - b*x)*c + 3*b)/(b*x^2)

Sympy [F]

\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx=\int \frac {\sqrt {x \left (c x - \sqrt {a x^{2} - b x}\right )}}{x^{3}}\, dx \]

[In]

integrate((c*x**2-x*(a*x**2-b*x)**(1/2))**(1/2)/x**3,x)

[Out]

Integral(sqrt(x*(c*x - sqrt(a*x**2 - b*x)))/x**3, x)

Maxima [F]

\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx=\int { \frac {\sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(c*x^2 - sqrt(a*x^2 - b*x)*x)/x^3, x)

Giac [F]

\[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx=\int { \frac {\sqrt {c x^{2} - \sqrt {a x^{2} - b x} x}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(sqrt(c*x^2 - sqrt(a*x^2 - b*x)*x)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c x^2-x \sqrt {-b x+a x^2}}}{x^3} \, dx=\int \frac {\sqrt {c\,x^2-x\,\sqrt {a\,x^2-b\,x}}}{x^3} \,d x \]

[In]

int((c*x^2 - x*(a*x^2 - b*x)^(1/2))^(1/2)/x^3,x)

[Out]

int((c*x^2 - x*(a*x^2 - b*x)^(1/2))^(1/2)/x^3, x)