\(\int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx\) [785]

Optimal result

Integrand size = 18, antiderivative size = 26 \[ \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx=-\frac {(a+b x)^2}{2 a x \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 37} \[ \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx=-\frac {(a+b x)^2}{2 a x \sqrt {c x^2}} \]

[In]

Int[(a + b*x)/(x^2*Sqrt[c*x^2]),x]

[Out]

-1/2*(a + b*x)^2/(a*x*Sqrt[c*x^2])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx=-\frac {c x (a+2 b x)}{2 \left (c x^2\right )^{3/2}} \]

[In]

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

[Out]

-1/2*(c*x*(a + 2*b*x))/(c*x^2)^(3/2)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73

method	result	size
gosper	\(-\frac {2 b x +a}{2 x \sqrt {c \,x^{2}}}\)	\(19\)
default	\(-\frac {2 b x +a}{2 x \sqrt {c \,x^{2}}}\)	\(19\)
risch	\(\frac {-b x -\frac {a}{2}}{x \sqrt {c \,x^{2}}}\)	\(20\)
trager	\(\frac {\left (-1+x \right ) \left (a x +2 b x +a \right ) \sqrt {c \,x^{2}}}{2 c \,x^{3}}\)	\(28\)

[In]

int((b*x+a)/x^2/(c*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx=-\frac {\sqrt {c x^{2}} {\left (2 \, b x + a\right )}}{2 \, c x^{3}} \]

[In]

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

[Out]

-1/2*sqrt(c*x^2)*(2*b*x + a)/(c*x^3)

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx=- \frac {a}{2 x \sqrt {c x^{2}}} - \frac {b}{\sqrt {c x^{2}}} \]

[In]

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

[Out]

-a/(2*x*sqrt(c*x**2)) - b/sqrt(c*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx=-\frac {b}{\sqrt {c} x} - \frac {a}{2 \, \sqrt {c} x^{2}} \]

[In]

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

[Out]

-b/(sqrt(c)*x) - 1/2*a/(sqrt(c)*x^2)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx=-\frac {2 \, b x + a}{2 \, \sqrt {c} x^{2} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

-1/2*(2*b*x + a)/(sqrt(c)*x^2*sgn(x))

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {a+b x}{x^2 \sqrt {c x^2}} \, dx=-\frac {2\,b\,x^3+a\,x^2}{2\,\sqrt {c}\,x\,{\left (x^2\right )}^{3/2}} \]

[In]

int((a + b*x)/(x^2*(c*x^2)^(1/2)),x)

[Out]

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