\(\int \frac {a+b x}{\sqrt {c x^2}} \, dx\) [783]

Optimal result

Integrand size = 15, antiderivative size = 29 \[ \int \frac {a+b x}{\sqrt {c x^2}} \, dx=\frac {b x^2}{\sqrt {c x^2}}+\frac {a x \log (x)}{\sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 45} \[ \int \frac {a+b x}{\sqrt {c x^2}} \, dx=\frac {a x \log (x)}{\sqrt {c x^2}}+\frac {b x^2}{\sqrt {c x^2}} \]

[In]

Int[(a + b*x)/Sqrt[c*x^2],x]

[Out]

(b*x^2)/Sqrt[c*x^2] + (a*x*Log[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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x}{\sqrt {c x^2}} \, dx=\frac {x (b x+a \log (x))}{\sqrt {c x^2}} \]

[In]

Integrate[(a + b*x)/Sqrt[c*x^2],x]

[Out]

(x*(b*x + a*Log[x]))/Sqrt[c*x^2]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.62

method	result	size
default	\(\frac {x \left (b x +a \ln \left (x \right )\right )}{\sqrt {c \,x^{2}}}\)	\(18\)
risch	\(\frac {b \,x^{2}}{\sqrt {c \,x^{2}}}+\frac {a x \ln \left (x \right )}{\sqrt {c \,x^{2}}}\)	\(26\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {a+b x}{\sqrt {c x^2}} \, dx=\frac {\sqrt {c x^{2}} {\left (b x + a \log \left (x\right )\right )}}{c x} \]

[In]

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

[Out]

sqrt(c*x^2)*(b*x + a*log(x))/(c*x)

Sympy [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {a+b x}{\sqrt {c x^2}} \, dx=\begin {cases} \frac {a x \log {\left (x \right )}}{\sqrt {c x^{2}}} + \frac {b \sqrt {c x^{2}}}{c} & \text {for}\: c \neq 0 \\\tilde {\infty } \left (a x + \frac {b x^{2}}{2}\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((a*x*log(x)/sqrt(c*x**2) + b*sqrt(c*x**2)/c, Ne(c, 0)), (zoo*(a*x + b*x**2/2), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {a+b x}{\sqrt {c x^2}} \, dx=\frac {a \log \left (x\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2}} b}{c} \]

[In]

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

[Out]

a*log(x)/sqrt(c) + sqrt(c*x^2)*b/c

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.79 \[ \int \frac {a+b x}{\sqrt {c x^2}} \, dx=\frac {b x}{\sqrt {c} \mathrm {sgn}\left (x\right )} + \frac {a \log \left ({\left | x \right |}\right )}{\sqrt {c} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

b*x/(sqrt(c)*sgn(x)) + a*log(abs(x))/(sqrt(c)*sgn(x))

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.59 \[ \int \frac {a+b x}{\sqrt {c x^2}} \, dx=\frac {b\,\left |x\right |+a\,\ln \left (c\,x\right )\,\mathrm {sign}\left (x\right )}{\sqrt {c}} \]

[In]

int((a + b*x)/(c*x^2)^(1/2),x)

[Out]

(b*abs(x) + a*log(c*x)*sign(x))/c^(1/2)