\(\int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx\) [762]

Optimal result

Integrand size = 18, antiderivative size = 32 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx=-\frac {a \sqrt {c x^2}}{x^2}+\frac {b \sqrt {c x^2} \log (x)}{x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 45} \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx=\frac {b \sqrt {c x^2} \log (x)}{x}-\frac {a \sqrt {c x^2}}{x^2} \]

[In]

Int[(Sqrt[c*x^2]*(a + b*x))/x^3,x]

[Out]

-((a*Sqrt[c*x^2])/x^2) + (b*Sqrt[c*x^2]*Log[x])/x

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx=\frac {c (-a+b x \log (x))}{\sqrt {c x^2}} \]

[In]

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

[Out]

(c*(-a + b*x*Log[x]))/Sqrt[c*x^2]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.66

method	result	size
default	\(\frac {\sqrt {c \,x^{2}}\, \left (b \ln \left (x \right ) x -a \right )}{x^{2}}\)	\(21\)
risch	\(-\frac {a \sqrt {c \,x^{2}}}{x^{2}}+\frac {b \ln \left (x \right ) \sqrt {c \,x^{2}}}{x}\)	\(29\)

[In]

int((b*x+a)*(c*x^2)^(1/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx=\frac {\sqrt {c x^{2}} {\left (b x \log \left (x\right ) - a\right )}}{x^{2}} \]

[In]

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

[Out]

sqrt(c*x^2)*(b*x*log(x) - a)/x^2

Sympy [A] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx=- \frac {a \sqrt {c x^{2}}}{x^{2}} + \frac {b \sqrt {c x^{2}} \log {\left (x \right )}}{x} \]

[In]

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

[Out]

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x^3} \, dx={\left (b \log \left ({\left | x \right |}\right ) \mathrm {sgn}\left (x\right ) - \frac {a \mathrm {sgn}\left (x\right )}{x}\right )} \sqrt {c} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(((c*x^2)^(1/2)*(a + b*x))/x^3,x)

[Out]

int(((c*x^2)^(1/2)*(a + b*x))/x^3, x)