\(\int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx\) [760]

Optimal result

Integrand size = 18, antiderivative size = 27 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx=a \sqrt {c x^2}+\frac {1}{2} b x \sqrt {c x^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {15} \[ \int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx=a \sqrt {c x^2}+\frac {1}{2} b x \sqrt {c x^2} \]

[In]

Int[(Sqrt[c*x^2]*(a + b*x))/x,x]

[Out]

a*Sqrt[c*x^2] + (b*x*Sqrt[c*x^2])/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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {c x^2} \int (a+b x) \, dx}{x} \\ & = a \sqrt {c x^2}+\frac {1}{2} b x \sqrt {c x^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx=\frac {c x^2 (2 a+b x)}{2 \sqrt {c x^2}} \]

[In]

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

[Out]

(c*x^2*(2*a + b*x))/(2*Sqrt[c*x^2])

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63

method	result	size
gosper	\(\frac {\left (b x +2 a \right ) \sqrt {c \,x^{2}}}{2}\)	\(17\)
default	\(\frac {\left (b x +2 a \right ) \sqrt {c \,x^{2}}}{2}\)	\(17\)
risch	\(a \sqrt {c \,x^{2}}+\frac {b x \sqrt {c \,x^{2}}}{2}\)	\(22\)
trager	\(\frac {\left (b x +2 a +b \right ) \left (-1+x \right ) \sqrt {c \,x^{2}}}{2 x}\)	\(24\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx=\frac {1}{2} \, \sqrt {c x^{2}} {\left (b x + 2 \, a\right )} \]

[In]

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

[Out]

1/2*sqrt(c*x^2)*(b*x + 2*a)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx=a \sqrt {c x^{2}} + \frac {b x \sqrt {c x^{2}}}{2} \]

[In]

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

[Out]

a*sqrt(c*x**2) + b*x*sqrt(c*x**2)/2

Maxima [F(-2)]

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

[In]

integrate((b*x+a)*(c*x^2)^(1/2)/x,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 = 17, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx=\frac {1}{2} \, {\left (b x^{2} + 2 \, a x\right )} \sqrt {c} \mathrm {sgn}\left (x\right ) \]

[In]

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

[Out]

1/2*(b*x^2 + 2*a*x)*sqrt(c)*sgn(x)

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {c x^2} (a+b x)}{x} \, dx=\frac {\sqrt {c}\,\left |x\right |\,\left (2\,a+b\,x\right )}{2} \]

[In]

int(((c*x^2)^(1/2)*(a + b*x))/x,x)

[Out]

(c^(1/2)*abs(x)*(2*a + b*x))/2