\(\int \frac {a+b x}{x (c x^2)^{3/2}} \, dx\) [792]

Optimal result

Integrand size = 18, antiderivative size = 41 \[ \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {a}{3 c x^2 \sqrt {c x^2}}-\frac {b}{2 c x \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 41, 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} \[ \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {a}{3 c x^2 \sqrt {c x^2}}-\frac {b}{2 c x \sqrt {c x^2}} \]

[In]

Int[(a + b*x)/(x*(c*x^2)^(3/2)),x]

[Out]

-1/3*a/(c*x^2*Sqrt[c*x^2]) - b/(2*c*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^4} \, dx}{c \sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {a}{x^4}+\frac {b}{x^3}\right ) \, dx}{c \sqrt {c x^2}} \\ & = -\frac {a}{3 c x^2 \sqrt {c x^2}}-\frac {b}{2 c x \sqrt {c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.61 \[ \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {c x^2 (2 a+3 b x)}{6 \left (c x^2\right )^{5/2}} \]

[In]

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

[Out]

-1/6*(c*x^2*(2*a + 3*b*x))/(c*x^2)^(5/2)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.44

method	result	size
gosper	\(-\frac {3 b x +2 a}{6 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(18\)
default	\(-\frac {3 b x +2 a}{6 \left (c \,x^{2}\right )^{\frac {3}{2}}}\)	\(18\)
risch	\(\frac {-\frac {b x}{2}-\frac {a}{3}}{c \,x^{2} \sqrt {c \,x^{2}}}\)	\(23\)
trager	\(\frac {\left (-1+x \right ) \left (2 a \,x^{2}+3 b \,x^{2}+2 a x +3 b x +2 a \right ) \sqrt {c \,x^{2}}}{6 c^{2} x^{4}}\)	\(43\)

[In]

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

[Out]

-1/6*(3*b*x+2*a)/(c*x^2)^(3/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.56 \[ \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {c x^{2}} {\left (3 \, b x + 2 \, a\right )}}{6 \, c^{2} x^{4}} \]

[In]

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

[Out]

-1/6*sqrt(c*x^2)*(3*b*x + 2*a)/(c^2*x^4)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx=- \frac {a}{3 \left (c x^{2}\right )^{\frac {3}{2}}} - \frac {b x}{2 \left (c x^{2}\right )^{\frac {3}{2}}} \]

[In]

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

[Out]

-a/(3*(c*x**2)**(3/2)) - b*x/(2*(c*x**2)**(3/2))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.46 \[ \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {b}{2 \, c^{\frac {3}{2}} x^{2}} - \frac {a}{3 \, c^{\frac {3}{2}} x^{3}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.49 \[ \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {3 \, b x + 2 \, a}{6 \, c^{\frac {3}{2}} x^{3} \mathrm {sgn}\left (x\right )} \]

[In]

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

[Out]

-1/6*(3*b*x + 2*a)/(c^(3/2)*x^3*sgn(x))

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.63 \[ \int \frac {a+b x}{x \left (c x^2\right )^{3/2}} \, dx=-\frac {2\,a\,\sqrt {x^2}+3\,b\,x\,\sqrt {x^2}}{6\,c^{3/2}\,x^4} \]

[In]

int((a + b*x)/(x*(c*x^2)^(3/2)),x)

[Out]

-(2*a*(x^2)^(1/2) + 3*b*x*(x^2)^(1/2))/(6*c^(3/2)*x^4)