\(\int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx\) [787]

Optimal result

Integrand size = 18, antiderivative size = 35 \[ \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx=-\frac {a}{4 x^3 \sqrt {c x^2}}-\frac {b}{3 x^2 \sqrt {c x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, 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} \[ \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx=-\frac {a}{4 x^3 \sqrt {c x^2}}-\frac {b}{3 x^2 \sqrt {c x^2}} \]

[In]

Int[(a + b*x)/(x^4*Sqrt[c*x^2]),x]

[Out]

-1/4*a/(x^3*Sqrt[c*x^2]) - b/(3*x^2*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^5} \, dx}{\sqrt {c x^2}} \\ & = \frac {x \int \left (\frac {a}{x^5}+\frac {b}{x^4}\right ) \, dx}{\sqrt {c x^2}} \\ & = -\frac {a}{4 x^3 \sqrt {c x^2}}-\frac {b}{3 x^2 \sqrt {c x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.69 \[ \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx=-\frac {3 a+4 b x}{12 x^3 \sqrt {c x^2}} \]

[In]

Integrate[(a + b*x)/(x^4*Sqrt[c*x^2]),x]

[Out]

-1/12*(3*a + 4*b*x)/(x^3*Sqrt[c*x^2])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57

method	result	size
risch	\(\frac {-\frac {b x}{3}-\frac {a}{4}}{x^{3} \sqrt {c \,x^{2}}}\)	\(20\)
gosper	\(-\frac {4 b x +3 a}{12 x^{3} \sqrt {c \,x^{2}}}\)	\(21\)
default	\(-\frac {4 b x +3 a}{12 x^{3} \sqrt {c \,x^{2}}}\)	\(21\)
trager	\(\frac {\left (-1+x \right ) \left (3 a \,x^{3}+4 b \,x^{3}+3 a \,x^{2}+4 b \,x^{2}+3 a x +4 b x +3 a \right ) \sqrt {c \,x^{2}}}{12 c \,x^{5}}\)	\(55\)

[In]

int((b*x+a)/x^4/(c*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.66 \[ \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx=-\frac {\sqrt {c x^{2}} {\left (4 \, b x + 3 \, a\right )}}{12 \, c x^{5}} \]

[In]

integrate((b*x+a)/x^4/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

-1/12*sqrt(c*x^2)*(4*b*x + 3*a)/(c*x^5)

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx=- \frac {a}{4 x^{3} \sqrt {c x^{2}}} - \frac {b}{3 x^{2} \sqrt {c x^{2}}} \]

[In]

integrate((b*x+a)/x**4/(c*x**2)**(1/2),x)

[Out]

-a/(4*x**3*sqrt(c*x**2)) - b/(3*x**2*sqrt(c*x**2))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.54 \[ \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx=-\frac {b}{3 \, \sqrt {c} x^{3}} - \frac {a}{4 \, \sqrt {c} x^{4}} \]

[In]

integrate((b*x+a)/x^4/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.57 \[ \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx=-\frac {4 \, b x + 3 \, a}{12 \, \sqrt {c} x^{4} \mathrm {sgn}\left (x\right )} \]

[In]

integrate((b*x+a)/x^4/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

-1/12*(4*b*x + 3*a)/(sqrt(c)*x^4*sgn(x))

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.74 \[ \int \frac {a+b x}{x^4 \sqrt {c x^2}} \, dx=-\frac {3\,a\,\sqrt {x^2}+4\,b\,x\,\sqrt {x^2}}{12\,\sqrt {c}\,x^5} \]

[In]

int((a + b*x)/(x^4*(c*x^2)^(1/2)),x)

[Out]

-(3*a*(x^2)^(1/2) + 4*b*x*(x^2)^(1/2))/(12*c^(1/2)*x^5)