\(\int \frac {x}{(a+b x)^4} \, dx\) [39]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int \frac {x}{(a+b x)^4} \, dx=\frac {-\frac {a}{6 b^2}-\frac {x}{2 b}}{(a+b x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45} \[ \int \frac {x}{(a+b x)^4} \, dx=\frac {a}{3 b^2 (a+b x)^3}-\frac {1}{2 b^2 (a+b x)^2} \]

[In]

Int[x/(a + b*x)^4,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {a+3 b x}{6 b^2 (a+b x)^3} \]

[In]

Integrate[x/(a + b*x)^4,x]

[Out]

-1/6*(a + 3*b*x)/(b^2*(a + b*x)^3)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76

method	result	size
gosper	\(-\frac {3 b x +a}{6 \left (b x +a \right )^{3} b^{2}}\)	\(19\)
norman	\(\frac {-\frac {x}{2 b}-\frac {a}{6 b^{2}}}{\left (b x +a \right )^{3}}\)	\(22\)
risch	\(\frac {-\frac {x}{2 b}-\frac {a}{6 b^{2}}}{\left (b x +a \right )^{3}}\)	\(22\)
parallelrisch	\(\frac {-3 b^{2} x -a b}{6 b^{3} \left (b x +a \right )^{3}}\)	\(24\)
default	\(-\frac {1}{2 b^{2} \left (b x +a \right )^{2}}+\frac {a}{3 b^{2} \left (b x +a \right )^{3}}\)	\(27\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).

Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

[In]

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

[Out]

-1/6*(3*b*x + a)/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).

Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {x}{(a+b x)^4} \, dx=\frac {- a - 3 b x}{6 a^{3} b^{2} + 18 a^{2} b^{3} x + 18 a b^{4} x^{2} + 6 b^{5} x^{3}} \]

[In]

integrate(x/(b*x+a)**4,x)

[Out]

(-a - 3*b*x)/(6*a**3*b**2 + 18*a**2*b**3*x + 18*a*b**4*x**2 + 6*b**5*x**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b^{5} x^{3} + 3 \, a b^{4} x^{2} + 3 \, a^{2} b^{3} x + a^{3} b^{2}\right )}} \]

[In]

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

[Out]

-1/6*(3*b*x + a)/(b^5*x^3 + 3*a*b^4*x^2 + 3*a^2*b^3*x + a^3*b^2)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {3 \, b x + a}{6 \, {\left (b x + a\right )}^{3} b^{2}} \]

[In]

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

[Out]

-1/6*(3*b*x + a)/((b*x + a)^3*b^2)

Mupad [B] (verification not implemented)

Time = 18.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {x}{(a+b x)^4} \, dx=-\frac {\frac {a}{6\,b^2}+\frac {x}{2\,b}}{a^3+3\,a^2\,b\,x+3\,a\,b^2\,x^2+b^3\,x^3} \]

[In]

int(x/(a + b*x)^4,x)

[Out]

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