\(\int \frac {x}{(a+b x)^5} \, dx\) [43]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int \frac {x}{(a+b x)^5} \, dx=\frac {-\frac {a}{12 b^2}-\frac {x}{3 b}}{(a+b x)^4} \]

[Out]

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

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} \[ \int \frac {x}{(a+b x)^5} \, dx=\frac {a}{4 b^2 (a+b x)^4}-\frac {1}{3 b^2 (a+b x)^3} \]

[In]

Int[x/(a + b*x)^5,x]

[Out]

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

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)^5}+\frac {1}{b (a+b x)^4}\right ) \, dx \\ & = \frac {a}{4 b^2 (a+b x)^4}-\frac {1}{3 b^2 (a+b x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {x}{(a+b x)^5} \, dx=-\frac {a+4 b x}{12 b^2 (a+b x)^4} \]

[In]

Integrate[x/(a + b*x)^5,x]

[Out]

-1/12*(a + 4*b*x)/(b^2*(a + b*x)^4)

Maple [A] (verified)

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

method	result	size
gosper	\(-\frac {4 b x +a}{12 \left (b x +a \right )^{4} b^{2}}\)	\(19\)
norman	\(\frac {-\frac {x}{3 b}-\frac {a}{12 b^{2}}}{\left (b x +a \right )^{4}}\)	\(22\)
risch	\(\frac {-\frac {x}{3 b}-\frac {a}{12 b^{2}}}{\left (b x +a \right )^{4}}\)	\(22\)
parallelrisch	\(\frac {-4 b^{3} x -a \,b^{2}}{12 b^{4} \left (b x +a \right )^{4}}\)	\(26\)
default	\(\frac {a}{4 b^{2} \left (b x +a \right )^{4}}-\frac {1}{3 b^{2} \left (b x +a \right )^{3}}\)	\(27\)

[In]

int(x/(b*x+a)^5,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).

Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {x}{(a+b x)^5} \, dx=-\frac {4 \, b x + a}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]

[In]

integrate(x/(b*x+a)^5,x, algorithm="fricas")

[Out]

-1/12*(4*b*x + a)/(b^6*x^4 + 4*a*b^5*x^3 + 6*a^2*b^4*x^2 + 4*a^3*b^3*x + a^4*b^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (19) = 38\).

Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.24 \[ \int \frac {x}{(a+b x)^5} \, dx=\frac {- a - 4 b x}{12 a^{4} b^{2} + 48 a^{3} b^{3} x + 72 a^{2} b^{4} x^{2} + 48 a b^{5} x^{3} + 12 b^{6} x^{4}} \]

[In]

integrate(x/(b*x+a)**5,x)

[Out]

(-a - 4*b*x)/(12*a**4*b**2 + 48*a**3*b**3*x + 72*a**2*b**4*x**2 + 48*a*b**5*x**3 + 12*b**6*x**4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).

Time = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16 \[ \int \frac {x}{(a+b x)^5} \, dx=-\frac {4 \, b x + a}{12 \, {\left (b^{6} x^{4} + 4 \, a b^{5} x^{3} + 6 \, a^{2} b^{4} x^{2} + 4 \, a^{3} b^{3} x + a^{4} b^{2}\right )}} \]

[In]

integrate(x/(b*x+a)^5,x, algorithm="maxima")

[Out]

-1/12*(4*b*x + a)/(b^6*x^4 + 4*a*b^5*x^3 + 6*a^2*b^4*x^2 + 4*a^3*b^3*x + a^4*b^2)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {x}{(a+b x)^5} \, dx=-\frac {\frac {4}{{\left (b x + a\right )}^{3} b} - \frac {3 \, a}{{\left (b x + a\right )}^{4} b}}{12 \, b} \]

[In]

integrate(x/(b*x+a)^5,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72 \[ \int \frac {x}{(a+b x)^5} \, dx=-\frac {a+4\,b\,x}{12\,b^2\,{\left (a+b\,x\right )}^4} \]

[In]

int(x/(a + b*x)^5,x)

[Out]

-(a + 4*b*x)/(12*b^2*(a + b*x)^4)