\(\int \frac {x^2}{(a+b x)^4} \, dx\) [199]

Optimal result

Integrand size = 11, antiderivative size = 17 \[ \int \frac {x^2}{(a+b x)^4} \, dx=\frac {x^3}{3 a (a+b x)^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {37} \[ \int \frac {x^2}{(a+b x)^4} \, dx=\frac {x^3}{3 a (a+b x)^3} \]

[In]

Int[x^2/(a + b*x)^4,x]

[Out]

x^3/(3*a*(a + b*x)^3)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x^3}{3 a (a+b x)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {x^2}{(a+b x)^4} \, dx=-\frac {a^2+3 a b x+3 b^2 x^2}{3 b^3 (a+b x)^3} \]

[In]

Integrate[x^2/(a + b*x)^4,x]

[Out]

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

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76

method	result	size
gosper	\(-\frac {3 b^{2} x^{2}+3 a b x +a^{2}}{3 b^{3} \left (b x +a \right )^{3}}\)	\(30\)
parallelrisch	\(\frac {-3 b^{2} x^{2}-3 a b x -a^{2}}{3 b^{3} \left (b x +a \right )^{3}}\)	\(32\)
norman	\(\frac {-\frac {x^{2}}{b}-\frac {a x}{b^{2}}-\frac {a^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\)	\(33\)
risch	\(\frac {-\frac {x^{2}}{b}-\frac {a x}{b^{2}}-\frac {a^{2}}{3 b^{3}}}{\left (b x +a \right )^{3}}\)	\(33\)
default	\(-\frac {a^{2}}{3 b^{3} \left (b x +a \right )^{3}}+\frac {a}{b^{3} \left (b x +a \right )^{2}}-\frac {1}{\left (b x +a \right ) b^{3}}\)	\(41\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (15) = 30\).

Time = 0.22 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.18 \[ \int \frac {x^2}{(a+b x)^4} \, dx=-\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (12) = 24\).

Time = 0.13 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.29 \[ \int \frac {x^2}{(a+b x)^4} \, dx=\frac {- a^{2} - 3 a b x - 3 b^{2} x^{2}}{3 a^{3} b^{3} + 9 a^{2} b^{4} x + 9 a b^{5} x^{2} + 3 b^{6} x^{3}} \]

[In]

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

[Out]

(-a**2 - 3*a*b*x - 3*b**2*x**2)/(3*a**3*b**3 + 9*a**2*b**4*x + 9*a*b**5*x**2 + 3*b**6*x**3)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (15) = 30\).

Time = 0.25 (sec) , antiderivative size = 54, normalized size of antiderivative = 3.18 \[ \int \frac {x^2}{(a+b x)^4} \, dx=-\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b^{6} x^{3} + 3 \, a b^{5} x^{2} + 3 \, a^{2} b^{4} x + a^{3} b^{3}\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int \frac {x^2}{(a+b x)^4} \, dx=-\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, {\left (b x + a\right )}^{3} b^{3}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.29 \[ \int \frac {x^2}{(a+b x)^4} \, dx=-\frac {a^2+3\,a\,b\,x+3\,b^2\,x^2}{3\,a^3\,b^3+9\,a^2\,b^4\,x+9\,a\,b^5\,x^2+3\,b^6\,x^3} \]

[In]

int(x^2/(a + b*x)^4,x)

[Out]

-(a^2 + 3*b^2*x^2 + 3*a*b*x)/(3*a^3*b^3 + 3*b^6*x^3 + 9*a^2*b^4*x + 9*a*b^5*x^2)