\(\int \frac {x^2}{(a+b x)^{10}} \, dx\) [232]

Optimal result

Integrand size = 11, antiderivative size = 47 \[ \int \frac {x^2}{(a+b x)^{10}} \, dx=-\frac {a^2}{9 b^3 (a+b x)^9}+\frac {a}{4 b^3 (a+b x)^8}-\frac {1}{7 b^3 (a+b x)^7} \]

[Out]

-1/9*a^2/b^3/(b*x+a)^9+1/4*a/b^3/(b*x+a)^8-1/7/b^3/(b*x+a)^7

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {x^2}{(a+b x)^{10}} \, dx=-\frac {a^2}{9 b^3 (a+b x)^9}+\frac {a}{4 b^3 (a+b x)^8}-\frac {1}{7 b^3 (a+b x)^7} \]

[In]

Int[x^2/(a + b*x)^10,x]

[Out]

-1/9*a^2/(b^3*(a + b*x)^9) + a/(4*b^3*(a + b*x)^8) - 1/(7*b^3*(a + b*x)^7)

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^2}{b^2 (a+b x)^{10}}-\frac {2 a}{b^2 (a+b x)^9}+\frac {1}{b^2 (a+b x)^8}\right ) \, dx \\ & = -\frac {a^2}{9 b^3 (a+b x)^9}+\frac {a}{4 b^3 (a+b x)^8}-\frac {1}{7 b^3 (a+b x)^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{(a+b x)^{10}} \, dx=-\frac {a^2+9 a b x+36 b^2 x^2}{252 b^3 (a+b x)^9} \]

[In]

Integrate[x^2/(a + b*x)^10,x]

[Out]

-1/252*(a^2 + 9*a*b*x + 36*b^2*x^2)/(b^3*(a + b*x)^9)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {36 b^{2} x^{2}+9 a b x +a^{2}}{252 \left (b x +a \right )^{9} b^{3}}\)	\(30\)
norman	\(\frac {-\frac {x^{2}}{7 b}-\frac {a x}{28 b^{2}}-\frac {a^{2}}{252 b^{3}}}{\left (b x +a \right )^{9}}\)	\(33\)
risch	\(\frac {-\frac {x^{2}}{7 b}-\frac {a x}{28 b^{2}}-\frac {a^{2}}{252 b^{3}}}{\left (b x +a \right )^{9}}\)	\(33\)
parallelrisch	\(\frac {-36 b^{8} x^{2}-9 a \,b^{7} x -a^{2} b^{6}}{252 b^{9} \left (b x +a \right )^{9}}\)	\(37\)
default	\(-\frac {a^{2}}{9 b^{3} \left (b x +a \right )^{9}}+\frac {a}{4 b^{3} \left (b x +a \right )^{8}}-\frac {1}{7 b^{3} \left (b x +a \right )^{7}}\)	\(42\)

[In]

int(x^2/(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

-1/252*(36*b^2*x^2+9*a*b*x+a^2)/(b*x+a)^9/b^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (41) = 82\).

Time = 0.22 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.55 \[ \int \frac {x^2}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{2} x^{2} + 9 \, a b x + a^{2}}{252 \, {\left (b^{12} x^{9} + 9 \, a b^{11} x^{8} + 36 \, a^{2} b^{10} x^{7} + 84 \, a^{3} b^{9} x^{6} + 126 \, a^{4} b^{8} x^{5} + 126 \, a^{5} b^{7} x^{4} + 84 \, a^{6} b^{6} x^{3} + 36 \, a^{7} b^{5} x^{2} + 9 \, a^{8} b^{4} x + a^{9} b^{3}\right )}} \]

[In]

integrate(x^2/(b*x+a)^10,x, algorithm="fricas")

[Out]

-1/252*(36*b^2*x^2 + 9*a*b*x + a^2)/(b^12*x^9 + 9*a*b^11*x^8 + 36*a^2*b^10*x^7 + 84*a^3*b^9*x^6 + 126*a^4*b^8*
x^5 + 126*a^5*b^7*x^4 + 84*a^6*b^6*x^3 + 36*a^7*b^5*x^2 + 9*a^8*b^4*x + a^9*b^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (41) = 82\).

Time = 0.44 (sec) , antiderivative size = 128, normalized size of antiderivative = 2.72 \[ \int \frac {x^2}{(a+b x)^{10}} \, dx=\frac {- a^{2} - 9 a b x - 36 b^{2} x^{2}}{252 a^{9} b^{3} + 2268 a^{8} b^{4} x + 9072 a^{7} b^{5} x^{2} + 21168 a^{6} b^{6} x^{3} + 31752 a^{5} b^{7} x^{4} + 31752 a^{4} b^{8} x^{5} + 21168 a^{3} b^{9} x^{6} + 9072 a^{2} b^{10} x^{7} + 2268 a b^{11} x^{8} + 252 b^{12} x^{9}} \]

[In]

integrate(x**2/(b*x+a)**10,x)

[Out]

(-a**2 - 9*a*b*x - 36*b**2*x**2)/(252*a**9*b**3 + 2268*a**8*b**4*x + 9072*a**7*b**5*x**2 + 21168*a**6*b**6*x**
3 + 31752*a**5*b**7*x**4 + 31752*a**4*b**8*x**5 + 21168*a**3*b**9*x**6 + 9072*a**2*b**10*x**7 + 2268*a*b**11*x
**8 + 252*b**12*x**9)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (41) = 82\).

Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.55 \[ \int \frac {x^2}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{2} x^{2} + 9 \, a b x + a^{2}}{252 \, {\left (b^{12} x^{9} + 9 \, a b^{11} x^{8} + 36 \, a^{2} b^{10} x^{7} + 84 \, a^{3} b^{9} x^{6} + 126 \, a^{4} b^{8} x^{5} + 126 \, a^{5} b^{7} x^{4} + 84 \, a^{6} b^{6} x^{3} + 36 \, a^{7} b^{5} x^{2} + 9 \, a^{8} b^{4} x + a^{9} b^{3}\right )}} \]

[In]

integrate(x^2/(b*x+a)^10,x, algorithm="maxima")

[Out]

-1/252*(36*b^2*x^2 + 9*a*b*x + a^2)/(b^12*x^9 + 9*a*b^11*x^8 + 36*a^2*b^10*x^7 + 84*a^3*b^9*x^6 + 126*a^4*b^8*
x^5 + 126*a^5*b^7*x^4 + 84*a^6*b^6*x^3 + 36*a^7*b^5*x^2 + 9*a^8*b^4*x + a^9*b^3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.62 \[ \int \frac {x^2}{(a+b x)^{10}} \, dx=-\frac {36 \, b^{2} x^{2} + 9 \, a b x + a^{2}}{252 \, {\left (b x + a\right )}^{9} b^{3}} \]

[In]

integrate(x^2/(b*x+a)^10,x, algorithm="giac")

[Out]

-1/252*(36*b^2*x^2 + 9*a*b*x + a^2)/((b*x + a)^9*b^3)

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \frac {x^2}{(a+b x)^{10}} \, dx=-\frac {8\,a^2+72\,a\,b\,x+288\,b^2\,x^2}{2016\,b^3\,{\left (a+b\,x\right )}^9} \]

[In]

int(x^2/(a + b*x)^10,x)

[Out]

-(8*a^2 + 288*b^2*x^2 + 72*a*b*x)/(2016*b^3*(a + b*x)^9)