\(\int \frac {(a^2+2 a b x^2+b^2 x^4)^3}{x^{18}} \, dx\) [470]

Optimal result

Integrand size = 24, antiderivative size = 82 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx=-\frac {a^6}{17 x^{17}}-\frac {2 a^5 b}{5 x^{15}}-\frac {15 a^4 b^2}{13 x^{13}}-\frac {20 a^3 b^3}{11 x^{11}}-\frac {5 a^2 b^4}{3 x^9}-\frac {6 a b^5}{7 x^7}-\frac {b^6}{5 x^5} \]

[Out]

-1/17*a^6/x^17-2/5*a^5*b/x^15-15/13*a^4*b^2/x^13-20/11*a^3*b^3/x^11-5/3*a^2*b^4/x^9-6/7*a*b^5/x^7-1/5*b^6/x^5

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 276} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx=-\frac {a^6}{17 x^{17}}-\frac {2 a^5 b}{5 x^{15}}-\frac {15 a^4 b^2}{13 x^{13}}-\frac {20 a^3 b^3}{11 x^{11}}-\frac {5 a^2 b^4}{3 x^9}-\frac {6 a b^5}{7 x^7}-\frac {b^6}{5 x^5} \]

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^18,x]

[Out]

-1/17*a^6/x^17 - (2*a^5*b)/(5*x^15) - (15*a^4*b^2)/(13*x^13) - (20*a^3*b^3)/(11*x^11) - (5*a^2*b^4)/(3*x^9) -
(6*a*b^5)/(7*x^7) - b^6/(5*x^5)

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^{18}} \, dx}{b^6} \\ & = \frac {\int \left (\frac {a^6 b^6}{x^{18}}+\frac {6 a^5 b^7}{x^{16}}+\frac {15 a^4 b^8}{x^{14}}+\frac {20 a^3 b^9}{x^{12}}+\frac {15 a^2 b^{10}}{x^{10}}+\frac {6 a b^{11}}{x^8}+\frac {b^{12}}{x^6}\right ) \, dx}{b^6} \\ & = -\frac {a^6}{17 x^{17}}-\frac {2 a^5 b}{5 x^{15}}-\frac {15 a^4 b^2}{13 x^{13}}-\frac {20 a^3 b^3}{11 x^{11}}-\frac {5 a^2 b^4}{3 x^9}-\frac {6 a b^5}{7 x^7}-\frac {b^6}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx=-\frac {a^6}{17 x^{17}}-\frac {2 a^5 b}{5 x^{15}}-\frac {15 a^4 b^2}{13 x^{13}}-\frac {20 a^3 b^3}{11 x^{11}}-\frac {5 a^2 b^4}{3 x^9}-\frac {6 a b^5}{7 x^7}-\frac {b^6}{5 x^5} \]

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^18,x]

[Out]

-1/17*a^6/x^17 - (2*a^5*b)/(5*x^15) - (15*a^4*b^2)/(13*x^13) - (20*a^3*b^3)/(11*x^11) - (5*a^2*b^4)/(3*x^9) -
(6*a*b^5)/(7*x^7) - b^6/(5*x^5)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {a^{6}}{17 x^{17}}-\frac {2 a^{5} b}{5 x^{15}}-\frac {15 a^{4} b^{2}}{13 x^{13}}-\frac {20 a^{3} b^{3}}{11 x^{11}}-\frac {5 a^{2} b^{4}}{3 x^{9}}-\frac {6 a \,b^{5}}{7 x^{7}}-\frac {b^{6}}{5 x^{5}}\)	\(69\)
norman	\(\frac {-\frac {1}{17} a^{6}-\frac {2}{5} a^{5} b \,x^{2}-\frac {15}{13} a^{4} b^{2} x^{4}-\frac {20}{11} a^{3} b^{3} x^{6}-\frac {5}{3} a^{2} b^{4} x^{8}-\frac {6}{7} a \,b^{5} x^{10}-\frac {1}{5} b^{6} x^{12}}{x^{17}}\)	\(70\)
risch	\(\frac {-\frac {1}{17} a^{6}-\frac {2}{5} a^{5} b \,x^{2}-\frac {15}{13} a^{4} b^{2} x^{4}-\frac {20}{11} a^{3} b^{3} x^{6}-\frac {5}{3} a^{2} b^{4} x^{8}-\frac {6}{7} a \,b^{5} x^{10}-\frac {1}{5} b^{6} x^{12}}{x^{17}}\)	\(70\)
gosper	\(-\frac {51051 b^{6} x^{12}+218790 a \,b^{5} x^{10}+425425 a^{2} b^{4} x^{8}+464100 a^{3} b^{3} x^{6}+294525 a^{4} b^{2} x^{4}+102102 a^{5} b \,x^{2}+15015 a^{6}}{255255 x^{17}}\)	\(71\)
parallelrisch	\(\frac {-51051 b^{6} x^{12}-218790 a \,b^{5} x^{10}-425425 a^{2} b^{4} x^{8}-464100 a^{3} b^{3} x^{6}-294525 a^{4} b^{2} x^{4}-102102 a^{5} b \,x^{2}-15015 a^{6}}{255255 x^{17}}\)	\(71\)

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^3/x^18,x,method=_RETURNVERBOSE)

[Out]

-1/17*a^6/x^17-2/5*a^5*b/x^15-15/13*a^4*b^2/x^13-20/11*a^3*b^3/x^11-5/3*a^2*b^4/x^9-6/7*a*b^5/x^7-1/5*b^6/x^5

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx=-\frac {51051 \, b^{6} x^{12} + 218790 \, a b^{5} x^{10} + 425425 \, a^{2} b^{4} x^{8} + 464100 \, a^{3} b^{3} x^{6} + 294525 \, a^{4} b^{2} x^{4} + 102102 \, a^{5} b x^{2} + 15015 \, a^{6}}{255255 \, x^{17}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^18,x, algorithm="fricas")

[Out]

-1/255255*(51051*b^6*x^12 + 218790*a*b^5*x^10 + 425425*a^2*b^4*x^8 + 464100*a^3*b^3*x^6 + 294525*a^4*b^2*x^4 +
102102*a^5*b*x^2 + 15015*a^6)/x^17

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx=\frac {- 15015 a^{6} - 102102 a^{5} b x^{2} - 294525 a^{4} b^{2} x^{4} - 464100 a^{3} b^{3} x^{6} - 425425 a^{2} b^{4} x^{8} - 218790 a b^{5} x^{10} - 51051 b^{6} x^{12}}{255255 x^{17}} \]

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**18,x)

[Out]

(-15015*a**6 - 102102*a**5*b*x**2 - 294525*a**4*b**2*x**4 - 464100*a**3*b**3*x**6 - 425425*a**2*b**4*x**8 - 21
8790*a*b**5*x**10 - 51051*b**6*x**12)/(255255*x**17)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx=-\frac {51051 \, b^{6} x^{12} + 218790 \, a b^{5} x^{10} + 425425 \, a^{2} b^{4} x^{8} + 464100 \, a^{3} b^{3} x^{6} + 294525 \, a^{4} b^{2} x^{4} + 102102 \, a^{5} b x^{2} + 15015 \, a^{6}}{255255 \, x^{17}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^18,x, algorithm="maxima")

[Out]

-1/255255*(51051*b^6*x^12 + 218790*a*b^5*x^10 + 425425*a^2*b^4*x^8 + 464100*a^3*b^3*x^6 + 294525*a^4*b^2*x^4 +
102102*a^5*b*x^2 + 15015*a^6)/x^17

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx=-\frac {51051 \, b^{6} x^{12} + 218790 \, a b^{5} x^{10} + 425425 \, a^{2} b^{4} x^{8} + 464100 \, a^{3} b^{3} x^{6} + 294525 \, a^{4} b^{2} x^{4} + 102102 \, a^{5} b x^{2} + 15015 \, a^{6}}{255255 \, x^{17}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^18,x, algorithm="giac")

[Out]

-1/255255*(51051*b^6*x^12 + 218790*a*b^5*x^10 + 425425*a^2*b^4*x^8 + 464100*a^3*b^3*x^6 + 294525*a^4*b^2*x^4 +
102102*a^5*b*x^2 + 15015*a^6)/x^17

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.85 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{18}} \, dx=-\frac {\frac {a^6}{17}+\frac {2\,a^5\,b\,x^2}{5}+\frac {15\,a^4\,b^2\,x^4}{13}+\frac {20\,a^3\,b^3\,x^6}{11}+\frac {5\,a^2\,b^4\,x^8}{3}+\frac {6\,a\,b^5\,x^{10}}{7}+\frac {b^6\,x^{12}}{5}}{x^{17}} \]

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^18,x)

[Out]

-(a^6/17 + (b^6*x^12)/5 + (2*a^5*b*x^2)/5 + (6*a*b^5*x^10)/7 + (15*a^4*b^2*x^4)/13 + (20*a^3*b^3*x^6)/11 + (5*
a^2*b^4*x^8)/3)/x^17