\(\int \frac {(a^2+2 a b x^2+b^2 x^4)^3}{x^{10}} \, dx\) [462]

Optimal result

Integrand size = 24, antiderivative size = 74 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx=-\frac {a^6}{9 x^9}-\frac {6 a^5 b}{7 x^7}-\frac {3 a^4 b^2}{x^5}-\frac {20 a^3 b^3}{3 x^3}-\frac {15 a^2 b^4}{x}+6 a b^5 x+\frac {b^6 x^3}{3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 74, 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} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx=-\frac {a^6}{9 x^9}-\frac {6 a^5 b}{7 x^7}-\frac {3 a^4 b^2}{x^5}-\frac {20 a^3 b^3}{3 x^3}-\frac {15 a^2 b^4}{x}+6 a b^5 x+\frac {b^6 x^3}{3} \]

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^10,x]

[Out]

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

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^{10}} \, dx}{b^6} \\ & = \frac {\int \left (6 a b^{11}+\frac {a^6 b^6}{x^{10}}+\frac {6 a^5 b^7}{x^8}+\frac {15 a^4 b^8}{x^6}+\frac {20 a^3 b^9}{x^4}+\frac {15 a^2 b^{10}}{x^2}+b^{12} x^2\right ) \, dx}{b^6} \\ & = -\frac {a^6}{9 x^9}-\frac {6 a^5 b}{7 x^7}-\frac {3 a^4 b^2}{x^5}-\frac {20 a^3 b^3}{3 x^3}-\frac {15 a^2 b^4}{x}+6 a b^5 x+\frac {b^6 x^3}{3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx=-\frac {a^6}{9 x^9}-\frac {6 a^5 b}{7 x^7}-\frac {3 a^4 b^2}{x^5}-\frac {20 a^3 b^3}{3 x^3}-\frac {15 a^2 b^4}{x}+6 a b^5 x+\frac {b^6 x^3}{3} \]

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^3/x^10,x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91

method	result	size
default	\(-\frac {a^{6}}{9 x^{9}}-\frac {6 a^{5} b}{7 x^{7}}-\frac {3 a^{4} b^{2}}{x^{5}}-\frac {20 a^{3} b^{3}}{3 x^{3}}-\frac {15 a^{2} b^{4}}{x}+6 a \,b^{5} x +\frac {b^{6} x^{3}}{3}\)	\(67\)
risch	\(\frac {b^{6} x^{3}}{3}+6 a \,b^{5} x +\frac {-15 a^{2} b^{4} x^{8}-\frac {20}{3} a^{3} b^{3} x^{6}-3 a^{4} b^{2} x^{4}-\frac {6}{7} a^{5} b \,x^{2}-\frac {1}{9} a^{6}}{x^{9}}\)	\(69\)
norman	\(\frac {\frac {1}{3} b^{6} x^{12}+6 a \,b^{5} x^{10}-15 a^{2} b^{4} x^{8}-\frac {20}{3} a^{3} b^{3} x^{6}-3 a^{4} b^{2} x^{4}-\frac {6}{7} a^{5} b \,x^{2}-\frac {1}{9} a^{6}}{x^{9}}\)	\(70\)
gosper	\(-\frac {-21 b^{6} x^{12}-378 a \,b^{5} x^{10}+945 a^{2} b^{4} x^{8}+420 a^{3} b^{3} x^{6}+189 a^{4} b^{2} x^{4}+54 a^{5} b \,x^{2}+7 a^{6}}{63 x^{9}}\)	\(71\)
parallelrisch	\(\frac {21 b^{6} x^{12}+378 a \,b^{5} x^{10}-945 a^{2} b^{4} x^{8}-420 a^{3} b^{3} x^{6}-189 a^{4} b^{2} x^{4}-54 a^{5} b \,x^{2}-7 a^{6}}{63 x^{9}}\)	\(71\)

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^3/x^10,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx=\frac {21 \, b^{6} x^{12} + 378 \, a b^{5} x^{10} - 945 \, a^{2} b^{4} x^{8} - 420 \, a^{3} b^{3} x^{6} - 189 \, a^{4} b^{2} x^{4} - 54 \, a^{5} b x^{2} - 7 \, a^{6}}{63 \, x^{9}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^10,x, algorithm="fricas")

[Out]

1/63*(21*b^6*x^12 + 378*a*b^5*x^10 - 945*a^2*b^4*x^8 - 420*a^3*b^3*x^6 - 189*a^4*b^2*x^4 - 54*a^5*b*x^2 - 7*a^
6)/x^9

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.99 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx=6 a b^{5} x + \frac {b^{6} x^{3}}{3} + \frac {- 7 a^{6} - 54 a^{5} b x^{2} - 189 a^{4} b^{2} x^{4} - 420 a^{3} b^{3} x^{6} - 945 a^{2} b^{4} x^{8}}{63 x^{9}} \]

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**3/x**10,x)

[Out]

6*a*b**5*x + b**6*x**3/3 + (-7*a**6 - 54*a**5*b*x**2 - 189*a**4*b**2*x**4 - 420*a**3*b**3*x**6 - 945*a**2*b**4
*x**8)/(63*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx=\frac {1}{3} \, b^{6} x^{3} + 6 \, a b^{5} x - \frac {945 \, a^{2} b^{4} x^{8} + 420 \, a^{3} b^{3} x^{6} + 189 \, a^{4} b^{2} x^{4} + 54 \, a^{5} b x^{2} + 7 \, a^{6}}{63 \, x^{9}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^10,x, algorithm="maxima")

[Out]

1/3*b^6*x^3 + 6*a*b^5*x - 1/63*(945*a^2*b^4*x^8 + 420*a^3*b^3*x^6 + 189*a^4*b^2*x^4 + 54*a^5*b*x^2 + 7*a^6)/x^
9

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx=\frac {1}{3} \, b^{6} x^{3} + 6 \, a b^{5} x - \frac {945 \, a^{2} b^{4} x^{8} + 420 \, a^{3} b^{3} x^{6} + 189 \, a^{4} b^{2} x^{4} + 54 \, a^{5} b x^{2} + 7 \, a^{6}}{63 \, x^{9}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^3/x^10,x, algorithm="giac")

[Out]

1/3*b^6*x^3 + 6*a*b^5*x - 1/63*(945*a^2*b^4*x^8 + 420*a^3*b^3*x^6 + 189*a^4*b^2*x^4 + 54*a^5*b*x^2 + 7*a^6)/x^
9

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^{10}} \, dx=-\frac {\frac {a^6}{9}+\frac {6\,a^5\,b\,x^2}{7}+3\,a^4\,b^2\,x^4+\frac {20\,a^3\,b^3\,x^6}{3}+15\,a^2\,b^4\,x^8-6\,a\,b^5\,x^{10}-\frac {b^6\,x^{12}}{3}}{x^9} \]

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^3/x^10,x)

[Out]

-(a^6/9 - (b^6*x^12)/3 + (6*a^5*b*x^2)/7 - 6*a*b^5*x^10 + 3*a^4*b^2*x^4 + (20*a^3*b^3*x^6)/3 + 15*a^2*b^4*x^8)
/x^9