\(\int \frac {(a^2+2 a b x^2+b^2 x^4)^2}{x^{14}} \, dx\) [441]

Optimal result

Integrand size = 24, antiderivative size = 56 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{14}} \, dx=-\frac {a^4}{13 x^{13}}-\frac {4 a^3 b}{11 x^{11}}-\frac {2 a^2 b^2}{3 x^9}-\frac {4 a b^3}{7 x^7}-\frac {b^4}{5 x^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, 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} \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{14}} \, dx=-\frac {a^4}{13 x^{13}}-\frac {4 a^3 b}{11 x^{11}}-\frac {2 a^2 b^2}{3 x^9}-\frac {4 a b^3}{7 x^7}-\frac {b^4}{5 x^5} \]

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^14,x]

[Out]

-1/13*a^4/x^13 - (4*a^3*b)/(11*x^11) - (2*a^2*b^2)/(3*x^9) - (4*a*b^3)/(7*x^7) - b^4/(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 )^4}{x^{14}} \, dx}{b^4} \\ & = \frac {\int \left (\frac {a^4 b^4}{x^{14}}+\frac {4 a^3 b^5}{x^{12}}+\frac {6 a^2 b^6}{x^{10}}+\frac {4 a b^7}{x^8}+\frac {b^8}{x^6}\right ) \, dx}{b^4} \\ & = -\frac {a^4}{13 x^{13}}-\frac {4 a^3 b}{11 x^{11}}-\frac {2 a^2 b^2}{3 x^9}-\frac {4 a b^3}{7 x^7}-\frac {b^4}{5 x^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{14}} \, dx=-\frac {a^4}{13 x^{13}}-\frac {4 a^3 b}{11 x^{11}}-\frac {2 a^2 b^2}{3 x^9}-\frac {4 a b^3}{7 x^7}-\frac {b^4}{5 x^5} \]

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^14,x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.84

method	result	size
default	\(-\frac {a^{4}}{13 x^{13}}-\frac {4 a^{3} b}{11 x^{11}}-\frac {2 a^{2} b^{2}}{3 x^{9}}-\frac {4 a \,b^{3}}{7 x^{7}}-\frac {b^{4}}{5 x^{5}}\)	\(47\)
norman	\(\frac {-\frac {1}{5} b^{4} x^{8}-\frac {4}{7} a \,b^{3} x^{6}-\frac {2}{3} a^{2} b^{2} x^{4}-\frac {4}{11} a^{3} b \,x^{2}-\frac {1}{13} a^{4}}{x^{13}}\)	\(48\)
risch	\(\frac {-\frac {1}{5} b^{4} x^{8}-\frac {4}{7} a \,b^{3} x^{6}-\frac {2}{3} a^{2} b^{2} x^{4}-\frac {4}{11} a^{3} b \,x^{2}-\frac {1}{13} a^{4}}{x^{13}}\)	\(48\)
gosper	\(-\frac {3003 b^{4} x^{8}+8580 a \,b^{3} x^{6}+10010 a^{2} b^{2} x^{4}+5460 a^{3} b \,x^{2}+1155 a^{4}}{15015 x^{13}}\)	\(49\)
parallelrisch	\(\frac {-3003 b^{4} x^{8}-8580 a \,b^{3} x^{6}-10010 a^{2} b^{2} x^{4}-5460 a^{3} b \,x^{2}-1155 a^{4}}{15015 x^{13}}\)	\(49\)

[In]

int((b^2*x^4+2*a*b*x^2+a^2)^2/x^14,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{14}} \, dx=-\frac {3003 \, b^{4} x^{8} + 8580 \, a b^{3} x^{6} + 10010 \, a^{2} b^{2} x^{4} + 5460 \, a^{3} b x^{2} + 1155 \, a^{4}}{15015 \, x^{13}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^2/x^14,x, algorithm="fricas")

[Out]

-1/15015*(3003*b^4*x^8 + 8580*a*b^3*x^6 + 10010*a^2*b^2*x^4 + 5460*a^3*b*x^2 + 1155*a^4)/x^13

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{14}} \, dx=\frac {- 1155 a^{4} - 5460 a^{3} b x^{2} - 10010 a^{2} b^{2} x^{4} - 8580 a b^{3} x^{6} - 3003 b^{4} x^{8}}{15015 x^{13}} \]

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)**2/x**14,x)

[Out]

(-1155*a**4 - 5460*a**3*b*x**2 - 10010*a**2*b**2*x**4 - 8580*a*b**3*x**6 - 3003*b**4*x**8)/(15015*x**13)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{14}} \, dx=-\frac {3003 \, b^{4} x^{8} + 8580 \, a b^{3} x^{6} + 10010 \, a^{2} b^{2} x^{4} + 5460 \, a^{3} b x^{2} + 1155 \, a^{4}}{15015 \, x^{13}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^2/x^14,x, algorithm="maxima")

[Out]

-1/15015*(3003*b^4*x^8 + 8580*a*b^3*x^6 + 10010*a^2*b^2*x^4 + 5460*a^3*b*x^2 + 1155*a^4)/x^13

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{14}} \, dx=-\frac {3003 \, b^{4} x^{8} + 8580 \, a b^{3} x^{6} + 10010 \, a^{2} b^{2} x^{4} + 5460 \, a^{3} b x^{2} + 1155 \, a^{4}}{15015 \, x^{13}} \]

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)^2/x^14,x, algorithm="giac")

[Out]

-1/15015*(3003*b^4*x^8 + 8580*a*b^3*x^6 + 10010*a^2*b^2*x^4 + 5460*a^3*b*x^2 + 1155*a^4)/x^13

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{14}} \, dx=-\frac {\frac {a^4}{13}+\frac {4\,a^3\,b\,x^2}{11}+\frac {2\,a^2\,b^2\,x^4}{3}+\frac {4\,a\,b^3\,x^6}{7}+\frac {b^4\,x^8}{5}}{x^{13}} \]

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)^2/x^14,x)

[Out]

-(a^4/13 + (b^4*x^8)/5 + (4*a^3*b*x^2)/11 + (4*a*b^3*x^6)/7 + (2*a^2*b^2*x^4)/3)/x^13