\(\int \frac {x^{14}}{(a^2+2 a b x^2+b^2 x^4)^3} \, dx\) [524]

Optimal result

Integrand size = 24, antiderivative size = 142 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {3003 a x}{256 b^7}+\frac {1001 x^3}{256 b^6}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {3003 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{15/2}} \]

[Out]

-3003/256*a*x/b^7+1001/256*x^3/b^6-1/10*x^13/b/(b*x^2+a)^5-13/80*x^11/b^2/(b*x^2+a)^4-143/480*x^9/b^3/(b*x^2+a
)^3-429/640*x^7/b^4/(b*x^2+a)^2-3003/1280*x^5/b^5/(b*x^2+a)+3003/256*a^(3/2)*arctan(x*b^(1/2)/a^(1/2))/b^(15/2
)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {28, 294, 308, 211} \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {3003 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{15/2}}-\frac {3003 a x}{256 b^7}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}+\frac {1001 x^3}{256 b^6} \]

[In]

Int[x^14/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

(-3003*a*x)/(256*b^7) + (1001*x^3)/(256*b^6) - x^13/(10*b*(a + b*x^2)^5) - (13*x^11)/(80*b^2*(a + b*x^2)^4) -
(143*x^9)/(480*b^3*(a + b*x^2)^3) - (429*x^7)/(640*b^4*(a + b*x^2)^2) - (3003*x^5)/(1280*b^5*(a + b*x^2)) + (3
003*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(256*b^(15/2))

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 308

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rubi steps \begin{align*} \text {integral}& = b^6 \int \frac {x^{14}}{\left (a b+b^2 x^2\right )^6} \, dx \\ & = -\frac {x^{13}}{10 b \left (a+b x^2\right )^5}+\frac {1}{10} \left (13 b^4\right ) \int \frac {x^{12}}{\left (a b+b^2 x^2\right )^5} \, dx \\ & = -\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}+\frac {1}{80} \left (143 b^2\right ) \int \frac {x^{10}}{\left (a b+b^2 x^2\right )^4} \, dx \\ & = -\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}+\frac {429}{160} \int \frac {x^8}{\left (a b+b^2 x^2\right )^3} \, dx \\ & = -\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}+\frac {3003 \int \frac {x^6}{\left (a b+b^2 x^2\right )^2} \, dx}{640 b^2} \\ & = -\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {3003 \int \frac {x^4}{a b+b^2 x^2} \, dx}{256 b^4} \\ & = -\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {3003 \int \left (-\frac {a}{b^3}+\frac {x^2}{b^2}+\frac {a^2}{b^2 \left (a b+b^2 x^2\right )}\right ) \, dx}{256 b^4} \\ & = -\frac {3003 a x}{256 b^7}+\frac {1001 x^3}{256 b^6}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {\left (3003 a^2\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{256 b^6} \\ & = -\frac {3003 a x}{256 b^7}+\frac {1001 x^3}{256 b^6}-\frac {x^{13}}{10 b \left (a+b x^2\right )^5}-\frac {13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac {143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac {429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac {3003 x^5}{1280 b^5 \left (a+b x^2\right )}+\frac {3003 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{256 b^{15/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.78 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {\frac {\sqrt {b} x \left (-45045 a^6-210210 a^5 b x^2-384384 a^4 b^2 x^4-338910 a^3 b^3 x^6-137995 a^2 b^4 x^8-16640 a b^5 x^{10}+1280 b^6 x^{12}\right )}{\left (a+b x^2\right )^5}+45045 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3840 b^{15/2}} \]

[In]

Integrate[x^14/(a^2 + 2*a*b*x^2 + b^2*x^4)^3,x]

[Out]

((Sqrt[b]*x*(-45045*a^6 - 210210*a^5*b*x^2 - 384384*a^4*b^2*x^4 - 338910*a^3*b^3*x^6 - 137995*a^2*b^4*x^8 - 16
640*a*b^5*x^10 + 1280*b^6*x^12))/(a + b*x^2)^5 + 45045*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3840*b^(15/2))

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.68

method	result	size
default	\(-\frac {-\frac {1}{3} b \,x^{3}+6 a x}{b^{7}}+\frac {a^{2} \left (\frac {-\frac {2373}{256} b^{4} x^{9}-\frac {12131}{384} a \,b^{3} x^{7}-\frac {1253}{30} a^{2} b^{2} x^{5}-\frac {9629}{384} a^{3} b \,x^{3}-\frac {1467}{256} a^{4} x}{\left (b \,x^{2}+a \right )^{5}}+\frac {3003 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \sqrt {a b}}\right )}{b^{7}}\)	\(96\)
risch	\(\frac {x^{3}}{3 b^{6}}-\frac {6 a x}{b^{7}}+\frac {-\frac {2373}{256} a^{2} b^{4} x^{9}-\frac {12131}{384} a^{3} b^{3} x^{7}-\frac {1253}{30} a^{4} b^{2} x^{5}-\frac {9629}{384} a^{5} b \,x^{3}-\frac {1467}{256} a^{6} x}{b^{7} \left (b \,x^{2}+a \right ) \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{2}}+\frac {3003 \sqrt {-a b}\, a \ln \left (-\sqrt {-a b}\, x +a \right )}{512 b^{8}}-\frac {3003 \sqrt {-a b}\, a \ln \left (\sqrt {-a b}\, x +a \right )}{512 b^{8}}\)	\(146\)

[In]

int(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x,method=_RETURNVERBOSE)

[Out]

-1/b^7*(-1/3*b*x^3+6*a*x)+1/b^7*a^2*((-2373/256*b^4*x^9-12131/384*a*b^3*x^7-1253/30*a^2*b^2*x^5-9629/384*a^3*b
*x^3-1467/256*a^4*x)/(b*x^2+a)^5+3003/256/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 428, normalized size of antiderivative = 3.01 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\left [\frac {2560 \, b^{6} x^{13} - 33280 \, a b^{5} x^{11} - 275990 \, a^{2} b^{4} x^{9} - 677820 \, a^{3} b^{3} x^{7} - 768768 \, a^{4} b^{2} x^{5} - 420420 \, a^{5} b x^{3} - 90090 \, a^{6} x + 45045 \, {\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{7680 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}, \frac {1280 \, b^{6} x^{13} - 16640 \, a b^{5} x^{11} - 137995 \, a^{2} b^{4} x^{9} - 338910 \, a^{3} b^{3} x^{7} - 384384 \, a^{4} b^{2} x^{5} - 210210 \, a^{5} b x^{3} - 45045 \, a^{6} x + 45045 \, {\left (a b^{5} x^{10} + 5 \, a^{2} b^{4} x^{8} + 10 \, a^{3} b^{3} x^{6} + 10 \, a^{4} b^{2} x^{4} + 5 \, a^{5} b x^{2} + a^{6}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{3840 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}}\right ] \]

[In]

integrate(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="fricas")

[Out]

[1/7680*(2560*b^6*x^13 - 33280*a*b^5*x^11 - 275990*a^2*b^4*x^9 - 677820*a^3*b^3*x^7 - 768768*a^4*b^2*x^5 - 420
420*a^5*b*x^3 - 90090*a^6*x + 45045*(a*b^5*x^10 + 5*a^2*b^4*x^8 + 10*a^3*b^3*x^6 + 10*a^4*b^2*x^4 + 5*a^5*b*x^
2 + a^6)*sqrt(-a/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^12*x^10 + 5*a*b^11*x^8 + 10*a^2*b^10*x
^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7), 1/3840*(1280*b^6*x^13 - 16640*a*b^5*x^11 - 137995*a^2*b^4*x^9
- 338910*a^3*b^3*x^7 - 384384*a^4*b^2*x^5 - 210210*a^5*b*x^3 - 45045*a^6*x + 45045*(a*b^5*x^10 + 5*a^2*b^4*x^8
+ 10*a^3*b^3*x^6 + 10*a^4*b^2*x^4 + 5*a^5*b*x^2 + a^6)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^12*x^10 + 5*a*b^
11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7)]

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.44 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=- \frac {6 a x}{b^{7}} - \frac {3003 \sqrt {- \frac {a^{3}}{b^{15}}} \log {\left (x - \frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}}}{a} \right )}}{512} + \frac {3003 \sqrt {- \frac {a^{3}}{b^{15}}} \log {\left (x + \frac {b^{7} \sqrt {- \frac {a^{3}}{b^{15}}}}{a} \right )}}{512} + \frac {- 22005 a^{6} x - 96290 a^{5} b x^{3} - 160384 a^{4} b^{2} x^{5} - 121310 a^{3} b^{3} x^{7} - 35595 a^{2} b^{4} x^{9}}{3840 a^{5} b^{7} + 19200 a^{4} b^{8} x^{2} + 38400 a^{3} b^{9} x^{4} + 38400 a^{2} b^{10} x^{6} + 19200 a b^{11} x^{8} + 3840 b^{12} x^{10}} + \frac {x^{3}}{3 b^{6}} \]

[In]

integrate(x**14/(b**2*x**4+2*a*b*x**2+a**2)**3,x)

[Out]

-6*a*x/b**7 - 3003*sqrt(-a**3/b**15)*log(x - b**7*sqrt(-a**3/b**15)/a)/512 + 3003*sqrt(-a**3/b**15)*log(x + b*
*7*sqrt(-a**3/b**15)/a)/512 + (-22005*a**6*x - 96290*a**5*b*x**3 - 160384*a**4*b**2*x**5 - 121310*a**3*b**3*x*
*7 - 35595*a**2*b**4*x**9)/(3840*a**5*b**7 + 19200*a**4*b**8*x**2 + 38400*a**3*b**9*x**4 + 38400*a**2*b**10*x*
*6 + 19200*a*b**11*x**8 + 3840*b**12*x**10) + x**3/(3*b**6)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.04 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=-\frac {35595 \, a^{2} b^{4} x^{9} + 121310 \, a^{3} b^{3} x^{7} + 160384 \, a^{4} b^{2} x^{5} + 96290 \, a^{5} b x^{3} + 22005 \, a^{6} x}{3840 \, {\left (b^{12} x^{10} + 5 \, a b^{11} x^{8} + 10 \, a^{2} b^{10} x^{6} + 10 \, a^{3} b^{9} x^{4} + 5 \, a^{4} b^{8} x^{2} + a^{5} b^{7}\right )}} + \frac {3003 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{7}} + \frac {b x^{3} - 18 \, a x}{3 \, b^{7}} \]

[In]

integrate(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="maxima")

[Out]

-1/3840*(35595*a^2*b^4*x^9 + 121310*a^3*b^3*x^7 + 160384*a^4*b^2*x^5 + 96290*a^5*b*x^3 + 22005*a^6*x)/(b^12*x^
10 + 5*a*b^11*x^8 + 10*a^2*b^10*x^6 + 10*a^3*b^9*x^4 + 5*a^4*b^8*x^2 + a^5*b^7) + 3003/256*a^2*arctan(b*x/sqrt
(a*b))/(sqrt(a*b)*b^7) + 1/3*(b*x^3 - 18*a*x)/b^7

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.75 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {3003 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{256 \, \sqrt {a b} b^{7}} - \frac {35595 \, a^{2} b^{4} x^{9} + 121310 \, a^{3} b^{3} x^{7} + 160384 \, a^{4} b^{2} x^{5} + 96290 \, a^{5} b x^{3} + 22005 \, a^{6} x}{3840 \, {\left (b x^{2} + a\right )}^{5} b^{7}} + \frac {b^{12} x^{3} - 18 \, a b^{11} x}{3 \, b^{18}} \]

[In]

integrate(x^14/(b^2*x^4+2*a*b*x^2+a^2)^3,x, algorithm="giac")

[Out]

3003/256*a^2*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^7) - 1/3840*(35595*a^2*b^4*x^9 + 121310*a^3*b^3*x^7 + 160384*a
^4*b^2*x^5 + 96290*a^5*b*x^3 + 22005*a^6*x)/((b*x^2 + a)^5*b^7) + 1/3*(b^12*x^3 - 18*a*b^11*x)/b^18

Mupad [B] (verification not implemented)

Time = 13.55 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01 \[ \int \frac {x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx=\frac {x^3}{3\,b^6}-\frac {\frac {1467\,a^6\,x}{256}+\frac {9629\,a^5\,b\,x^3}{384}+\frac {1253\,a^4\,b^2\,x^5}{30}+\frac {12131\,a^3\,b^3\,x^7}{384}+\frac {2373\,a^2\,b^4\,x^9}{256}}{a^5\,b^7+5\,a^4\,b^8\,x^2+10\,a^3\,b^9\,x^4+10\,a^2\,b^{10}\,x^6+5\,a\,b^{11}\,x^8+b^{12}\,x^{10}}+\frac {3003\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{256\,b^{15/2}}-\frac {6\,a\,x}{b^7} \]

[In]

int(x^14/(a^2 + b^2*x^4 + 2*a*b*x^2)^3,x)

[Out]

x^3/(3*b^6) - ((1467*a^6*x)/256 + (9629*a^5*b*x^3)/384 + (1253*a^4*b^2*x^5)/30 + (12131*a^3*b^3*x^7)/384 + (23
73*a^2*b^4*x^9)/256)/(a^5*b^7 + b^12*x^10 + 5*a*b^11*x^8 + 5*a^4*b^8*x^2 + 10*a^3*b^9*x^4 + 10*a^2*b^10*x^6) +
(3003*a^(3/2)*atan((b^(1/2)*x)/a^(1/2)))/(256*b^(15/2)) - (6*a*x)/b^7