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

Optimal. Leaf size=142 \[ \frac{3003 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{15/2}}-\frac{13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac{143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac{429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac{3003 x^5}{1280 b^5 \left (a+b x^2\right )}-\frac{3003 a x}{256 b^7}-\frac{x^{13}}{10 b \left (a+b x^2\right )^5}+\frac{1001 x^3}{256 b^6} \]

[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))

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Rubi [A]  time = 0.0927995, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {28, 288, 302, 205} \[ \frac{3003 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{256 b^{15/2}}-\frac{13 x^{11}}{80 b^2 \left (a+b x^2\right )^4}-\frac{143 x^9}{480 b^3 \left (a+b x^2\right )^3}-\frac{429 x^7}{640 b^4 \left (a+b x^2\right )^2}-\frac{3003 x^5}{1280 b^5 \left (a+b x^2\right )}-\frac{3003 a x}{256 b^7}-\frac{x^{13}}{10 b \left (a+b x^2\right )^5}+\frac{1001 x^3}{256 b^6} \]

Antiderivative was successfully verified.

[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 288

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 302

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]

Rule 205

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

Rubi steps

\begin{align*} \int \frac{x^{14}}{\left (a^2+2 a b x^2+b^2 x^4\right )^3} \, dx &=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]  time = 0.0585399, size = 111, normalized size = 0.78 \[ \frac{\frac{\sqrt{b} x \left (-137995 a^2 b^4 x^8-338910 a^3 b^3 x^6-384384 a^4 b^2 x^4-210210 a^5 b x^2-45045 a^6-16640 a b^5 x^{10}+1280 b^6 x^{12}\right )}{\left (a+b x^2\right )^5}+45045 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3840 b^{15/2}} \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 0.055, size = 137, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{3\,{b}^{6}}}-6\,{\frac{ax}{{b}^{7}}}-{\frac{2373\,{a}^{2}{x}^{9}}{256\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{12131\,{a}^{3}{x}^{7}}{384\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1253\,{a}^{4}{x}^{5}}{30\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{9629\,{a}^{5}{x}^{3}}{384\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{5}}}-{\frac{1467\,{a}^{6}x}{256\,{b}^{7} \left ( b{x}^{2}+a \right ) ^{5}}}+{\frac{3003\,{a}^{2}}{256\,{b}^{7}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53815, size = 990, normalized size = 6.97 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [A]  time = 1.46659, size = 202, normalized size = 1.42 \begin{align*} - \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.16593, size = 143, normalized size = 1.01 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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