3.2.46 \(\int \frac {x^{12}}{(a+b x^2)^2} \, dx\) [146]

Optimal. Leaf size=105 \[ \frac {11 a^4 x}{2 b^6}-\frac {11 a^3 x^3}{6 b^5}+\frac {11 a^2 x^5}{10 b^4}-\frac {11 a x^7}{14 b^3}+\frac {11 x^9}{18 b^2}-\frac {x^{11}}{2 b \left (a+b x^2\right )}-\frac {11 a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{13/2}} \]

[Out]

11/2*a^4*x/b^6-11/6*a^3*x^3/b^5+11/10*a^2*x^5/b^4-11/14*a*x^7/b^3+11/18*x^9/b^2-1/2*x^11/b/(b*x^2+a)-11/2*a^(9
/2)*arctan(x*b^(1/2)/a^(1/2))/b^(13/2)

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Rubi [A]
time = 0.03, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {294, 308, 211} \begin {gather*} -\frac {11 a^{9/2} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{13/2}}+\frac {11 a^4 x}{2 b^6}-\frac {11 a^3 x^3}{6 b^5}+\frac {11 a^2 x^5}{10 b^4}-\frac {11 a x^7}{14 b^3}-\frac {x^{11}}{2 b \left (a+b x^2\right )}+\frac {11 x^9}{18 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^12/(a + b*x^2)^2,x]

[Out]

(11*a^4*x)/(2*b^6) - (11*a^3*x^3)/(6*b^5) + (11*a^2*x^5)/(10*b^4) - (11*a*x^7)/(14*b^3) + (11*x^9)/(18*b^2) -
x^11/(2*b*(a + b*x^2)) - (11*a^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))

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*} \int \frac {x^{12}}{\left (a+b x^2\right )^2} \, dx &=-\frac {x^{11}}{2 b \left (a+b x^2\right )}+\frac {11 \int \frac {x^{10}}{a+b x^2} \, dx}{2 b}\\ &=-\frac {x^{11}}{2 b \left (a+b x^2\right )}+\frac {11 \int \left (\frac {a^4}{b^5}-\frac {a^3 x^2}{b^4}+\frac {a^2 x^4}{b^3}-\frac {a x^6}{b^2}+\frac {x^8}{b}-\frac {a^5}{b^5 \left (a+b x^2\right )}\right ) \, dx}{2 b}\\ &=\frac {11 a^4 x}{2 b^6}-\frac {11 a^3 x^3}{6 b^5}+\frac {11 a^2 x^5}{10 b^4}-\frac {11 a x^7}{14 b^3}+\frac {11 x^9}{18 b^2}-\frac {x^{11}}{2 b \left (a+b x^2\right )}-\frac {\left (11 a^5\right ) \int \frac {1}{a+b x^2} \, dx}{2 b^6}\\ &=\frac {11 a^4 x}{2 b^6}-\frac {11 a^3 x^3}{6 b^5}+\frac {11 a^2 x^5}{10 b^4}-\frac {11 a x^7}{14 b^3}+\frac {11 x^9}{18 b^2}-\frac {x^{11}}{2 b \left (a+b x^2\right )}-\frac {11 a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{13/2}}\\ \end {align*}

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Mathematica [A]
time = 0.05, size = 93, normalized size = 0.89 \begin {gather*} \frac {x \left (3150 a^4-840 a^3 b x^2+378 a^2 b^2 x^4-180 a b^3 x^6+70 b^4 x^8+\frac {315 a^5}{a+b x^2}\right )}{630 b^6}-\frac {11 a^{9/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{13/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^12/(a + b*x^2)^2,x]

[Out]

(x*(3150*a^4 - 840*a^3*b*x^2 + 378*a^2*b^2*x^4 - 180*a*b^3*x^6 + 70*b^4*x^8 + (315*a^5)/(a + b*x^2)))/(630*b^6
) - (11*a^(9/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))

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Maple [A]
time = 0.05, size = 87, normalized size = 0.83

method	result	size
default	\(\frac {\frac {1}{9} b^{4} x^{9}-\frac {2}{7} a \,b^{3} x^{7}+\frac {3}{5} a^{2} b^{2} x^{5}-\frac {4}{3} a^{3} b \,x^{3}+5 a^{4} x}{b^{6}}-\frac {a^{5} \left (-\frac {x}{2 \left (b \,x^{2}+a \right )}+\frac {11 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{b^{6}}\)	\(87\)
risch	\(\frac {x^{9}}{9 b^{2}}-\frac {2 a \,x^{7}}{7 b^{3}}+\frac {3 a^{2} x^{5}}{5 b^{4}}-\frac {4 a^{3} x^{3}}{3 b^{5}}+\frac {5 a^{4} x}{b^{6}}+\frac {a^{5} x}{2 b^{6} \left (b \,x^{2}+a \right )}+\frac {11 \sqrt {-a b}\, a^{4} \ln \left (-\sqrt {-a b}\, x -a \right )}{4 b^{7}}-\frac {11 \sqrt {-a b}\, a^{4} \ln \left (\sqrt {-a b}\, x -a \right )}{4 b^{7}}\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

1/b^6*(1/9*b^4*x^9-2/7*a*b^3*x^7+3/5*a^2*b^2*x^5-4/3*a^3*b*x^3+5*a^4*x)-a^5/b^6*(-1/2*x/(b*x^2+a)+11/2/(a*b)^(
1/2)*arctan(b*x/(a*b)^(1/2)))

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Maxima [A]
time = 0.49, size = 93, normalized size = 0.89 \begin {gather*} \frac {a^{5} x}{2 \, {\left (b^{7} x^{2} + a b^{6}\right )}} - \frac {11 \, a^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{6}} + \frac {35 \, b^{4} x^{9} - 90 \, a b^{3} x^{7} + 189 \, a^{2} b^{2} x^{5} - 420 \, a^{3} b x^{3} + 1575 \, a^{4} x}{315 \, b^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

1/2*a^5*x/(b^7*x^2 + a*b^6) - 11/2*a^5*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/315*(35*b^4*x^9 - 90*a*b^3*x^
7 + 189*a^2*b^2*x^5 - 420*a^3*b*x^3 + 1575*a^4*x)/b^6

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Fricas [A]
time = 1.18, size = 234, normalized size = 2.23 \begin {gather*} \left [\frac {140 \, b^{5} x^{11} - 220 \, a b^{4} x^{9} + 396 \, a^{2} b^{3} x^{7} - 924 \, a^{3} b^{2} x^{5} + 4620 \, a^{4} b x^{3} + 6930 \, a^{5} x + 3465 \, {\left (a^{4} b x^{2} + a^{5}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{1260 \, {\left (b^{7} x^{2} + a b^{6}\right )}}, \frac {70 \, b^{5} x^{11} - 110 \, a b^{4} x^{9} + 198 \, a^{2} b^{3} x^{7} - 462 \, a^{3} b^{2} x^{5} + 2310 \, a^{4} b x^{3} + 3465 \, a^{5} x - 3465 \, {\left (a^{4} b x^{2} + a^{5}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{630 \, {\left (b^{7} x^{2} + a b^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/1260*(140*b^5*x^11 - 220*a*b^4*x^9 + 396*a^2*b^3*x^7 - 924*a^3*b^2*x^5 + 4620*a^4*b*x^3 + 6930*a^5*x + 3465
*(a^4*b*x^2 + a^5)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^7*x^2 + a*b^6), 1/630*(70*b^
5*x^11 - 110*a*b^4*x^9 + 198*a^2*b^3*x^7 - 462*a^3*b^2*x^5 + 2310*a^4*b*x^3 + 3465*a^5*x - 3465*(a^4*b*x^2 + a
^5)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^7*x^2 + a*b^6)]

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Sympy [A]
time = 0.14, size = 151, normalized size = 1.44 \begin {gather*} \frac {a^{5} x}{2 a b^{6} + 2 b^{7} x^{2}} + \frac {5 a^{4} x}{b^{6}} - \frac {4 a^{3} x^{3}}{3 b^{5}} + \frac {3 a^{2} x^{5}}{5 b^{4}} - \frac {2 a x^{7}}{7 b^{3}} + \frac {11 \sqrt {- \frac {a^{9}}{b^{13}}} \log {\left (x - \frac {b^{6} \sqrt {- \frac {a^{9}}{b^{13}}}}{a^{4}} \right )}}{4} - \frac {11 \sqrt {- \frac {a^{9}}{b^{13}}} \log {\left (x + \frac {b^{6} \sqrt {- \frac {a^{9}}{b^{13}}}}{a^{4}} \right )}}{4} + \frac {x^{9}}{9 b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12/(b*x**2+a)**2,x)

[Out]

a**5*x/(2*a*b**6 + 2*b**7*x**2) + 5*a**4*x/b**6 - 4*a**3*x**3/(3*b**5) + 3*a**2*x**5/(5*b**4) - 2*a*x**7/(7*b*
*3) + 11*sqrt(-a**9/b**13)*log(x - b**6*sqrt(-a**9/b**13)/a**4)/4 - 11*sqrt(-a**9/b**13)*log(x + b**6*sqrt(-a*
*9/b**13)/a**4)/4 + x**9/(9*b**2)

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Giac [A]
time = 0.49, size = 95, normalized size = 0.90 \begin {gather*} -\frac {11 \, a^{5} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{6}} + \frac {a^{5} x}{2 \, {\left (b x^{2} + a\right )} b^{6}} + \frac {35 \, b^{16} x^{9} - 90 \, a b^{15} x^{7} + 189 \, a^{2} b^{14} x^{5} - 420 \, a^{3} b^{13} x^{3} + 1575 \, a^{4} b^{12} x}{315 \, b^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^2,x, algorithm="giac")

[Out]

-11/2*a^5*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/2*a^5*x/((b*x^2 + a)*b^6) + 1/315*(35*b^16*x^9 - 90*a*b^15
*x^7 + 189*a^2*b^14*x^5 - 420*a^3*b^13*x^3 + 1575*a^4*b^12*x)/b^18

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Mupad [B]
time = 0.07, size = 88, normalized size = 0.84 \begin {gather*} \frac {x^9}{9\,b^2}-\frac {2\,a\,x^7}{7\,b^3}+\frac {5\,a^4\,x}{b^6}-\frac {11\,a^{9/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{13/2}}+\frac {3\,a^2\,x^5}{5\,b^4}-\frac {4\,a^3\,x^3}{3\,b^5}+\frac {a^5\,x}{2\,\left (b^7\,x^2+a\,b^6\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12/(a + b*x^2)^2,x)

[Out]

x^9/(9*b^2) - (2*a*x^7)/(7*b^3) + (5*a^4*x)/b^6 - (11*a^(9/2)*atan((b^(1/2)*x)/a^(1/2)))/(2*b^(13/2)) + (3*a^2
*x^5)/(5*b^4) - (4*a^3*x^3)/(3*b^5) + (a^5*x)/(2*(a*b^6 + b^7*x^2))

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