3.181 \(\int \frac {x^{12}}{(a+b x^2)^3} \, dx\)

Optimal. Leaf size=111 \[ \frac {99 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}-\frac {99 a^3 x}{8 b^6}+\frac {33 a^2 x^3}{8 b^5}-\frac {99 a x^5}{40 b^4}-\frac {11 x^9}{8 b^2 \left (a+b x^2\right )}-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}+\frac {99 x^7}{56 b^3} \]

[Out]

-99/8*a^3*x/b^6+33/8*a^2*x^3/b^5-99/40*a*x^5/b^4+99/56*x^7/b^3-1/4*x^11/b/(b*x^2+a)^2-11/8*x^9/b^2/(b*x^2+a)+9
9/8*a^(7/2)*arctan(x*b^(1/2)/a^(1/2))/b^(13/2)

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Rubi [A]  time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac {33 a^2 x^3}{8 b^5}-\frac {99 a^3 x}{8 b^6}+\frac {99 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}-\frac {11 x^9}{8 b^2 \left (a+b x^2\right )}-\frac {99 a x^5}{40 b^4}-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}+\frac {99 x^7}{56 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^12/(a + b*x^2)^3,x]

[Out]

(-99*a^3*x)/(8*b^6) + (33*a^2*x^3)/(8*b^5) - (99*a*x^5)/(40*b^4) + (99*x^7)/(56*b^3) - x^11/(4*b*(a + b*x^2)^2
) - (11*x^9)/(8*b^2*(a + b*x^2)) + (99*a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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]

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]

Rubi steps

\begin {align*} \int \frac {x^{12}}{\left (a+b x^2\right )^3} \, dx &=-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}+\frac {11 \int \frac {x^{10}}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}-\frac {11 x^9}{8 b^2 \left (a+b x^2\right )}+\frac {99 \int \frac {x^8}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}-\frac {11 x^9}{8 b^2 \left (a+b x^2\right )}+\frac {99 \int \left (-\frac {a^3}{b^4}+\frac {a^2 x^2}{b^3}-\frac {a x^4}{b^2}+\frac {x^6}{b}+\frac {a^4}{b^4 \left (a+b x^2\right )}\right ) \, dx}{8 b^2}\\ &=-\frac {99 a^3 x}{8 b^6}+\frac {33 a^2 x^3}{8 b^5}-\frac {99 a x^5}{40 b^4}+\frac {99 x^7}{56 b^3}-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}-\frac {11 x^9}{8 b^2 \left (a+b x^2\right )}+\frac {\left (99 a^4\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^6}\\ &=-\frac {99 a^3 x}{8 b^6}+\frac {33 a^2 x^3}{8 b^5}-\frac {99 a x^5}{40 b^4}+\frac {99 x^7}{56 b^3}-\frac {x^{11}}{4 b \left (a+b x^2\right )^2}-\frac {11 x^9}{8 b^2 \left (a+b x^2\right )}+\frac {99 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 99, normalized size = 0.89 \[ \frac {99 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}-\frac {3465 a^5 x+5775 a^4 b x^3+1848 a^3 b^2 x^5-264 a^2 b^3 x^7+88 a b^4 x^9-40 b^5 x^{11}}{280 b^6 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^12/(a + b*x^2)^3,x]

[Out]

-1/280*(3465*a^5*x + 5775*a^4*b*x^3 + 1848*a^3*b^2*x^5 - 264*a^2*b^3*x^7 + 88*a*b^4*x^9 - 40*b^5*x^11)/(b^6*(a
+ b*x^2)^2) + (99*a^(7/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

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fricas [A]  time = 0.86, size = 278, normalized size = 2.50 \[ \left [\frac {80 \, b^{5} x^{11} - 176 \, a b^{4} x^{9} + 528 \, a^{2} b^{3} x^{7} - 3696 \, a^{3} b^{2} x^{5} - 11550 \, a^{4} b x^{3} - 6930 \, a^{5} x + 3465 \, {\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{560 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac {40 \, b^{5} x^{11} - 88 \, a b^{4} x^{9} + 264 \, a^{2} b^{3} x^{7} - 1848 \, a^{3} b^{2} x^{5} - 5775 \, a^{4} b x^{3} - 3465 \, a^{5} x + 3465 \, {\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{280 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/560*(80*b^5*x^11 - 176*a*b^4*x^9 + 528*a^2*b^3*x^7 - 3696*a^3*b^2*x^5 - 11550*a^4*b*x^3 - 6930*a^5*x + 3465
*(a^3*b^2*x^4 + 2*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^8*x^4 + 2*a*
b^7*x^2 + a^2*b^6), 1/280*(40*b^5*x^11 - 88*a*b^4*x^9 + 264*a^2*b^3*x^7 - 1848*a^3*b^2*x^5 - 5775*a^4*b*x^3 -
3465*a^5*x + 3465*(a^3*b^2*x^4 + 2*a^4*b*x^2 + a^5)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^8*x^4 + 2*a*b^7*x^2
+ a^2*b^6)]

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giac [A]  time = 0.61, size = 96, normalized size = 0.86 \[ \frac {99 \, a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} - \frac {21 \, a^{4} b x^{3} + 19 \, a^{5} x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {5 \, b^{18} x^{7} - 21 \, a b^{17} x^{5} + 70 \, a^{2} b^{16} x^{3} - 350 \, a^{3} b^{15} x}{35 \, b^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^3,x, algorithm="giac")

[Out]

99/8*a^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) - 1/8*(21*a^4*b*x^3 + 19*a^5*x)/((b*x^2 + a)^2*b^6) + 1/35*(5*b
^18*x^7 - 21*a*b^17*x^5 + 70*a^2*b^16*x^3 - 350*a^3*b^15*x)/b^21

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maple [A]  time = 0.01, size = 99, normalized size = 0.89 \[ \frac {x^{7}}{7 b^{3}}-\frac {21 a^{4} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{5}}-\frac {3 a \,x^{5}}{5 b^{4}}-\frac {19 a^{5} x}{8 \left (b \,x^{2}+a \right )^{2} b^{6}}+\frac {2 a^{2} x^{3}}{b^{5}}+\frac {99 a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{6}}-\frac {10 a^{3} x}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12/(b*x^2+a)^3,x)

[Out]

1/7*x^7/b^3-3/5*a*x^5/b^4+2*a^2*x^3/b^5-10*a^3*x/b^6-21/8/b^5*a^4/(b*x^2+a)^2*x^3-19/8/b^6*a^5/(b*x^2+a)^2*x+9
9/8/b^6*a^4/(a*b)^(1/2)*arctan(1/(a*b)^(1/2)*b*x)

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maxima [A]  time = 2.95, size = 105, normalized size = 0.95 \[ -\frac {21 \, a^{4} b x^{3} + 19 \, a^{5} x}{8 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} + \frac {99 \, a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {5 \, b^{3} x^{7} - 21 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} - 350 \, a^{3} x}{35 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^12/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/8*(21*a^4*b*x^3 + 19*a^5*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6) + 99/8*a^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b
^6) + 1/35*(5*b^3*x^7 - 21*a*b^2*x^5 + 70*a^2*b*x^3 - 350*a^3*x)/b^6

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mupad [B]  time = 0.08, size = 99, normalized size = 0.89 \[ \frac {x^7}{7\,b^3}-\frac {\frac {19\,a^5\,x}{8}+\frac {21\,b\,a^4\,x^3}{8}}{a^2\,b^6+2\,a\,b^7\,x^2+b^8\,x^4}-\frac {3\,a\,x^5}{5\,b^4}-\frac {10\,a^3\,x}{b^6}+\frac {99\,a^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,b^{13/2}}+\frac {2\,a^2\,x^3}{b^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^12/(a + b*x^2)^3,x)

[Out]

x^7/(7*b^3) - ((19*a^5*x)/8 + (21*a^4*b*x^3)/8)/(a^2*b^6 + b^8*x^4 + 2*a*b^7*x^2) - (3*a*x^5)/(5*b^4) - (10*a^
3*x)/b^6 + (99*a^(7/2)*atan((b^(1/2)*x)/a^(1/2)))/(8*b^(13/2)) + (2*a^2*x^3)/b^5

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sympy [A]  time = 0.49, size = 162, normalized size = 1.46 \[ - \frac {10 a^{3} x}{b^{6}} + \frac {2 a^{2} x^{3}}{b^{5}} - \frac {3 a x^{5}}{5 b^{4}} - \frac {99 \sqrt {- \frac {a^{7}}{b^{13}}} \log {\left (x - \frac {b^{6} \sqrt {- \frac {a^{7}}{b^{13}}}}{a^{3}} \right )}}{16} + \frac {99 \sqrt {- \frac {a^{7}}{b^{13}}} \log {\left (x + \frac {b^{6} \sqrt {- \frac {a^{7}}{b^{13}}}}{a^{3}} \right )}}{16} + \frac {- 19 a^{5} x - 21 a^{4} b x^{3}}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} + \frac {x^{7}}{7 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**12/(b*x**2+a)**3,x)

[Out]

-10*a**3*x/b**6 + 2*a**2*x**3/b**5 - 3*a*x**5/(5*b**4) - 99*sqrt(-a**7/b**13)*log(x - b**6*sqrt(-a**7/b**13)/a
**3)/16 + 99*sqrt(-a**7/b**13)*log(x + b**6*sqrt(-a**7/b**13)/a**3)/16 + (-19*a**5*x - 21*a**4*b*x**3)/(8*a**2
*b**6 + 16*a*b**7*x**2 + 8*b**8*x**4) + x**7/(7*b**3)

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