3.148 \(\int \frac {x^{10}}{(a+b x^2)^2} \, dx\)

Optimal. Leaf size=92 \[ \frac {9 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}}-\frac {9 a^3 x}{2 b^5}+\frac {3 a^2 x^3}{2 b^4}-\frac {9 a x^5}{10 b^3}-\frac {x^9}{2 b \left (a+b x^2\right )}+\frac {9 x^7}{14 b^2} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 92, 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 = {288, 302, 205} \[ \frac {3 a^2 x^3}{2 b^4}-\frac {9 a^3 x}{2 b^5}+\frac {9 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}}-\frac {9 a x^5}{10 b^3}-\frac {x^9}{2 b \left (a+b x^2\right )}+\frac {9 x^7}{14 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^10/(a + b*x^2)^2,x]

[Out]

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

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

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Mathematica [A]  time = 0.06, size = 82, normalized size = 0.89 \[ \frac {9 a^{7/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 b^{11/2}}+\frac {x \left (-\frac {35 a^4}{a+b x^2}-280 a^3+70 a^2 b x^2-28 a b^2 x^4+10 b^3 x^6\right )}{70 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10/(a + b*x^2)^2,x]

[Out]

(x*(-280*a^3 + 70*a^2*b*x^2 - 28*a*b^2*x^4 + 10*b^3*x^6 - (35*a^4)/(a + b*x^2)))/(70*b^5) + (9*a^(7/2)*ArcTan[
(Sqrt[b]*x)/Sqrt[a]])/(2*b^(11/2))

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fricas [A]  time = 1.00, size = 212, normalized size = 2.30 \[ \left [\frac {20 \, b^{4} x^{9} - 36 \, a b^{3} x^{7} + 84 \, a^{2} b^{2} x^{5} - 420 \, a^{3} b x^{3} - 630 \, a^{4} x + 315 \, {\left (a^{3} b x^{2} + a^{4}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{140 \, {\left (b^{6} x^{2} + a b^{5}\right )}}, \frac {10 \, b^{4} x^{9} - 18 \, a b^{3} x^{7} + 42 \, a^{2} b^{2} x^{5} - 210 \, a^{3} b x^{3} - 315 \, a^{4} x + 315 \, {\left (a^{3} b x^{2} + a^{4}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{70 \, {\left (b^{6} x^{2} + a b^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/140*(20*b^4*x^9 - 36*a*b^3*x^7 + 84*a^2*b^2*x^5 - 420*a^3*b*x^3 - 630*a^4*x + 315*(a^3*b*x^2 + a^4)*sqrt(-a
/b)*log((b*x^2 + 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)))/(b^6*x^2 + a*b^5), 1/70*(10*b^4*x^9 - 18*a*b^3*x^7 + 42*a
^2*b^2*x^5 - 210*a^3*b*x^3 - 315*a^4*x + 315*(a^3*b*x^2 + a^4)*sqrt(a/b)*arctan(b*x*sqrt(a/b)/a))/(b^6*x^2 + a
*b^5)]

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giac [A]  time = 0.64, size = 84, normalized size = 0.91 \[ \frac {9 \, a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} - \frac {a^{4} x}{2 \, {\left (b x^{2} + a\right )} b^{5}} + \frac {5 \, b^{12} x^{7} - 14 \, a b^{11} x^{5} + 35 \, a^{2} b^{10} x^{3} - 140 \, a^{3} b^{9} x}{35 \, b^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^2+a)^2,x, algorithm="giac")

[Out]

9/2*a^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) - 1/2*a^4*x/((b*x^2 + a)*b^5) + 1/35*(5*b^12*x^7 - 14*a*b^11*x^5
+ 35*a^2*b^10*x^3 - 140*a^3*b^9*x)/b^14

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maple [A]  time = 0.01, size = 78, normalized size = 0.85 \[ \frac {x^{7}}{7 b^{2}}-\frac {2 a \,x^{5}}{5 b^{3}}+\frac {a^{2} x^{3}}{b^{4}}-\frac {a^{4} x}{2 \left (b \,x^{2}+a \right ) b^{5}}+\frac {9 a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, b^{5}}-\frac {4 a^{3} x}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(b*x^2+a)^2,x)

[Out]

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

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maxima [A]  time = 3.01, size = 82, normalized size = 0.89 \[ -\frac {a^{4} x}{2 \, {\left (b^{6} x^{2} + a b^{5}\right )}} + \frac {9 \, a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{5}} + \frac {5 \, b^{3} x^{7} - 14 \, a b^{2} x^{5} + 35 \, a^{2} b x^{3} - 140 \, a^{3} x}{35 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^2+a)^2,x, algorithm="maxima")

[Out]

-1/2*a^4*x/(b^6*x^2 + a*b^5) + 9/2*a^4*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^5) + 1/35*(5*b^3*x^7 - 14*a*b^2*x^5
+ 35*a^2*b*x^3 - 140*a^3*x)/b^5

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mupad [B]  time = 4.56, size = 77, normalized size = 0.84 \[ \frac {x^7}{7\,b^2}-\frac {2\,a\,x^5}{5\,b^3}-\frac {4\,a^3\,x}{b^5}+\frac {9\,a^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,b^{11/2}}+\frac {a^2\,x^3}{b^4}-\frac {a^4\,x}{2\,\left (b^6\,x^2+a\,b^5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10/(a + b*x^2)^2,x)

[Out]

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

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sympy [A]  time = 0.33, size = 134, normalized size = 1.46 \[ - \frac {a^{4} x}{2 a b^{5} + 2 b^{6} x^{2}} - \frac {4 a^{3} x}{b^{5}} + \frac {a^{2} x^{3}}{b^{4}} - \frac {2 a x^{5}}{5 b^{3}} - \frac {9 \sqrt {- \frac {a^{7}}{b^{11}}} \log {\left (x - \frac {b^{5} \sqrt {- \frac {a^{7}}{b^{11}}}}{a^{3}} \right )}}{4} + \frac {9 \sqrt {- \frac {a^{7}}{b^{11}}} \log {\left (x + \frac {b^{5} \sqrt {- \frac {a^{7}}{b^{11}}}}{a^{3}} \right )}}{4} + \frac {x^{7}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10/(b*x**2+a)**2,x)

[Out]

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

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