\(\int \frac {x^{10} (A+B x^2)}{(a+b x^2)^3} \, dx\) [98]

Optimal result

Integrand size = 20, antiderivative size = 158 \[ \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {2 a^2 (3 A b-5 a B) x}{b^6}-\frac {a (A b-2 a B) x^3}{b^5}+\frac {(A b-3 a B) x^5}{5 b^4}+\frac {B x^7}{7 b^3}-\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}-\frac {9 a^{5/2} (7 A b-11 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \]

[Out]

2*a^2*(3*A*b-5*B*a)*x/b^6-a*(A*b-2*B*a)*x^3/b^5+1/5*(A*b-3*B*a)*x^5/b^4+1/7*B*x^7/b^3-1/4*a^4*(A*b-B*a)*x/b^6/
(b*x^2+a)^2+1/8*a^3*(17*A*b-21*B*a)*x/b^6/(b*x^2+a)-9/8*a^(5/2)*(7*A*b-11*B*a)*arctan(x*b^(1/2)/a^(1/2))/b^(13
/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {466, 1828, 1824, 211} \[ \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {9 a^{5/2} (7 A b-11 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}}-\frac {a^4 x (A b-a B)}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 x (17 A b-21 a B)}{8 b^6 \left (a+b x^2\right )}+\frac {2 a^2 x (3 A b-5 a B)}{b^6}-\frac {a x^3 (A b-2 a B)}{b^5}+\frac {x^5 (A b-3 a B)}{5 b^4}+\frac {B x^7}{7 b^3} \]

[In]

Int[(x^10*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(2*a^2*(3*A*b - 5*a*B)*x)/b^6 - (a*(A*b - 2*a*B)*x^3)/b^5 + ((A*b - 3*a*B)*x^5)/(5*b^4) + (B*x^7)/(7*b^3) - (a
^4*(A*b - a*B)*x)/(4*b^6*(a + b*x^2)^2) + (a^3*(17*A*b - 21*a*B)*x)/(8*b^6*(a + b*x^2)) - (9*a^(5/2)*(7*A*b -
11*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*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 466

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[(a + b*x^2)^(p + 1)*E
xpandToSum[2*b*(p + 1)*x^2*Together[(b^(m/2)*x^(m - 2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d))/(a + b*x^2)]
- (-a)^(m/2 - 1)*(b*c - a*d), x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IGtQ[
m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 1824

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}-\frac {\int \frac {-a^4 (A b-a B)+4 a^3 b (A b-a B) x^2-4 a^2 b^2 (A b-a B) x^4+4 a b^3 (A b-a B) x^6-4 b^4 (A b-a B) x^8-4 b^5 B x^{10}}{\left (a+b x^2\right )^2} \, dx}{4 b^6} \\ & = -\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}+\frac {\int \frac {-a^4 (15 A b-19 a B)+8 a^3 b (3 A b-4 a B) x^2-8 a^2 b^2 (2 A b-3 a B) x^4+8 a b^3 (A b-2 a B) x^6+8 a b^4 B x^8}{a+b x^2} \, dx}{8 a b^6} \\ & = -\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}+\frac {\int \left (16 a^3 (3 A b-5 a B)-24 a^2 b (A b-2 a B) x^2+8 a b^2 (A b-3 a B) x^4+8 a b^3 B x^6+\frac {9 \left (-7 a^4 A b+11 a^5 B\right )}{a+b x^2}\right ) \, dx}{8 a b^6} \\ & = \frac {2 a^2 (3 A b-5 a B) x}{b^6}-\frac {a (A b-2 a B) x^3}{b^5}+\frac {(A b-3 a B) x^5}{5 b^4}+\frac {B x^7}{7 b^3}-\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}-\frac {\left (9 a^3 (7 A b-11 a B)\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^6} \\ & = \frac {2 a^2 (3 A b-5 a B) x}{b^6}-\frac {a (A b-2 a B) x^3}{b^5}+\frac {(A b-3 a B) x^5}{5 b^4}+\frac {B x^7}{7 b^3}-\frac {a^4 (A b-a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}-\frac {9 a^{5/2} (7 A b-11 a B) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00 \[ \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {2 a^2 (-3 A b+5 a B) x}{b^6}+\frac {a (-A b+2 a B) x^3}{b^5}+\frac {(A b-3 a B) x^5}{5 b^4}+\frac {B x^7}{7 b^3}+\frac {a^4 (-A b+a B) x}{4 b^6 \left (a+b x^2\right )^2}+\frac {a^3 (17 A b-21 a B) x}{8 b^6 \left (a+b x^2\right )}+\frac {9 a^{5/2} (-7 A b+11 a B) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{13/2}} \]

[In]

Integrate[(x^10*(A + B*x^2))/(a + b*x^2)^3,x]

[Out]

(-2*a^2*(-3*A*b + 5*a*B)*x)/b^6 + (a*(-(A*b) + 2*a*B)*x^3)/b^5 + ((A*b - 3*a*B)*x^5)/(5*b^4) + (B*x^7)/(7*b^3)
+ (a^4*(-(A*b) + a*B)*x)/(4*b^6*(a + b*x^2)^2) + (a^3*(17*A*b - 21*a*B)*x)/(8*b^6*(a + b*x^2)) + (9*a^(5/2)*(
-7*A*b + 11*a*B)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*b^(13/2))

Maple [A] (verified)

Time = 2.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {\frac {1}{7} b^{3} B \,x^{7}+\frac {1}{5} A \,b^{3} x^{5}-\frac {3}{5} B a \,b^{2} x^{5}-a A \,b^{2} x^{3}+2 B \,a^{2} b \,x^{3}+6 a^{2} A b x -10 a^{3} B x}{b^{6}}-\frac {a^{3} \left (\frac {\left (-\frac {17}{8} b^{2} A +\frac {21}{8} a b B \right ) x^{3}-\frac {a \left (15 A b -19 B a \right ) x}{8}}{\left (b \,x^{2}+a \right )^{2}}+\frac {9 \left (7 A b -11 B a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}\right )}{b^{6}}\)	\(144\)
risch	\(\frac {B \,x^{7}}{7 b^{3}}+\frac {A \,x^{5}}{5 b^{3}}-\frac {3 B a \,x^{5}}{5 b^{4}}-\frac {a A \,x^{3}}{b^{4}}+\frac {2 B \,a^{2} x^{3}}{b^{5}}+\frac {6 a^{2} A x}{b^{5}}-\frac {10 a^{3} B x}{b^{6}}+\frac {\left (\frac {17}{8} a^{3} b^{2} A -\frac {21}{8} a^{4} b B \right ) x^{3}+\frac {a^{4} \left (15 A b -19 B a \right ) x}{8}}{b^{6} \left (b \,x^{2}+a \right )^{2}}+\frac {63 \sqrt {-a b}\, a^{2} \ln \left (-\sqrt {-a b}\, x -a \right ) A}{16 b^{6}}-\frac {99 \sqrt {-a b}\, a^{3} \ln \left (-\sqrt {-a b}\, x -a \right ) B}{16 b^{7}}-\frac {63 \sqrt {-a b}\, a^{2} \ln \left (\sqrt {-a b}\, x -a \right ) A}{16 b^{6}}+\frac {99 \sqrt {-a b}\, a^{3} \ln \left (\sqrt {-a b}\, x -a \right ) B}{16 b^{7}}\)	\(236\)

[In]

int(x^10*(B*x^2+A)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/b^6*(1/7*b^3*B*x^7+1/5*A*b^3*x^5-3/5*B*a*b^2*x^5-a*A*b^2*x^3+2*B*a^2*b*x^3+6*a^2*A*b*x-10*a^3*B*x)-a^3/b^6*(
((-17/8*b^2*A+21/8*a*b*B)*x^3-1/8*a*(15*A*b-19*B*a)*x)/(b*x^2+a)^2+9/8*(7*A*b-11*B*a)/(a*b)^(1/2)*arctan(b*x/(
a*b)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.96 \[ \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\left [\frac {80 \, B b^{5} x^{11} - 16 \, {\left (11 \, B a b^{4} - 7 \, A b^{5}\right )} x^{9} + 48 \, {\left (11 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{7} - 336 \, {\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{5} - 1050 \, {\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{3} - 315 \, {\left (11 \, B a^{5} - 7 \, A a^{4} b + {\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{4} + 2 \, {\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} - 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) - 630 \, {\left (11 \, B a^{5} - 7 \, A a^{4} b\right )} x}{560 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac {40 \, B b^{5} x^{11} - 8 \, {\left (11 \, B a b^{4} - 7 \, A b^{5}\right )} x^{9} + 24 \, {\left (11 \, B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{7} - 168 \, {\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{5} - 525 \, {\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{3} + 315 \, {\left (11 \, B a^{5} - 7 \, A a^{4} b + {\left (11 \, B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{4} + 2 \, {\left (11 \, B a^{4} b - 7 \, A a^{3} b^{2}\right )} x^{2}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) - 315 \, {\left (11 \, B a^{5} - 7 \, A a^{4} b\right )} x}{280 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \]

[In]

integrate(x^10*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/560*(80*B*b^5*x^11 - 16*(11*B*a*b^4 - 7*A*b^5)*x^9 + 48*(11*B*a^2*b^3 - 7*A*a*b^4)*x^7 - 336*(11*B*a^3*b^2
- 7*A*a^2*b^3)*x^5 - 1050*(11*B*a^4*b - 7*A*a^3*b^2)*x^3 - 315*(11*B*a^5 - 7*A*a^4*b + (11*B*a^3*b^2 - 7*A*a^2
*b^3)*x^4 + 2*(11*B*a^4*b - 7*A*a^3*b^2)*x^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 630
*(11*B*a^5 - 7*A*a^4*b)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6), 1/280*(40*B*b^5*x^11 - 8*(11*B*a*b^4 - 7*A*b^5)*
x^9 + 24*(11*B*a^2*b^3 - 7*A*a*b^4)*x^7 - 168*(11*B*a^3*b^2 - 7*A*a^2*b^3)*x^5 - 525*(11*B*a^4*b - 7*A*a^3*b^2
)*x^3 + 315*(11*B*a^5 - 7*A*a^4*b + (11*B*a^3*b^2 - 7*A*a^2*b^3)*x^4 + 2*(11*B*a^4*b - 7*A*a^3*b^2)*x^2)*sqrt(
a/b)*arctan(b*x*sqrt(a/b)/a) - 315*(11*B*a^5 - 7*A*a^4*b)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6)]

Sympy [A] (verification not implemented)

Time = 0.80 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.77 \[ \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {B x^{7}}{7 b^{3}} + x^{5} \left (\frac {A}{5 b^{3}} - \frac {3 B a}{5 b^{4}}\right ) + x^{3} \left (- \frac {A a}{b^{4}} + \frac {2 B a^{2}}{b^{5}}\right ) + x \left (\frac {6 A a^{2}}{b^{5}} - \frac {10 B a^{3}}{b^{6}}\right ) - \frac {9 \sqrt {- \frac {a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right ) \log {\left (- \frac {9 b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right )}{- 63 A a^{2} b + 99 B a^{3}} + x \right )}}{16} + \frac {9 \sqrt {- \frac {a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right ) \log {\left (\frac {9 b^{6} \sqrt {- \frac {a^{5}}{b^{13}}} \left (- 7 A b + 11 B a\right )}{- 63 A a^{2} b + 99 B a^{3}} + x \right )}}{16} + \frac {x^{3} \cdot \left (17 A a^{3} b^{2} - 21 B a^{4} b\right ) + x \left (15 A a^{4} b - 19 B a^{5}\right )}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} \]

[In]

integrate(x**10*(B*x**2+A)/(b*x**2+a)**3,x)

[Out]

B*x**7/(7*b**3) + x**5*(A/(5*b**3) - 3*B*a/(5*b**4)) + x**3*(-A*a/b**4 + 2*B*a**2/b**5) + x*(6*A*a**2/b**5 - 1
0*B*a**3/b**6) - 9*sqrt(-a**5/b**13)*(-7*A*b + 11*B*a)*log(-9*b**6*sqrt(-a**5/b**13)*(-7*A*b + 11*B*a)/(-63*A*
a**2*b + 99*B*a**3) + x)/16 + 9*sqrt(-a**5/b**13)*(-7*A*b + 11*B*a)*log(9*b**6*sqrt(-a**5/b**13)*(-7*A*b + 11*
B*a)/(-63*A*a**2*b + 99*B*a**3) + x)/16 + (x**3*(17*A*a**3*b**2 - 21*B*a**4*b) + x*(15*A*a**4*b - 19*B*a**5))/
(8*a**2*b**6 + 16*a*b**7*x**2 + 8*b**8*x**4)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.08 \[ \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=-\frac {{\left (21 \, B a^{4} b - 17 \, A a^{3} b^{2}\right )} x^{3} + {\left (19 \, B a^{5} - 15 \, A a^{4} b\right )} x}{8 \, {\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}} + \frac {9 \, {\left (11 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} + \frac {5 \, B b^{3} x^{7} - 7 \, {\left (3 \, B a b^{2} - A b^{3}\right )} x^{5} + 35 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x^{3} - 70 \, {\left (5 \, B a^{3} - 3 \, A a^{2} b\right )} x}{35 \, b^{6}} \]

[In]

integrate(x^10*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="maxima")

[Out]

-1/8*((21*B*a^4*b - 17*A*a^3*b^2)*x^3 + (19*B*a^5 - 15*A*a^4*b)*x)/(b^8*x^4 + 2*a*b^7*x^2 + a^2*b^6) + 9/8*(11
*B*a^4 - 7*A*a^3*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) + 1/35*(5*B*b^3*x^7 - 7*(3*B*a*b^2 - A*b^3)*x^5 + 35
*(2*B*a^2*b - A*a*b^2)*x^3 - 70*(5*B*a^3 - 3*A*a^2*b)*x)/b^6

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.03 \[ \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=\frac {9 \, {\left (11 \, B a^{4} - 7 \, A a^{3} b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{6}} - \frac {21 \, B a^{4} b x^{3} - 17 \, A a^{3} b^{2} x^{3} + 19 \, B a^{5} x - 15 \, A a^{4} b x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{6}} + \frac {5 \, B b^{18} x^{7} - 21 \, B a b^{17} x^{5} + 7 \, A b^{18} x^{5} + 70 \, B a^{2} b^{16} x^{3} - 35 \, A a b^{17} x^{3} - 350 \, B a^{3} b^{15} x + 210 \, A a^{2} b^{16} x}{35 \, b^{21}} \]

[In]

integrate(x^10*(B*x^2+A)/(b*x^2+a)^3,x, algorithm="giac")

[Out]

9/8*(11*B*a^4 - 7*A*a^3*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^6) - 1/8*(21*B*a^4*b*x^3 - 17*A*a^3*b^2*x^3 + 19
*B*a^5*x - 15*A*a^4*b*x)/((b*x^2 + a)^2*b^6) + 1/35*(5*B*b^18*x^7 - 21*B*a*b^17*x^5 + 7*A*b^18*x^5 + 70*B*a^2*
b^16*x^3 - 35*A*a*b^17*x^3 - 350*B*a^3*b^15*x + 210*A*a^2*b^16*x)/b^21

Mupad [B] (verification not implemented)

Time = 4.95 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.56 \[ \int \frac {x^{10} \left (A+B x^2\right )}{\left (a+b x^2\right )^3} \, dx=x^5\,\left (\frac {A}{5\,b^3}-\frac {3\,B\,a}{5\,b^4}\right )-\frac {x\,\left (\frac {19\,B\,a^5}{8}-\frac {15\,A\,a^4\,b}{8}\right )-x^3\,\left (\frac {17\,A\,a^3\,b^2}{8}-\frac {21\,B\,a^4\,b}{8}\right )}{a^2\,b^6+2\,a\,b^7\,x^2+b^8\,x^4}-x^3\,\left (\frac {a\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )}{b}+\frac {B\,a^2}{b^5}\right )-x\,\left (\frac {B\,a^3}{b^6}-\frac {3\,a\,\left (\frac {3\,a\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )}{b}+\frac {3\,B\,a^2}{b^5}\right )}{b}+\frac {3\,a^2\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )}{b^2}\right )+\frac {B\,x^7}{7\,b^3}+\frac {9\,a^{5/2}\,\mathrm {atan}\left (\frac {a^{5/2}\,\sqrt {b}\,x\,\left (7\,A\,b-11\,B\,a\right )}{11\,B\,a^4-7\,A\,a^3\,b}\right )\,\left (7\,A\,b-11\,B\,a\right )}{8\,b^{13/2}} \]

[In]

int((x^10*(A + B*x^2))/(a + b*x^2)^3,x)

[Out]

x^5*(A/(5*b^3) - (3*B*a)/(5*b^4)) - (x*((19*B*a^5)/8 - (15*A*a^4*b)/8) - x^3*((17*A*a^3*b^2)/8 - (21*B*a^4*b)/
8))/(a^2*b^6 + b^8*x^4 + 2*a*b^7*x^2) - x^3*((a*(A/b^3 - (3*B*a)/b^4))/b + (B*a^2)/b^5) - x*((B*a^3)/b^6 - (3*
a*((3*a*(A/b^3 - (3*B*a)/b^4))/b + (3*B*a^2)/b^5))/b + (3*a^2*(A/b^3 - (3*B*a)/b^4))/b^2) + (B*x^7)/(7*b^3) +
(9*a^(5/2)*atan((a^(5/2)*b^(1/2)*x*(7*A*b - 11*B*a))/(11*B*a^4 - 7*A*a^3*b))*(7*A*b - 11*B*a))/(8*b^(13/2))