Integrand size = 26, antiderivative size = 101 \[ \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {D x}{b^2}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}+\frac {(b B-3 a D) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}+\frac {C \log \left (a+b x^2\right )}{2 b^2} \]
D*x/b^2-1/2*x*(a*(B-a*D/b)-(A*b-C*a)*x)/a/b/(b*x^2+a)+1/2*C*ln(b*x^2+a)/b^ 2+1/2*(B*b-3*D*a)*arctan(x*b^(1/2)/a^(1/2))/b^(5/2)/a^(1/2)
Time = 0.03 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.91 \[ \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {D x}{b^2}+\frac {-A b+a C-b B x+a D x}{2 b^2 \left (a+b x^2\right )}-\frac {(-b B+3 a D) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 \sqrt {a} b^{5/2}}+\frac {C \log \left (a+b x^2\right )}{2 b^2} \]
(D*x)/b^2 + (-(A*b) + a*C - b*B*x + a*D*x)/(2*b^2*(a + b*x^2)) - ((-(b*B) + 3*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sqrt[a]*b^(5/2)) + (C*Log[a + b*x ^2])/(2*b^2)
Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2335, 25, 2341, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 2335 |
\(\displaystyle -\frac {\int -\frac {2 a D x^2+2 a C x+\frac {a (b B-a D)}{b}}{b x^2+a}dx}{2 a b}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {2 a D x^2+2 a C x+\frac {a (b B-a D)}{b}}{b x^2+a}dx}{2 a b}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2341 |
\(\displaystyle \frac {\int \left (\frac {2 a D}{b}+\frac {a (b B-3 a D)+2 a b C x}{b \left (b x^2+a\right )}\right )dx}{2 a b}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {\sqrt {a} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (b B-3 a D)}{b^{3/2}}+\frac {a C \log \left (a+b x^2\right )}{b}+\frac {2 a D x}{b}}{2 a b}-\frac {x \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}\) |
-1/2*(x*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(a*b*(a + b*x^2)) + ((2*a*D*x)/ b + (Sqrt[a]*(b*B - 3*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) + (a*C*Log [a + b*x^2])/b)/(2*a*b)
3.1.97.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq , a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(c*x)^m*(a + b*x^2)^(p + 1)*((a*g - b*f*x)/(2*a*b*(p + 1))), x] + Simp[c/(2*a*b*(p + 1)) Int[(c*x)^(m - 1)*(a + b*x^2)^(p + 1)*ExpandToSu m[2*a*b*(p + 1)*x*Q - a*g*m + b*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && GtQ[m, 0]
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]
Time = 3.44 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {D x}{b^{2}}+\frac {\frac {\left (-\frac {B b}{2}+\frac {D a}{2}\right ) x -\frac {A b}{2}+\frac {C a}{2}}{b \,x^{2}+a}+\frac {C \ln \left (b \,x^{2}+a \right )}{2}+\frac {\left (B b -3 D a \right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{b^{2}}\) | \(78\) |
D*x/b^2+1/b^2*(((-1/2*B*b+1/2*D*a)*x-1/2*A*b+1/2*C*a)/(b*x^2+a)+1/2*C*ln(b *x^2+a)+1/2*(B*b-3*D*a)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.28 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.84 \[ \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\left [\frac {4 \, D a b^{2} x^{3} + 2 \, C a^{2} b - 2 \, A a b^{2} - {\left (3 \, D a^{2} - B a b + {\left (3 \, D a b - B b^{2}\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (3 \, D a^{2} b - B a b^{2}\right )} x + 2 \, {\left (C a b^{2} x^{2} + C a^{2} b\right )} \log \left (b x^{2} + a\right )}{4 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}, \frac {2 \, D a b^{2} x^{3} + C a^{2} b - A a b^{2} - {\left (3 \, D a^{2} - B a b + {\left (3 \, D a b - B b^{2}\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (3 \, D a^{2} b - B a b^{2}\right )} x + {\left (C a b^{2} x^{2} + C a^{2} b\right )} \log \left (b x^{2} + a\right )}{2 \, {\left (a b^{4} x^{2} + a^{2} b^{3}\right )}}\right ] \]
[1/4*(4*D*a*b^2*x^3 + 2*C*a^2*b - 2*A*a*b^2 - (3*D*a^2 - B*a*b + (3*D*a*b - B*b^2)*x^2)*sqrt(-a*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a)) + 2 *(3*D*a^2*b - B*a*b^2)*x + 2*(C*a*b^2*x^2 + C*a^2*b)*log(b*x^2 + a))/(a*b^ 4*x^2 + a^2*b^3), 1/2*(2*D*a*b^2*x^3 + C*a^2*b - A*a*b^2 - (3*D*a^2 - B*a* b + (3*D*a*b - B*b^2)*x^2)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + (3*D*a^2*b - B*a*b^2)*x + (C*a*b^2*x^2 + C*a^2*b)*log(b*x^2 + a))/(a*b^4*x^2 + a^2*b^3) ]
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (87) = 174\).
Time = 1.25 (sec) , antiderivative size = 212, normalized size of antiderivative = 2.10 \[ \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {D x}{b^{2}} + \left (\frac {C}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right ) \log {\left (x + \frac {2 C a - 4 a b^{2} \left (\frac {C}{2 b^{2}} - \frac {\sqrt {- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right )}{- B b + 3 D a} \right )} + \left (\frac {C}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right ) \log {\left (x + \frac {2 C a - 4 a b^{2} \left (\frac {C}{2 b^{2}} + \frac {\sqrt {- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right )}{- B b + 3 D a} \right )} + \frac {- A b + C a + x \left (- B b + D a\right )}{2 a b^{2} + 2 b^{3} x^{2}} \]
D*x/b**2 + (C/(2*b**2) - sqrt(-a*b**5)*(-B*b + 3*D*a)/(4*a*b**5))*log(x + (2*C*a - 4*a*b**2*(C/(2*b**2) - sqrt(-a*b**5)*(-B*b + 3*D*a)/(4*a*b**5)))/ (-B*b + 3*D*a)) + (C/(2*b**2) + sqrt(-a*b**5)*(-B*b + 3*D*a)/(4*a*b**5))*l og(x + (2*C*a - 4*a*b**2*(C/(2*b**2) + sqrt(-a*b**5)*(-B*b + 3*D*a)/(4*a*b **5)))/(-B*b + 3*D*a)) + (-A*b + C*a + x*(-B*b + D*a))/(2*a*b**2 + 2*b**3* x**2)
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {C a - A b + {\left (D a - B b\right )} x}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} + \frac {D x}{b^{2}} + \frac {C \log \left (b x^{2} + a\right )}{2 \, b^{2}} - \frac {{\left (3 \, D a - B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} \]
1/2*(C*a - A*b + (D*a - B*b)*x)/(b^3*x^2 + a*b^2) + D*x/b^2 + 1/2*C*log(b* x^2 + a)/b^2 - 1/2*(3*D*a - B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2)
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.80 \[ \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\frac {D x}{b^{2}} + \frac {C \log \left (b x^{2} + a\right )}{2 \, b^{2}} - \frac {{\left (3 \, D a - B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} b^{2}} + \frac {C a - A b + {\left (D a - B b\right )} x}{2 \, {\left (b x^{2} + a\right )} b^{2}} \]
D*x/b^2 + 1/2*C*log(b*x^2 + a)/b^2 - 1/2*(3*D*a - B*b)*arctan(b*x/sqrt(a*b ))/(sqrt(a*b)*b^2) + 1/2*(C*a - A*b + (D*a - B*b)*x)/((b*x^2 + a)*b^2)
Timed out. \[ \int \frac {x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx=\int \frac {x\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{{\left (b\,x^2+a\right )}^2} \,d x \]