Integrand size = 28, antiderivative size = 174 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^3} \, dx=-\frac {A}{2 a^3 x^2}-\frac {B}{a^3 x}-\frac {\frac {A b}{a}-C+\left (\frac {b B}{a}-D\right ) x}{4 a \left (a+b x^2\right )^2}-\frac {4 (2 A b-a C)+(7 b B-3 a D) x}{8 a^3 \left (a+b x^2\right )}-\frac {3 (5 b B-a D) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{7/2} \sqrt {b}}-\frac {(3 A b-a C) \log (x)}{a^4}+\frac {(3 A b-a C) \log \left (a+b x^2\right )}{2 a^4} \]
-1/2*A/a^3/x^2-B/a^3/x+1/4*(-A*b/a+C-(b*B/a-D)*x)/a/(b*x^2+a)^2+1/8*(-8*A* b+4*C*a-(7*B*b-3*D*a)*x)/a^3/(b*x^2+a)-(3*A*b-C*a)*ln(x)/a^4+1/2*(3*A*b-C* a)*ln(b*x^2+a)/a^4-3/8*(5*B*b-D*a)*arctan(x*b^(1/2)/a^(1/2))/a^(7/2)/b^(1/ 2)
Time = 0.10 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.84 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 a A}{x^2}-\frac {8 a B}{x}+\frac {a (-8 A b+4 a C-7 b B x+3 a D x)}{a+b x^2}+\frac {2 a^2 (-A b-b B x+a (C+D x))}{\left (a+b x^2\right )^2}+\frac {3 \sqrt {a} (-5 b B+a D) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+8 (-3 A b+a C) \log (x)+4 (3 A b-a C) \log \left (a+b x^2\right )}{8 a^4} \]
((-4*a*A)/x^2 - (8*a*B)/x + (a*(-8*A*b + 4*a*C - 7*b*B*x + 3*a*D*x))/(a + b*x^2) + (2*a^2*(-(A*b) - b*B*x + a*(C + D*x)))/(a + b*x^2)^2 + (3*Sqrt[a] *(-5*b*B + a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] + 8*(-3*A*b + a*C)*Lo g[x] + 4*(3*A*b - a*C)*Log[a + b*x^2])/(8*a^4)
Time = 0.65 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2336, 25, 2336, 25, 2333, 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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle -\frac {\int -\frac {-3 \left (\frac {b B}{a}-D\right ) x^3-4 \left (\frac {A b}{a}-C\right ) x^2+4 B x+4 A}{x^3 \left (b x^2+a\right )^2}dx}{4 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-3 \left (\frac {b B}{a}-D\right ) x^3-4 \left (\frac {A b}{a}-C\right ) x^2+4 B x+4 A}{x^3 \left (b x^2+a\right )^2}dx}{4 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {-\frac {\int -\frac {-\left (\left (\frac {7 b B}{a}-3 D\right ) x^3\right )-8 \left (\frac {2 A b}{a}-C\right ) x^2+8 B x+8 A}{x^3 \left (b x^2+a\right )}dx}{2 a}-\frac {4 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {7 b B}{a}-3 D\right )}{2 a \left (a+b x^2\right )}}{4 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-\left (\left (\frac {7 b B}{a}-3 D\right ) x^3\right )-8 \left (\frac {2 A b}{a}-C\right ) x^2+8 B x+8 A}{x^3 \left (b x^2+a\right )}dx}{2 a}-\frac {4 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {7 b B}{a}-3 D\right )}{2 a \left (a+b x^2\right )}}{4 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\frac {\int \left (\frac {8 A}{a x^3}+\frac {8 (a C-3 A b)}{a^2 x}+\frac {8 b (3 A b-a C) x-3 a (5 b B-a D)}{a^2 \left (b x^2+a\right )}+\frac {8 B}{a x^2}\right )dx}{2 a}-\frac {4 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {7 b B}{a}-3 D\right )}{2 a \left (a+b x^2\right )}}{4 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) (5 b B-a D)}{a^{3/2} \sqrt {b}}+\frac {4 (3 A b-a C) \log \left (a+b x^2\right )}{a^2}-\frac {8 \log (x) (3 A b-a C)}{a^2}-\frac {4 A}{a x^2}-\frac {8 B}{a x}}{2 a}-\frac {4 \left (\frac {2 A b}{a}-C\right )+x \left (\frac {7 b B}{a}-3 D\right )}{2 a \left (a+b x^2\right )}}{4 a}-\frac {\frac {A b}{a}+x \left (\frac {b B}{a}-D\right )-C}{4 a \left (a+b x^2\right )^2}\) |
-1/4*((A*b)/a - C + ((b*B)/a - D)*x)/(a*(a + b*x^2)^2) + (-1/2*(4*((2*A*b) /a - C) + ((7*b*B)/a - 3*D)*x)/(a*(a + b*x^2)) + ((-4*A)/(a*x^2) - (8*B)/( a*x) - (3*(5*b*B - a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[b]) - ( 8*(3*A*b - a*C)*Log[x])/a^2 + (4*(3*A*b - a*C)*Log[a + b*x^2])/a^2)/(2*a)) /(4*a)
3.2.9.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*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] + Simp[1/(2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Time = 3.60 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.97
| method | result | size |
| default | \(-\frac {A}{2 a^{3} x^{2}}-\frac {B}{a^{3} x}+\frac {\left (-3 A b +C a \right ) \ln \left (x \right )}{a^{4}}+\frac {\frac {\left (-\frac {7}{8} a \,b^{2} B +\frac {3}{8} D a^{2} b \right ) x^{3}+\left (-a \,b^{2} A +\frac {1}{2} C \,a^{2} b \right ) x^{2}-\frac {a^{2} \left (9 B b -5 D a \right ) x}{8}-\frac {5 a^{2} b A}{4}+\frac {3 C \,a^{3}}{4}}{\left (b \,x^{2}+a \right )^{2}}+\frac {\left (24 b^{2} A -8 C a b \right ) \ln \left (b \,x^{2}+a \right )}{16 b}+\frac {\left (-15 a b B +3 D a^{2}\right ) \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}}}{a^{4}}\) | \(169\) |
-1/2*A/a^3/x^2-B/a^3/x+(-3*A*b+C*a)/a^4*ln(x)+1/a^4*(((-7/8*a*b^2*B+3/8*D* a^2*b)*x^3+(-a*b^2*A+1/2*C*a^2*b)*x^2-1/8*a^2*(9*B*b-5*D*a)*x-5/4*a^2*b*A+ 3/4*C*a^3)/(b*x^2+a)^2+1/16*(24*A*b^2-8*C*a*b)/b*ln(b*x^2+a)+1/8*(-15*B*a* b+3*D*a^2)/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (147) = 294\).
Time = 0.34 (sec) , antiderivative size = 696, normalized size of antiderivative = 4.00 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^3} \, dx=\left [-\frac {16 \, B a^{3} b x - 6 \, {\left (D a^{2} b^{2} - 5 \, B a b^{3}\right )} x^{5} + 8 \, A a^{3} b - 8 \, {\left (C a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} - 10 \, {\left (D a^{3} b - 5 \, B a^{2} b^{2}\right )} x^{3} - 12 \, {\left (C a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} + 3 \, {\left ({\left (D a b^{2} - 5 \, B b^{3}\right )} x^{6} + 2 \, {\left (D a^{2} b - 5 \, B a b^{2}\right )} x^{4} + {\left (D a^{3} - 5 \, B a^{2} b\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 8 \, {\left ({\left (C a b^{3} - 3 \, A b^{4}\right )} x^{6} + 2 \, {\left (C a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + {\left (C a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 16 \, {\left ({\left (C a b^{3} - 3 \, A b^{4}\right )} x^{6} + 2 \, {\left (C a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + {\left (C a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{16 \, {\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{4} + a^{6} b x^{2}\right )}}, -\frac {8 \, B a^{3} b x - 3 \, {\left (D a^{2} b^{2} - 5 \, B a b^{3}\right )} x^{5} + 4 \, A a^{3} b - 4 \, {\left (C a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} - 5 \, {\left (D a^{3} b - 5 \, B a^{2} b^{2}\right )} x^{3} - 6 \, {\left (C a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2} - 3 \, {\left ({\left (D a b^{2} - 5 \, B b^{3}\right )} x^{6} + 2 \, {\left (D a^{2} b - 5 \, B a b^{2}\right )} x^{4} + {\left (D a^{3} - 5 \, B a^{2} b\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + 4 \, {\left ({\left (C a b^{3} - 3 \, A b^{4}\right )} x^{6} + 2 \, {\left (C a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + {\left (C a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) - 8 \, {\left ({\left (C a b^{3} - 3 \, A b^{4}\right )} x^{6} + 2 \, {\left (C a^{2} b^{2} - 3 \, A a b^{3}\right )} x^{4} + {\left (C a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{2}\right )} \log \left (x\right )}{8 \, {\left (a^{4} b^{3} x^{6} + 2 \, a^{5} b^{2} x^{4} + a^{6} b x^{2}\right )}}\right ] \]
[-1/16*(16*B*a^3*b*x - 6*(D*a^2*b^2 - 5*B*a*b^3)*x^5 + 8*A*a^3*b - 8*(C*a^ 2*b^2 - 3*A*a*b^3)*x^4 - 10*(D*a^3*b - 5*B*a^2*b^2)*x^3 - 12*(C*a^3*b - 3* A*a^2*b^2)*x^2 + 3*((D*a*b^2 - 5*B*b^3)*x^6 + 2*(D*a^2*b - 5*B*a*b^2)*x^4 + (D*a^3 - 5*B*a^2*b)*x^2)*sqrt(-a*b)*log((b*x^2 - 2*sqrt(-a*b)*x - a)/(b* x^2 + a)) + 8*((C*a*b^3 - 3*A*b^4)*x^6 + 2*(C*a^2*b^2 - 3*A*a*b^3)*x^4 + ( C*a^3*b - 3*A*a^2*b^2)*x^2)*log(b*x^2 + a) - 16*((C*a*b^3 - 3*A*b^4)*x^6 + 2*(C*a^2*b^2 - 3*A*a*b^3)*x^4 + (C*a^3*b - 3*A*a^2*b^2)*x^2)*log(x))/(a^4 *b^3*x^6 + 2*a^5*b^2*x^4 + a^6*b*x^2), -1/8*(8*B*a^3*b*x - 3*(D*a^2*b^2 - 5*B*a*b^3)*x^5 + 4*A*a^3*b - 4*(C*a^2*b^2 - 3*A*a*b^3)*x^4 - 5*(D*a^3*b - 5*B*a^2*b^2)*x^3 - 6*(C*a^3*b - 3*A*a^2*b^2)*x^2 - 3*((D*a*b^2 - 5*B*b^3)* x^6 + 2*(D*a^2*b - 5*B*a*b^2)*x^4 + (D*a^3 - 5*B*a^2*b)*x^2)*sqrt(a*b)*arc tan(sqrt(a*b)*x/a) + 4*((C*a*b^3 - 3*A*b^4)*x^6 + 2*(C*a^2*b^2 - 3*A*a*b^3 )*x^4 + (C*a^3*b - 3*A*a^2*b^2)*x^2)*log(b*x^2 + a) - 8*((C*a*b^3 - 3*A*b^ 4)*x^6 + 2*(C*a^2*b^2 - 3*A*a*b^3)*x^4 + (C*a^3*b - 3*A*a^2*b^2)*x^2)*log( x))/(a^4*b^3*x^6 + 2*a^5*b^2*x^4 + a^6*b*x^2)]
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (D a b - 5 \, B b^{2}\right )} x^{5} + 4 \, {\left (C a b - 3 \, A b^{2}\right )} x^{4} - 8 \, B a^{2} x + 5 \, {\left (D a^{2} - 5 \, B a b\right )} x^{3} - 4 \, A a^{2} + 6 \, {\left (C a^{2} - 3 \, A a b\right )} x^{2}}{8 \, {\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} + \frac {3 \, {\left (D a - 5 \, B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} - \frac {{\left (C a - 3 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac {{\left (C a - 3 \, A b\right )} \log \left (x\right )}{a^{4}} \]
1/8*(3*(D*a*b - 5*B*b^2)*x^5 + 4*(C*a*b - 3*A*b^2)*x^4 - 8*B*a^2*x + 5*(D* a^2 - 5*B*a*b)*x^3 - 4*A*a^2 + 6*(C*a^2 - 3*A*a*b)*x^2)/(a^3*b^2*x^6 + 2*a ^4*b*x^4 + a^5*x^2) + 3/8*(D*a - 5*B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a ^3) - 1/2*(C*a - 3*A*b)*log(b*x^2 + a)/a^4 + (C*a - 3*A*b)*log(x)/a^4
Time = 0.31 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (D a - 5 \, B b\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{3}} - \frac {{\left (C a - 3 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac {{\left (C a - 3 \, A b\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {3 \, D a b x^{5} - 15 \, B b^{2} x^{5} + 4 \, C a b x^{4} - 12 \, A b^{2} x^{4} + 5 \, D a^{2} x^{3} - 25 \, B a b x^{3} + 6 \, C a^{2} x^{2} - 18 \, A a b x^{2} - 8 \, B a^{2} x - 4 \, A a^{2}}{8 \, {\left (b x^{3} + a x\right )}^{2} a^{3}} \]
3/8*(D*a - 5*B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a^3) - 1/2*(C*a - 3*A*b )*log(b*x^2 + a)/a^4 + (C*a - 3*A*b)*log(abs(x))/a^4 + 1/8*(3*D*a*b*x^5 - 15*B*b^2*x^5 + 4*C*a*b*x^4 - 12*A*b^2*x^4 + 5*D*a^2*x^3 - 25*B*a*b*x^3 + 6 *C*a^2*x^2 - 18*A*a*b*x^2 - 8*B*a^2*x - 4*A*a^2)/((b*x^3 + a*x)^2*a^3)
Time = 6.39 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.32 \[ \int \frac {A+B x+C x^2+D x^3}{x^3 \left (a+b x^2\right )^3} \, dx=\frac {\frac {3\,C}{4\,a}+\frac {C\,b\,x^2}{2\,a^2}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {\frac {A}{2\,a}+\frac {9\,A\,b\,x^2}{4\,a^2}+\frac {3\,A\,b^2\,x^4}{2\,a^3}}{a^2\,x^2+2\,a\,b\,x^4+b^2\,x^6}-\frac {\frac {B}{a}+\frac {25\,B\,b\,x^2}{8\,a^2}+\frac {15\,B\,b^2\,x^4}{8\,a^3}}{a^2\,x+2\,a\,b\,x^3+b^2\,x^5}-\frac {C\,\ln \left (b\,x^2+a\right )}{2\,a^3}+\frac {C\,\ln \left (x\right )}{a^3}+\frac {3\,A\,b\,\ln \left (b\,x^2+a\right )}{2\,a^4}-\frac {3\,A\,b\,\ln \left (x\right )}{a^4}+\frac {x\,D\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},3;\ \frac {3}{2};\ -\frac {b\,x^2}{a}\right )}{a^3}-\frac {15\,B\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,a^{7/2}} \]
((3*C)/(4*a) + (C*b*x^2)/(2*a^2))/(a^2 + b^2*x^4 + 2*a*b*x^2) - (A/(2*a) + (9*A*b*x^2)/(4*a^2) + (3*A*b^2*x^4)/(2*a^3))/(a^2*x^2 + b^2*x^6 + 2*a*b*x ^4) - (B/a + (25*B*b*x^2)/(8*a^2) + (15*B*b^2*x^4)/(8*a^3))/(a^2*x + b^2*x ^5 + 2*a*b*x^3) - (C*log(a + b*x^2))/(2*a^3) + (C*log(x))/a^3 + (3*A*b*log (a + b*x^2))/(2*a^4) - (3*A*b*log(x))/a^4 + (x*D*hypergeom([1/2, 3], 3/2, -(b*x^2)/a))/a^3 - (15*B*b^(1/2)*atan((b^(1/2)*x)/a^(1/2)))/(8*a^(7/2))