Integrand size = 34, antiderivative size = 120 \[ \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx=-\frac {(b c C-b B d-A c d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {b}}\right )}{\sqrt {b} d^{3/2} \left (c^2+b d\right )}+\frac {\left (c^2 C-B c d+A d^2\right ) \log (c+d x)}{d^2 \left (c^2+b d\right )}+\frac {(B c+b C-A d) \log \left (b+d x^2\right )}{2 d \left (c^2+b d\right )} \] Output:
-(-A*c*d-B*b*d+C*b*c)*arctan(d^(1/2)*x/b^(1/2))/b^(1/2)/d^(3/2)/(b*d+c^2)+ (A*d^2-B*c*d+C*c^2)*ln(d*x+c)/d^2/(b*d+c^2)+1/2*(-A*d+B*c+C*b)*ln(d*x^2+b) /d/(b*d+c^2)
Time = 0.06 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx=\frac {2 \sqrt {d} (-b c C+b B d+A c d) \arctan \left (\frac {\sqrt {d} x}{\sqrt {b}}\right )+\sqrt {b} \left (2 \left (c^2 C-B c d+A d^2\right ) \log (c+d x)+d (B c+b C-A d) \log \left (b+d x^2\right )\right )}{2 \sqrt {b} d^2 \left (c^2+b d\right )} \] Input:
Integrate[(A + B*x + C*x^2)/(b*c + b*d*x + c*d*x^2 + d^2*x^3),x]
Output:
(2*Sqrt[d]*(-(b*c*C) + b*B*d + A*c*d)*ArcTan[(Sqrt[d]*x)/Sqrt[b]] + Sqrt[b ]*(2*(c^2*C - B*c*d + A*d^2)*Log[c + d*x] + d*(B*c + b*C - A*d)*Log[b + d* x^2]))/(2*Sqrt[b]*d^2*(c^2 + b*d))
Time = 0.59 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {2462, 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.
\(\displaystyle \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle \int \left (\frac {A d^2-B c d+c^2 C}{d \left (b d+c^2\right ) (c+d x)}+\frac {d x (-A d+b C+B c)+A c d+b B d-b c C}{d \left (b d+c^2\right ) \left (b+d x^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\arctan \left (\frac {\sqrt {d} x}{\sqrt {b}}\right ) (-A c d-b B d+b c C)}{\sqrt {b} d^{3/2} \left (b d+c^2\right )}+\frac {\log (c+d x) \left (A d^2-B c d+c^2 C\right )}{d^2 \left (b d+c^2\right )}+\frac {\log \left (b+d x^2\right ) (-A d+b C+B c)}{2 d \left (b d+c^2\right )}\) |
Input:
Int[(A + B*x + C*x^2)/(b*c + b*d*x + c*d*x^2 + d^2*x^3),x]
Output:
-(((b*c*C - b*B*d - A*c*d)*ArcTan[(Sqrt[d]*x)/Sqrt[b]])/(Sqrt[b]*d^(3/2)*( c^2 + b*d))) + ((c^2*C - B*c*d + A*d^2)*Log[c + d*x])/(d^2*(c^2 + b*d)) + ((B*c + b*C - A*d)*Log[b + d*x^2])/(2*d*(c^2 + b*d))
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u*Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.13 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (d x +c \right )}{d^{2} \left (b d +c^{2}\right )}+\frac {\frac {\left (-A \,d^{2}+B c d +C b d \right ) \ln \left (d \,x^{2}+b \right )}{2 d}+\frac {\left (A c d +B b d -C b c \right ) \arctan \left (\frac {d x}{\sqrt {b d}}\right )}{\sqrt {b d}}}{d \left (b d +c^{2}\right )}\) | \(108\) |
risch | \(\text {Expression too large to display}\) | \(5978\) |
Input:
int((C*x^2+B*x+A)/(d^2*x^3+c*d*x^2+b*d*x+b*c),x,method=_RETURNVERBOSE)
Output:
(A*d^2-B*c*d+C*c^2)*ln(d*x+c)/d^2/(b*d+c^2)+1/d/(b*d+c^2)*(1/2*(-A*d^2+B*c *d+C*b*d)/d*ln(d*x^2+b)+(A*c*d+B*b*d-C*b*c)/(b*d)^(1/2)*arctan(d*x/(b*d)^( 1/2)))
Time = 0.58 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.00 \[ \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx=\left [\frac {{\left (C b c - {\left (B b + A c\right )} d\right )} \sqrt {-b d} \log \left (\frac {d x^{2} - 2 \, \sqrt {-b d} x - b}{d x^{2} + b}\right ) - {\left (A b d^{2} - {\left (C b^{2} + B b c\right )} d\right )} \log \left (d x^{2} + b\right ) + 2 \, {\left (C b c^{2} - B b c d + A b d^{2}\right )} \log \left (d x + c\right )}{2 \, {\left (b c^{2} d^{2} + b^{2} d^{3}\right )}}, -\frac {2 \, {\left (C b c - {\left (B b + A c\right )} d\right )} \sqrt {b d} \arctan \left (\frac {\sqrt {b d} x}{b}\right ) + {\left (A b d^{2} - {\left (C b^{2} + B b c\right )} d\right )} \log \left (d x^{2} + b\right ) - 2 \, {\left (C b c^{2} - B b c d + A b d^{2}\right )} \log \left (d x + c\right )}{2 \, {\left (b c^{2} d^{2} + b^{2} d^{3}\right )}}\right ] \] Input:
integrate((C*x^2+B*x+A)/(d^2*x^3+c*d*x^2+b*d*x+b*c),x, algorithm="fricas")
Output:
[1/2*((C*b*c - (B*b + A*c)*d)*sqrt(-b*d)*log((d*x^2 - 2*sqrt(-b*d)*x - b)/ (d*x^2 + b)) - (A*b*d^2 - (C*b^2 + B*b*c)*d)*log(d*x^2 + b) + 2*(C*b*c^2 - B*b*c*d + A*b*d^2)*log(d*x + c))/(b*c^2*d^2 + b^2*d^3), -1/2*(2*(C*b*c - (B*b + A*c)*d)*sqrt(b*d)*arctan(sqrt(b*d)*x/b) + (A*b*d^2 - (C*b^2 + B*b*c )*d)*log(d*x^2 + b) - 2*(C*b*c^2 - B*b*c*d + A*b*d^2)*log(d*x + c))/(b*c^2 *d^2 + b^2*d^3)]
Timed out. \[ \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx=\text {Timed out} \] Input:
integrate((C*x**2+B*x+A)/(d**2*x**3+c*d*x**2+b*d*x+b*c),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx=\frac {{\left (C b + B c - A d\right )} \log \left (d x^{2} + b\right )}{2 \, {\left (c^{2} d + b d^{2}\right )}} + \frac {{\left (C c^{2} - B c d + A d^{2}\right )} \log \left (d x + c\right )}{c^{2} d^{2} + b d^{3}} - \frac {{\left (C b c - {\left (B b + A c\right )} d\right )} \arctan \left (\frac {d x}{\sqrt {b d}}\right )}{{\left (c^{2} d + b d^{2}\right )} \sqrt {b d}} \] Input:
integrate((C*x^2+B*x+A)/(d^2*x^3+c*d*x^2+b*d*x+b*c),x, algorithm="maxima")
Output:
1/2*(C*b + B*c - A*d)*log(d*x^2 + b)/(c^2*d + b*d^2) + (C*c^2 - B*c*d + A* d^2)*log(d*x + c)/(c^2*d^2 + b*d^3) - (C*b*c - (B*b + A*c)*d)*arctan(d*x/s qrt(b*d))/((c^2*d + b*d^2)*sqrt(b*d))
Time = 0.11 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx=\frac {{\left (C b + B c - A d\right )} \log \left (d x^{2} + b\right )}{2 \, {\left (c^{2} d + b d^{2}\right )}} + \frac {{\left (C c^{2} - B c d + A d^{2}\right )} \log \left ({\left | d x + c \right |}\right )}{c^{2} d^{2} + b d^{3}} - \frac {{\left (C b c - B b d - A c d\right )} \arctan \left (\frac {d x}{\sqrt {b d}}\right )}{{\left (c^{2} d + b d^{2}\right )} \sqrt {b d}} \] Input:
integrate((C*x^2+B*x+A)/(d^2*x^3+c*d*x^2+b*d*x+b*c),x, algorithm="giac")
Output:
1/2*(C*b + B*c - A*d)*log(d*x^2 + b)/(c^2*d + b*d^2) + (C*c^2 - B*c*d + A* d^2)*log(abs(d*x + c))/(c^2*d^2 + b*d^3) - (C*b*c - B*b*d - A*c*d)*arctan( d*x/sqrt(b*d))/((c^2*d + b*d^2)*sqrt(b*d))
Time = 14.64 (sec) , antiderivative size = 793, normalized size of antiderivative = 6.61 \[ \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx =\text {Too large to display} \] Input:
int((A + B*x + C*x^2)/(b*c + d^2*x^3 + b*d*x + c*d*x^2),x)
Output:
(log(x*(B^2*d^2 - A*C*d^2 + C^2*b*d - B*C*c*d) + ((d^2*((C*b^2)/2 + (B*b*c )/2) + d*((A*c*(-b*d^3)^(1/2))/2 + (B*b*(-b*d^3)^(1/2))/2) - (A*b*d^3)/2 - (C*b*c*(-b*d^3)^(1/2))/2)*(x*(3*A*d^4 + 2*C*c^2*d^2 - B*c*d^3 - 5*C*b*d^3 ) + A*c*d^3 - B*b*d^3 + ((x*(6*b*d^5 - 2*c^2*d^4) + 8*b*c*d^4)*(d^2*((C*b^ 2)/2 + (B*b*c)/2) + d*((A*c*(-b*d^3)^(1/2))/2 + (B*b*(-b*d^3)^(1/2))/2) - (A*b*d^3)/2 - (C*b*c*(-b*d^3)^(1/2))/2))/(b^2*d^4 + b*c^2*d^3) - 5*C*b*c*d ^2))/(b^2*d^4 + b*c^2*d^3) + A*B*d^2 + C^2*b*c - A*C*c*d)*(d^2*((C*b^2)/2 + (B*b*c)/2) + d*((A*c*(-b*d^3)^(1/2))/2 + (B*b*(-b*d^3)^(1/2))/2) - (A*b* d^3)/2 - (C*b*c*(-b*d^3)^(1/2))/2))/(b^2*d^4 + b*c^2*d^3) + (log(x*(B^2*d^ 2 - A*C*d^2 + C^2*b*d - B*C*c*d) + ((d^2*((C*b^2)/2 + (B*b*c)/2) - d*((A*c *(-b*d^3)^(1/2))/2 + (B*b*(-b*d^3)^(1/2))/2) - (A*b*d^3)/2 + (C*b*c*(-b*d^ 3)^(1/2))/2)*(x*(3*A*d^4 + 2*C*c^2*d^2 - B*c*d^3 - 5*C*b*d^3) + A*c*d^3 - B*b*d^3 + ((x*(6*b*d^5 - 2*c^2*d^4) + 8*b*c*d^4)*(d^2*((C*b^2)/2 + (B*b*c) /2) - d*((A*c*(-b*d^3)^(1/2))/2 + (B*b*(-b*d^3)^(1/2))/2) - (A*b*d^3)/2 + (C*b*c*(-b*d^3)^(1/2))/2))/(b^2*d^4 + b*c^2*d^3) - 5*C*b*c*d^2))/(b^2*d^4 + b*c^2*d^3) + A*B*d^2 + C^2*b*c - A*C*c*d)*(d^2*((C*b^2)/2 + (B*b*c)/2) - d*((A*c*(-b*d^3)^(1/2))/2 + (B*b*(-b*d^3)^(1/2))/2) - (A*b*d^3)/2 + (C*b* c*(-b*d^3)^(1/2))/2))/(b^2*d^4 + b*c^2*d^3) + (log(c + d*x)*(A*d^2 + C*c^2 - B*c*d))/(d^2*(b*d + c^2))
Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.26 \[ \int \frac {A+B x+C x^2}{b c+b d x+c d x^2+d^2 x^3} \, dx=\frac {2 \sqrt {d}\, \sqrt {b}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {b}}\right ) a c d +2 \sqrt {d}\, \sqrt {b}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {b}}\right ) b^{2} d -2 \sqrt {d}\, \sqrt {b}\, \mathit {atan} \left (\frac {d x}{\sqrt {d}\, \sqrt {b}}\right ) b \,c^{2}-\mathrm {log}\left (d \,x^{2}+b \right ) a b \,d^{2}+2 \,\mathrm {log}\left (d \,x^{2}+b \right ) b^{2} c d +2 \,\mathrm {log}\left (d x +c \right ) a b \,d^{2}-2 \,\mathrm {log}\left (d x +c \right ) b^{2} c d +2 \,\mathrm {log}\left (d x +c \right ) b \,c^{3}}{2 b \,d^{2} \left (b d +c^{2}\right )} \] Input:
int((C*x^2+B*x+A)/(d^2*x^3+c*d*x^2+b*d*x+b*c),x)
Output:
(2*sqrt(d)*sqrt(b)*atan((d*x)/(sqrt(d)*sqrt(b)))*a*c*d + 2*sqrt(d)*sqrt(b) *atan((d*x)/(sqrt(d)*sqrt(b)))*b**2*d - 2*sqrt(d)*sqrt(b)*atan((d*x)/(sqrt (d)*sqrt(b)))*b*c**2 - log(b + d*x**2)*a*b*d**2 + 2*log(b + d*x**2)*b**2*c *d + 2*log(c + d*x)*a*b*d**2 - 2*log(c + d*x)*b**2*c*d + 2*log(c + d*x)*b* c**3)/(2*b*d**2*(b*d + c**2))