Integrand size = 20, antiderivative size = 100 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {b c \left (A+\frac {a C}{c}\right )+\left (2 A c^2+\left (b^2-2 a c\right ) C\right ) x}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {4 (A c+a C) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
(-b*c*(A+a*C/c)-(2*A*c^2+(-2*a*c+b^2)*C)*x)/c/(-4*a*c+b^2)/(c*x^2+b*x+a)+4 *(A*c+C*a)*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)
Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.98 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {b^2 C x+a C (b-2 c x)+A c (b+2 c x)}{c \left (-b^2+4 a c\right ) (a+x (b+c x))}+\frac {4 (A c+a C) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}} \]
(b^2*C*x + a*C*(b - 2*c*x) + A*c*(b + 2*c*x))/(c*(-b^2 + 4*a*c)*(a + x*(b + c*x))) + (4*(A*c + a*C)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2)
Time = 0.25 (sec) , antiderivative size = 100, 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 = {2191, 27, 1083, 219}
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.
\(\displaystyle \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 2191 |
\(\displaystyle -\frac {\int \frac {2 (A c+a C)}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 (a C+A c) \int \frac {1}{c x^2+b x+a}dx}{b^2-4 a c}-\frac {x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {4 (a C+A c) \int \frac {1}{b^2-(b+2 c x)^2-4 a c}d(b+2 c x)}{b^2-4 a c}-\frac {x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {4 (a C+A c) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {x \left (C \left (b^2-2 a c\right )+2 A c^2\right )+b c \left (\frac {a C}{c}+A\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}\) |
-((b*c*(A + (a*C)/c) + (2*A*c^2 + (b^2 - 2*a*c)*C)*x)/(c*(b^2 - 4*a*c)*(a + b*x + c*x^2))) + (4*(A*c + a*C)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/ (b^2 - 4*a*c)^(3/2)
3.2.45.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Time = 0.60 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.15
method | result | size |
default | \(\frac {\frac {\left (2 A \,c^{2}-2 C a c +C \,b^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {b \left (A c +C a \right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {4 \left (A c +C a \right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}\) | \(115\) |
risch | \(\frac {\frac {\left (2 A \,c^{2}-2 C a c +C \,b^{2}\right ) x}{c \left (4 a c -b^{2}\right )}+\frac {b \left (A c +C a \right )}{c \left (4 a c -b^{2}\right )}}{c \,x^{2}+b x +a}+\frac {2 \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) A c}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}+\frac {2 \ln \left (\left (-8 a \,c^{2}+2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}-4 a b c +b^{3}\right ) C a}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) A c}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}-\frac {2 \ln \left (\left (8 a \,c^{2}-2 b^{2} c \right ) x +\left (-4 a c +b^{2}\right )^{\frac {3}{2}}+4 a b c -b^{3}\right ) C a}{\left (-4 a c +b^{2}\right )^{\frac {3}{2}}}\) | \(274\) |
((2*A*c^2-2*C*a*c+C*b^2)/c/(4*a*c-b^2)*x+b/c*(A*c+C*a)/(4*a*c-b^2))/(c*x^2 +b*x+a)+4*(A*c+C*a)/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (96) = 192\).
Time = 0.31 (sec) , antiderivative size = 511, normalized size of antiderivative = 5.11 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx=\left [-\frac {C a b^{3} - 4 \, A a b c^{2} + 2 \, {\left (C a^{2} c + A a c^{2} + {\left (C a c^{2} + A c^{3}\right )} x^{2} + {\left (C a b c + A b c^{2}\right )} x\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) - {\left (4 \, C a^{2} b - A b^{3}\right )} c + {\left (C b^{4} - 6 \, C a b^{2} c - 8 \, A a c^{3} + 2 \, {\left (4 \, C a^{2} + A b^{2}\right )} c^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}, -\frac {C a b^{3} - 4 \, A a b c^{2} - 4 \, {\left (C a^{2} c + A a c^{2} + {\left (C a c^{2} + A c^{3}\right )} x^{2} + {\left (C a b c + A b c^{2}\right )} x\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - {\left (4 \, C a^{2} b - A b^{3}\right )} c + {\left (C b^{4} - 6 \, C a b^{2} c - 8 \, A a c^{3} + 2 \, {\left (4 \, C a^{2} + A b^{2}\right )} c^{2}\right )} x}{a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}\right ] \]
[-(C*a*b^3 - 4*A*a*b*c^2 + 2*(C*a^2*c + A*a*c^2 + (C*a*c^2 + A*c^3)*x^2 + (C*a*b*c + A*b*c^2)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c - sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) - (4*C*a^2*b - A *b^3)*c + (C*b^4 - 6*C*a*b^2*c - 8*A*a*c^3 + 2*(4*C*a^2 + A*b^2)*c^2)*x)/( a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4 )*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x), -(C*a*b^3 - 4*A*a*b*c^2 - 4*(C*a^2*c + A*a*c^2 + (C*a*c^2 + A*c^3)*x^2 + (C*a*b*c + A*b*c^2)*x)*sqr t(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (4 *C*a^2*b - A*b^3)*c + (C*b^4 - 6*C*a*b^2*c - 8*A*a*c^3 + 2*(4*C*a^2 + A*b^ 2)*c^2)*x)/(a*b^4*c - 8*a^2*b^2*c^2 + 16*a^3*c^3 + (b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + (b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x)]
Leaf count of result is larger than twice the leaf count of optimal. 376 vs. \(2 (92) = 184\).
Time = 0.59 (sec) , antiderivative size = 376, normalized size of antiderivative = 3.76 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx=- 2 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) \log {\left (x + \frac {2 A b c + 2 C a b - 32 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) + 16 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) - 2 b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right )}{4 A c^{2} + 4 C a c} \right )} + 2 \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) \log {\left (x + \frac {2 A b c + 2 C a b + 32 a^{2} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) - 16 a b^{2} c \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right ) + 2 b^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{3}}} \left (A c + C a\right )}{4 A c^{2} + 4 C a c} \right )} + \frac {A b c + C a b + x \left (2 A c^{2} - 2 C a c + C b^{2}\right )}{4 a^{2} c^{2} - a b^{2} c + x^{2} \cdot \left (4 a c^{3} - b^{2} c^{2}\right ) + x \left (4 a b c^{2} - b^{3} c\right )} \]
-2*sqrt(-1/(4*a*c - b**2)**3)*(A*c + C*a)*log(x + (2*A*b*c + 2*C*a*b - 32* a**2*c**2*sqrt(-1/(4*a*c - b**2)**3)*(A*c + C*a) + 16*a*b**2*c*sqrt(-1/(4* a*c - b**2)**3)*(A*c + C*a) - 2*b**4*sqrt(-1/(4*a*c - b**2)**3)*(A*c + C*a ))/(4*A*c**2 + 4*C*a*c)) + 2*sqrt(-1/(4*a*c - b**2)**3)*(A*c + C*a)*log(x + (2*A*b*c + 2*C*a*b + 32*a**2*c**2*sqrt(-1/(4*a*c - b**2)**3)*(A*c + C*a) - 16*a*b**2*c*sqrt(-1/(4*a*c - b**2)**3)*(A*c + C*a) + 2*b**4*sqrt(-1/(4* a*c - b**2)**3)*(A*c + C*a))/(4*A*c**2 + 4*C*a*c)) + (A*b*c + C*a*b + x*(2 *A*c**2 - 2*C*a*c + C*b**2))/(4*a**2*c**2 - a*b**2*c + x**2*(4*a*c**3 - b* *2*c**2) + x*(4*a*b*c**2 - b**3*c))
Exception generated. \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx=-\frac {4 \, {\left (C a + A c\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {C b^{2} x - 2 \, C a c x + 2 \, A c^{2} x + C a b + A b c}{{\left (b^{2} c - 4 \, a c^{2}\right )} {\left (c x^{2} + b x + a\right )}} \]
-4*(C*a + A*c)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*a*c)*sqrt( -b^2 + 4*a*c)) - (C*b^2*x - 2*C*a*c*x + 2*A*c^2*x + C*a*b + A*b*c)/((b^2*c - 4*a*c^2)*(c*x^2 + b*x + a))
Time = 13.07 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.72 \[ \int \frac {A+C x^2}{\left (a+b x+c x^2\right )^2} \, dx=\frac {\frac {A\,b\,c+C\,a\,b}{c\,\left (4\,a\,c-b^2\right )}+\frac {x\,\left (C\,b^2+2\,A\,c^2-2\,C\,a\,c\right )}{c\,\left (4\,a\,c-b^2\right )}}{c\,x^2+b\,x+a}-\frac {4\,\mathrm {atan}\left (\frac {\left (\frac {2\,\left (A\,c+C\,a\right )\,\left (b^3-4\,a\,b\,c\right )}{{\left (4\,a\,c-b^2\right )}^{5/2}}-\frac {4\,c\,x\,\left (A\,c+C\,a\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}}\right )\,\left (4\,a\,c-b^2\right )}{2\,A\,c+2\,C\,a}\right )\,\left (A\,c+C\,a\right )}{{\left (4\,a\,c-b^2\right )}^{3/2}} \]