Integrand size = 19, antiderivative size = 89 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx=-\frac {b \text {b1}-a \text {c1}+(\text {b1} c-b \text {c1}) x}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}+\frac {(\text {b1} c-b \text {c1}) \text {arctanh}\left (\frac {b+c x}{\sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}} \]
1/2*(-b*b1+a*c1-(-b*c1+b1*c)*x)/(-a*c+b^2)/(c*x^2+2*b*x+a)+1/2*(-b*c1+b1*c )*arctanh((c*x+b)/(-a*c+b^2)^(1/2))/(-a*c+b^2)^(3/2)
Time = 0.07 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.99 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx=\frac {\frac {-b \text {b1}+a \text {c1}-\text {b1} c x+b \text {c1} x}{a+x (2 b+c x)}+\frac {(-\text {b1} c+b \text {c1}) \arctan \left (\frac {b+c x}{\sqrt {-b^2+a c}}\right )}{\sqrt {-b^2+a c}}}{2 \left (b^2-a c\right )} \]
((-(b*b1) + a*c1 - b1*c*x + b*c1*x)/(a + x*(2*b + c*x)) + ((-(b1*c) + b*c1 )*ArcTan[(b + c*x)/Sqrt[-b^2 + a*c]])/Sqrt[-b^2 + a*c])/(2*(b^2 - a*c))
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1159, 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.
\(\displaystyle \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 1159 |
\(\displaystyle -\frac {(\text {b1} c-b \text {c1}) \int \frac {1}{c x^2+2 b x+a}dx}{2 \left (b^2-a c\right )}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {(\text {b1} c-b \text {c1}) \int \frac {1}{4 \left (b^2-a c\right )-(2 b+2 c x)^2}d(2 b+2 c x)}{b^2-a c}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(\text {b1} c-b \text {c1}) \text {arctanh}\left (\frac {2 b+2 c x}{2 \sqrt {b^2-a c}}\right )}{2 \left (b^2-a c\right )^{3/2}}-\frac {-a \text {c1}+x (\text {b1} c-b \text {c1})+b \text {b1}}{2 \left (b^2-a c\right ) \left (a+2 b x+c x^2\right )}\) |
-1/2*(b*b1 - a*c1 + (b1*c - b*c1)*x)/((b^2 - a*c)*(a + 2*b*x + c*x^2)) + ( (b1*c - b*c1)*ArcTanh[(2*b + 2*c*x)/(2*Sqrt[b^2 - a*c])])/(2*(b^2 - a*c)^( 3/2))
3.2.95.3.1 Defintions of rubi rules used
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[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[((b*d - 2*a*e + (2*c*d - b*e)*x)/((p + 1)*(b^2 - 4*a*c)))*(a + b* x + c*x^2)^(p + 1), x] - Simp[(2*p + 3)*((2*c*d - b*e)/((p + 1)*(b^2 - 4*a* c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] & & LtQ[p, -1] && NeQ[p, -3/2]
Time = 0.46 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16
method | result | size |
default | \(\frac {\left (-2 b \operatorname {c1} +2 \operatorname {b1} c \right ) x +2 b \operatorname {b1} -2 a \operatorname {c1}}{\left (4 a c -4 b^{2}\right ) \left (c \,x^{2}+2 b x +a \right )}+\frac {\left (-2 b \operatorname {c1} +2 \operatorname {b1} c \right ) \arctan \left (\frac {2 c x +2 b}{2 \sqrt {a c -b^{2}}}\right )}{\left (4 a c -4 b^{2}\right ) \sqrt {a c -b^{2}}}\) | \(103\) |
risch | \(\frac {-\frac {\left (b \operatorname {c1} -\operatorname {b1} c \right ) x}{2 \left (a c -b^{2}\right )}-\frac {a \operatorname {c1} -b \operatorname {b1}}{2 \left (a c -b^{2}\right )}}{c \,x^{2}+2 b x +a}+\frac {\ln \left (\left (-c^{2} a +b^{2} c \right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}-a b c +b^{3}\right ) b \operatorname {c1}}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (-c^{2} a +b^{2} c \right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}-a b c +b^{3}\right ) \operatorname {b1} c}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}-\frac {\ln \left (\left (c^{2} a -b^{2} c \right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}+a b c -b^{3}\right ) b \operatorname {c1}}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}+\frac {\ln \left (\left (c^{2} a -b^{2} c \right ) x -\left (-a c +b^{2}\right )^{\frac {3}{2}}+a b c -b^{3}\right ) \operatorname {b1} c}{4 \left (-a c +b^{2}\right )^{\frac {3}{2}}}\) | \(262\) |
((-2*b*c1+2*b1*c)*x+2*b*b1-2*a*c1)/(4*a*c-4*b^2)/(c*x^2+2*b*x+a)+(-2*b*c1+ 2*b1*c)/(4*a*c-4*b^2)/(a*c-b^2)^(1/2)*arctan(1/2*(2*c*x+2*b)/(a*c-b^2)^(1/ 2))
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (81) = 162\).
Time = 0.26 (sec) , antiderivative size = 447, normalized size of antiderivative = 5.02 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx=\left [-\frac {2 \, b^{3} b_{1} - 2 \, a b b_{1} c - {\left (a b_{1} c - a b c_{1} + {\left (b_{1} c^{2} - b c c_{1}\right )} x^{2} + 2 \, {\left (b b_{1} c - b^{2} c_{1}\right )} x\right )} \sqrt {b^{2} - a c} \log \left (\frac {c^{2} x^{2} + 2 \, b c x + 2 \, b^{2} - a c + 2 \, \sqrt {b^{2} - a c} {\left (c x + b\right )}}{c x^{2} + 2 \, b x + a}\right ) - 2 \, {\left (a b^{2} - a^{2} c\right )} c_{1} + 2 \, {\left (b^{2} b_{1} c - a b_{1} c^{2} - {\left (b^{3} - a b c\right )} c_{1}\right )} x}{4 \, {\left (a b^{4} - 2 \, a^{2} b^{2} c + a^{3} c^{2} + {\left (b^{4} c - 2 \, a b^{2} c^{2} + a^{2} c^{3}\right )} x^{2} + 2 \, {\left (b^{5} - 2 \, a b^{3} c + a^{2} b c^{2}\right )} x\right )}}, -\frac {b^{3} b_{1} - a b b_{1} c - {\left (a b_{1} c - a b c_{1} + {\left (b_{1} c^{2} - b c c_{1}\right )} x^{2} + 2 \, {\left (b b_{1} c - b^{2} c_{1}\right )} x\right )} \sqrt {-b^{2} + a c} \arctan \left (-\frac {\sqrt {-b^{2} + a c} {\left (c x + b\right )}}{b^{2} - a c}\right ) - {\left (a b^{2} - a^{2} c\right )} c_{1} + {\left (b^{2} b_{1} c - a b_{1} c^{2} - {\left (b^{3} - a b c\right )} c_{1}\right )} x}{2 \, {\left (a b^{4} - 2 \, a^{2} b^{2} c + a^{3} c^{2} + {\left (b^{4} c - 2 \, a b^{2} c^{2} + a^{2} c^{3}\right )} x^{2} + 2 \, {\left (b^{5} - 2 \, a b^{3} c + a^{2} b c^{2}\right )} x\right )}}\right ] \]
[-1/4*(2*b^3*b1 - 2*a*b*b1*c - (a*b1*c - a*b*c1 + (b1*c^2 - b*c*c1)*x^2 + 2*(b*b1*c - b^2*c1)*x)*sqrt(b^2 - a*c)*log((c^2*x^2 + 2*b*c*x + 2*b^2 - a* c + 2*sqrt(b^2 - a*c)*(c*x + b))/(c*x^2 + 2*b*x + a)) - 2*(a*b^2 - a^2*c)* c1 + 2*(b^2*b1*c - a*b1*c^2 - (b^3 - a*b*c)*c1)*x)/(a*b^4 - 2*a^2*b^2*c + a^3*c^2 + (b^4*c - 2*a*b^2*c^2 + a^2*c^3)*x^2 + 2*(b^5 - 2*a*b^3*c + a^2*b *c^2)*x), -1/2*(b^3*b1 - a*b*b1*c - (a*b1*c - a*b*c1 + (b1*c^2 - b*c*c1)*x ^2 + 2*(b*b1*c - b^2*c1)*x)*sqrt(-b^2 + a*c)*arctan(-sqrt(-b^2 + a*c)*(c*x + b)/(b^2 - a*c)) - (a*b^2 - a^2*c)*c1 + (b^2*b1*c - a*b1*c^2 - (b^3 - a* b*c)*c1)*x)/(a*b^4 - 2*a^2*b^2*c + a^3*c^2 + (b^4*c - 2*a*b^2*c^2 + a^2*c^ 3)*x^2 + 2*(b^5 - 2*a*b^3*c + a^2*b*c^2)*x)]
Leaf count of result is larger than twice the leaf count of optimal. 323 vs. \(2 (75) = 150\).
Time = 0.52 (sec) , antiderivative size = 323, normalized size of antiderivative = 3.63 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx=\frac {\sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) \log {\left (x + \frac {- a^{2} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) + 2 a b^{2} c \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) - b^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) + b^{2} c_{1} - b b_{1} c}{b c c_{1} - b_{1} c^{2}} \right )}}{4} - \frac {\sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) \log {\left (x + \frac {a^{2} c^{2} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) - 2 a b^{2} c \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) + b^{4} \sqrt {- \frac {1}{\left (a c - b^{2}\right )^{3}}} \left (b c_{1} - b_{1} c\right ) + b^{2} c_{1} - b b_{1} c}{b c c_{1} - b_{1} c^{2}} \right )}}{4} + \frac {- a c_{1} + b b_{1} + x \left (- b c_{1} + b_{1} c\right )}{2 a^{2} c - 2 a b^{2} + x^{2} \cdot \left (2 a c^{2} - 2 b^{2} c\right ) + x \left (4 a b c - 4 b^{3}\right )} \]
sqrt(-1/(a*c - b**2)**3)*(b*c1 - b1*c)*log(x + (-a**2*c**2*sqrt(-1/(a*c - b**2)**3)*(b*c1 - b1*c) + 2*a*b**2*c*sqrt(-1/(a*c - b**2)**3)*(b*c1 - b1*c ) - b**4*sqrt(-1/(a*c - b**2)**3)*(b*c1 - b1*c) + b**2*c1 - b*b1*c)/(b*c*c 1 - b1*c**2))/4 - sqrt(-1/(a*c - b**2)**3)*(b*c1 - b1*c)*log(x + (a**2*c** 2*sqrt(-1/(a*c - b**2)**3)*(b*c1 - b1*c) - 2*a*b**2*c*sqrt(-1/(a*c - b**2) **3)*(b*c1 - b1*c) + b**4*sqrt(-1/(a*c - b**2)**3)*(b*c1 - b1*c) + b**2*c1 - b*b1*c)/(b*c*c1 - b1*c**2))/4 + (-a*c1 + b*b1 + x*(-b*c1 + b1*c))/(2*a* *2*c - 2*a*b**2 + x**2*(2*a*c**2 - 2*b**2*c) + x*(4*a*b*c - 4*b**3))
Exception generated. \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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*b^2-4*a*c>0)', see `assume?` f or more de
Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx=-\frac {{\left (b_{1} c - b c_{1}\right )} \arctan \left (\frac {c x + b}{\sqrt {-b^{2} + a c}}\right )}{2 \, {\left (b^{2} - a c\right )} \sqrt {-b^{2} + a c}} - \frac {b_{1} c x - b c_{1} x + b b_{1} - a c_{1}}{2 \, {\left (c x^{2} + 2 \, b x + a\right )} {\left (b^{2} - a c\right )}} \]
-1/2*(b1*c - b*c1)*arctan((c*x + b)/sqrt(-b^2 + a*c))/((b^2 - a*c)*sqrt(-b ^2 + a*c)) - 1/2*(b1*c*x - b*c1*x + b*b1 - a*c1)/((c*x^2 + 2*b*x + a)*(b^2 - a*c))
Time = 0.30 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.79 \[ \int \frac {\text {b1}+\text {c1} x}{\left (a+2 b x+c x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {2\,\left (\frac {\left (4\,b^3-4\,a\,b\,c\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{8\,{\left (a\,c-b^2\right )}^{5/2}}-\frac {c\,x\,\left (b\,c_{1}-b_{1}\,c\right )}{2\,{\left (a\,c-b^2\right )}^{3/2}}\right )\,\left (a\,c-b^2\right )}{b\,c_{1}-b_{1}\,c}\right )\,\left (b\,c_{1}-b_{1}\,c\right )}{2\,{\left (a\,c-b^2\right )}^{3/2}}-\frac {\frac {a\,c_{1}-b\,b_{1}}{2\,\left (a\,c-b^2\right )}+\frac {x\,\left (b\,c_{1}-b_{1}\,c\right )}{2\,\left (a\,c-b^2\right )}}{c\,x^2+2\,b\,x+a} \]