Integrand size = 18, antiderivative size = 73 \[ \int \frac {d+e x}{x \left (a+c x^2\right )^2} \, dx=\frac {d+e x}{2 a \left (a+c x^2\right )}+\frac {e \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {c}}+\frac {d \log (x)}{a^2}-\frac {d \log \left (a+c x^2\right )}{2 a^2} \]
1/2*(e*x+d)/a/(c*x^2+a)+d*ln(x)/a^2-1/2*d*ln(c*x^2+a)/a^2+1/2*e*arctan(x*c ^(1/2)/a^(1/2))/a^(3/2)/c^(1/2)
Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x}{x \left (a+c x^2\right )^2} \, dx=\frac {\frac {a (d+e x)}{a+c x^2}+\frac {\sqrt {a} e \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {c}}+2 d \log (x)-d \log \left (a+c x^2\right )}{2 a^2} \]
((a*(d + e*x))/(a + c*x^2) + (Sqrt[a]*e*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/Sqrt[ c] + 2*d*Log[x] - d*Log[a + c*x^2])/(2*a^2)
Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {532, 25, 523, 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.
\(\displaystyle \int \frac {d+e x}{x \left (a+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {d+e x}{2 a \left (a+c x^2\right )}-\frac {\int -\frac {2 d+e x}{x \left (c x^2+a\right )}dx}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {2 d+e x}{x \left (c x^2+a\right )}dx}{2 a}+\frac {d+e x}{2 a \left (a+c x^2\right )}\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \frac {\int \left (\frac {2 d}{a x}+\frac {a e-2 c d x}{a \left (c x^2+a\right )}\right )dx}{2 a}+\frac {d+e x}{2 a \left (a+c x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {e \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {d \log \left (a+c x^2\right )}{a}+\frac {2 d \log (x)}{a}}{2 a}+\frac {d+e x}{2 a \left (a+c x^2\right )}\) |
(d + e*x)/(2*a*(a + c*x^2)) + ((e*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*Sq rt[c]) + (2*d*Log[x])/a - (d*Log[a + c*x^2])/a)/(2*a)
3.3.94.3.1 Defintions of rubi rules used
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {\frac {\frac {1}{2} a e x +\frac {1}{2} a d}{c \,x^{2}+a}-\frac {d \ln \left (c \,x^{2}+a \right )}{2}+\frac {a e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}}}{a^{2}}+\frac {d \ln \left (x \right )}{a^{2}}\) | \(63\) |
risch | \(\frac {\frac {e x}{2 a}+\frac {d}{2 a}}{c \,x^{2}+a}+\frac {\ln \left (\left (-a c e -6 \sqrt {-a c}\, c d \right ) x -\sqrt {-a c}\, a e +6 a c d \right ) e \sqrt {-a c}}{4 c \,a^{2}}-\frac {\ln \left (\left (-a c e -6 \sqrt {-a c}\, c d \right ) x -\sqrt {-a c}\, a e +6 a c d \right ) d}{2 a^{2}}-\frac {\ln \left (\left (-a c e +6 \sqrt {-a c}\, c d \right ) x +\sqrt {-a c}\, a e +6 a c d \right ) e \sqrt {-a c}}{4 c \,a^{2}}-\frac {\ln \left (\left (-a c e +6 \sqrt {-a c}\, c d \right ) x +\sqrt {-a c}\, a e +6 a c d \right ) d}{2 a^{2}}+\frac {d \ln \left (x \right )}{a^{2}}\) | \(213\) |
1/a^2*((1/2*a*e*x+1/2*a*d)/(c*x^2+a)-1/2*d*ln(c*x^2+a)+1/2*a*e/(a*c)^(1/2) *arctan(c*x/(a*c)^(1/2)))+d*ln(x)/a^2
Time = 0.67 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.97 \[ \int \frac {d+e x}{x \left (a+c x^2\right )^2} \, dx=\left [\frac {2 \, a c e x + 2 \, a c d - {\left (c e x^{2} + a e\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (c^{2} d x^{2} + a c d\right )} \log \left (c x^{2} + a\right ) + 4 \, {\left (c^{2} d x^{2} + a c d\right )} \log \left (x\right )}{4 \, {\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}, \frac {a c e x + a c d + {\left (c e x^{2} + a e\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (c^{2} d x^{2} + a c d\right )} \log \left (c x^{2} + a\right ) + 2 \, {\left (c^{2} d x^{2} + a c d\right )} \log \left (x\right )}{2 \, {\left (a^{2} c^{2} x^{2} + a^{3} c\right )}}\right ] \]
[1/4*(2*a*c*e*x + 2*a*c*d - (c*e*x^2 + a*e)*sqrt(-a*c)*log((c*x^2 - 2*sqrt (-a*c)*x - a)/(c*x^2 + a)) - 2*(c^2*d*x^2 + a*c*d)*log(c*x^2 + a) + 4*(c^2 *d*x^2 + a*c*d)*log(x))/(a^2*c^2*x^2 + a^3*c), 1/2*(a*c*e*x + a*c*d + (c*e *x^2 + a*e)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) - (c^2*d*x^2 + a*c*d)*log(c*x^ 2 + a) + 2*(c^2*d*x^2 + a*c*d)*log(x))/(a^2*c^2*x^2 + a^3*c)]
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (66) = 132\).
Time = 1.07 (sec) , antiderivative size = 359, normalized size of antiderivative = 4.92 \[ \int \frac {d+e x}{x \left (a+c x^2\right )^2} \, dx=\left (- \frac {d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{4 a^{4} c}\right ) \log {\left (x + \frac {- 96 a^{4} c d \left (- \frac {d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{4 a^{4} c}\right )^{2} + 4 a^{3} e^{2} \left (- \frac {d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{4 a^{4} c}\right ) + 48 a^{2} c d^{2} \left (- \frac {d}{2 a^{2}} - \frac {e \sqrt {- a^{5} c}}{4 a^{4} c}\right ) - 4 a d e^{2} + 48 c d^{3}}{a e^{3} + 36 c d^{2} e} \right )} + \left (- \frac {d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{4 a^{4} c}\right ) \log {\left (x + \frac {- 96 a^{4} c d \left (- \frac {d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{4 a^{4} c}\right )^{2} + 4 a^{3} e^{2} \left (- \frac {d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{4 a^{4} c}\right ) + 48 a^{2} c d^{2} \left (- \frac {d}{2 a^{2}} + \frac {e \sqrt {- a^{5} c}}{4 a^{4} c}\right ) - 4 a d e^{2} + 48 c d^{3}}{a e^{3} + 36 c d^{2} e} \right )} + \frac {d + e x}{2 a^{2} + 2 a c x^{2}} + \frac {d \log {\left (x \right )}}{a^{2}} \]
(-d/(2*a**2) - e*sqrt(-a**5*c)/(4*a**4*c))*log(x + (-96*a**4*c*d*(-d/(2*a* *2) - e*sqrt(-a**5*c)/(4*a**4*c))**2 + 4*a**3*e**2*(-d/(2*a**2) - e*sqrt(- a**5*c)/(4*a**4*c)) + 48*a**2*c*d**2*(-d/(2*a**2) - e*sqrt(-a**5*c)/(4*a** 4*c)) - 4*a*d*e**2 + 48*c*d**3)/(a*e**3 + 36*c*d**2*e)) + (-d/(2*a**2) + e *sqrt(-a**5*c)/(4*a**4*c))*log(x + (-96*a**4*c*d*(-d/(2*a**2) + e*sqrt(-a* *5*c)/(4*a**4*c))**2 + 4*a**3*e**2*(-d/(2*a**2) + e*sqrt(-a**5*c)/(4*a**4* c)) + 48*a**2*c*d**2*(-d/(2*a**2) + e*sqrt(-a**5*c)/(4*a**4*c)) - 4*a*d*e* *2 + 48*c*d**3)/(a*e**3 + 36*c*d**2*e)) + (d + e*x)/(2*a**2 + 2*a*c*x**2) + d*log(x)/a**2
Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x}{x \left (a+c x^2\right )^2} \, dx=\frac {e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a} + \frac {e x + d}{2 \, {\left (a c x^{2} + a^{2}\right )}} - \frac {d \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac {d \log \left (x\right )}{a^{2}} \]
1/2*e*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a) + 1/2*(e*x + d)/(a*c*x^2 + a^2) - 1/2*d*log(c*x^2 + a)/a^2 + d*log(x)/a^2
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x}{x \left (a+c x^2\right )^2} \, dx=\frac {e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a} - \frac {d \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac {d \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {a e x + a d}{2 \, {\left (c x^{2} + a\right )} a^{2}} \]
1/2*e*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a) - 1/2*d*log(c*x^2 + a)/a^2 + d*l og(abs(x))/a^2 + 1/2*(a*e*x + a*d)/((c*x^2 + a)*a^2)
Time = 10.41 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.26 \[ \int \frac {d+e x}{x \left (a+c x^2\right )^2} \, dx=\frac {\frac {d}{2\,a}+\frac {e\,x}{2\,a}}{c\,x^2+a}+\frac {d\,\ln \left (x\right )}{a^2}+\frac {\ln \left (a\,e\,\sqrt {-a^5\,c}-6\,a^3\,c\,d+a^3\,c\,e\,x+6\,c\,d\,x\,\sqrt {-a^5\,c}\right )\,\left (e\,\sqrt {-a^5\,c}-2\,a^2\,c\,d\right )}{4\,a^4\,c}-\frac {\ln \left (a\,e\,\sqrt {-a^5\,c}+6\,a^3\,c\,d-a^3\,c\,e\,x+6\,c\,d\,x\,\sqrt {-a^5\,c}\right )\,\left (e\,\sqrt {-a^5\,c}+2\,a^2\,c\,d\right )}{4\,a^4\,c} \]