Integrand size = 18, antiderivative size = 87 \[ \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx=-\frac {3 d}{2 a^2 x}+\frac {d+e x}{2 a x \left (a+c x^2\right )}-\frac {3 \sqrt {c} d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{5/2}}+\frac {e \log (x)}{a^2}-\frac {e \log \left (a+c x^2\right )}{2 a^2} \]
-3/2*d/a^2/x+1/2*(e*x+d)/a/x/(c*x^2+a)+e*ln(x)/a^2-1/2*e*ln(c*x^2+a)/a^2-3 /2*d*arctan(x*c^(1/2)/a^(1/2))*c^(1/2)/a^(5/2)
Time = 0.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.93 \[ \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx=-\frac {3 \sqrt {c} d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {\frac {2 a d-a e x+3 c d x^2}{a x+c x^3}-2 e \log (x)+e \log \left (a+c x^2\right )}{2 a^2} \]
(-3*Sqrt[c]*d*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(5/2)) - ((2*a*d - a*e*x + 3*c*d*x^2)/(a*x + c*x^3) - 2*e*Log[x] + e*Log[a + c*x^2])/(2*a^2)
Time = 0.32 (sec) , antiderivative size = 91, 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, 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.
\(\displaystyle \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {a e-c d x}{2 a^2 \left (a+c x^2\right )}-\frac {\int -\frac {-\frac {c d x^2}{a}+2 e x+2 d}{x^2 \left (c x^2+a\right )}dx}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {-\frac {c d x^2}{a}+2 e x+2 d}{x^2 \left (c x^2+a\right )}dx}{2 a}+\frac {a e-c d x}{2 a^2 \left (a+c x^2\right )}\) |
\(\Big \downarrow \) 2333 |
\(\displaystyle \frac {\int \left (\frac {2 d}{a x^2}+\frac {2 e}{a x}+\frac {-3 c d-2 c e x}{a \left (c x^2+a\right )}\right )dx}{2 a}+\frac {a e-c d x}{2 a^2 \left (a+c x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {3 \sqrt {c} d \arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{a^{3/2}}-\frac {e \log \left (a+c x^2\right )}{a}-\frac {2 d}{a x}+\frac {2 e \log (x)}{a}}{2 a}+\frac {a e-c d x}{2 a^2 \left (a+c x^2\right )}\) |
(a*e - c*d*x)/(2*a^2*(a + c*x^2)) + ((-2*d)/(a*x) - (3*Sqrt[c]*d*ArcTan[(S qrt[c]*x)/Sqrt[a]])/a^(3/2) + (2*e*Log[x])/a - (e*Log[a + c*x^2])/a)/(2*a)
3.3.95.3.1 Defintions of rubi rules used
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]
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]
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {c \left (\frac {\frac {d x}{2}-\frac {a e}{2 c}}{c \,x^{2}+a}+\frac {e \ln \left (c \,x^{2}+a \right )}{2 c}+\frac {3 d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}}\right )}{a^{2}}-\frac {d}{a^{2} x}+\frac {e \ln \left (x \right )}{a^{2}}\) | \(78\) |
risch | \(\frac {-\frac {3 c d \,x^{2}}{2 a^{2}}+\frac {e x}{2 a}-\frac {d}{a}}{x \left (c \,x^{2}+a \right )}+\frac {e \ln \left (x \right )}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{5} \textit {\_Z}^{2}+4 a^{3} e \textit {\_Z} +4 e^{2} a +9 c \,d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\textit {\_R}^{2} a^{5}+2 a^{3} e \textit {\_R} +6 c \,d^{2}\right ) x +a^{3} d \textit {\_R} -4 a d e \right )\right )}{4}\) | \(117\) |
-c/a^2*((1/2*d*x-1/2*a/c*e)/(c*x^2+a)+1/2*e*ln(c*x^2+a)/c+3/2*d/(a*c)^(1/2 )*arctan(c*x/(a*c)^(1/2)))-d/a^2/x+e*ln(x)/a^2
Time = 0.37 (sec) , antiderivative size = 225, normalized size of antiderivative = 2.59 \[ \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx=\left [-\frac {6 \, c d x^{2} - 2 \, a e x - 3 \, {\left (c d x^{3} + a d x\right )} \sqrt {-\frac {c}{a}} \log \left (\frac {c x^{2} - 2 \, a x \sqrt {-\frac {c}{a}} - a}{c x^{2} + a}\right ) + 4 \, a d + 2 \, {\left (c e x^{3} + a e x\right )} \log \left (c x^{2} + a\right ) - 4 \, {\left (c e x^{3} + a e x\right )} \log \left (x\right )}{4 \, {\left (a^{2} c x^{3} + a^{3} x\right )}}, -\frac {3 \, c d x^{2} - a e x + 3 \, {\left (c d x^{3} + a d x\right )} \sqrt {\frac {c}{a}} \arctan \left (x \sqrt {\frac {c}{a}}\right ) + 2 \, a d + {\left (c e x^{3} + a e x\right )} \log \left (c x^{2} + a\right ) - 2 \, {\left (c e x^{3} + a e x\right )} \log \left (x\right )}{2 \, {\left (a^{2} c x^{3} + a^{3} x\right )}}\right ] \]
[-1/4*(6*c*d*x^2 - 2*a*e*x - 3*(c*d*x^3 + a*d*x)*sqrt(-c/a)*log((c*x^2 - 2 *a*x*sqrt(-c/a) - a)/(c*x^2 + a)) + 4*a*d + 2*(c*e*x^3 + a*e*x)*log(c*x^2 + a) - 4*(c*e*x^3 + a*e*x)*log(x))/(a^2*c*x^3 + a^3*x), -1/2*(3*c*d*x^2 - a*e*x + 3*(c*d*x^3 + a*d*x)*sqrt(c/a)*arctan(x*sqrt(c/a)) + 2*a*d + (c*e*x ^3 + a*e*x)*log(c*x^2 + a) - 2*(c*e*x^3 + a*e*x)*log(x))/(a^2*c*x^3 + a^3* x)]
Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (80) = 160\).
Time = 1.16 (sec) , antiderivative size = 389, normalized size of antiderivative = 4.47 \[ \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx=\left (- \frac {e}{2 a^{2}} - \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) \log {\left (x + \frac {32 a^{6} e \left (- \frac {e}{2 a^{2}} - \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right )^{2} - 16 a^{4} e^{2} \left (- \frac {e}{2 a^{2}} - \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) - 12 a^{3} c d^{2} \left (- \frac {e}{2 a^{2}} - \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) - 16 a^{2} e^{3} + 12 a c d^{2} e}{36 a c d e^{2} + 9 c^{2} d^{3}} \right )} + \left (- \frac {e}{2 a^{2}} + \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) \log {\left (x + \frac {32 a^{6} e \left (- \frac {e}{2 a^{2}} + \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right )^{2} - 16 a^{4} e^{2} \left (- \frac {e}{2 a^{2}} + \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) - 12 a^{3} c d^{2} \left (- \frac {e}{2 a^{2}} + \frac {3 d \sqrt {- a^{5} c}}{4 a^{5}}\right ) - 16 a^{2} e^{3} + 12 a c d^{2} e}{36 a c d e^{2} + 9 c^{2} d^{3}} \right )} + \frac {- 2 a d + a e x - 3 c d x^{2}}{2 a^{3} x + 2 a^{2} c x^{3}} + \frac {e \log {\left (x \right )}}{a^{2}} \]
(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a**5))*log(x + (32*a**6*e*(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a**5))**2 - 16*a**4*e**2*(-e/(2*a**2) - 3*d*sqrt(- a**5*c)/(4*a**5)) - 12*a**3*c*d**2*(-e/(2*a**2) - 3*d*sqrt(-a**5*c)/(4*a** 5)) - 16*a**2*e**3 + 12*a*c*d**2*e)/(36*a*c*d*e**2 + 9*c**2*d**3)) + (-e/( 2*a**2) + 3*d*sqrt(-a**5*c)/(4*a**5))*log(x + (32*a**6*e*(-e/(2*a**2) + 3* d*sqrt(-a**5*c)/(4*a**5))**2 - 16*a**4*e**2*(-e/(2*a**2) + 3*d*sqrt(-a**5* c)/(4*a**5)) - 12*a**3*c*d**2*(-e/(2*a**2) + 3*d*sqrt(-a**5*c)/(4*a**5)) - 16*a**2*e**3 + 12*a*c*d**2*e)/(36*a*c*d*e**2 + 9*c**2*d**3)) + (-2*a*d + a*e*x - 3*c*d*x**2)/(2*a**3*x + 2*a**2*c*x**3) + e*log(x)/a**2
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx=-\frac {3 \, c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} - \frac {3 \, c d x^{2} - a e x + 2 \, a d}{2 \, {\left (a^{2} c x^{3} + a^{3} x\right )}} - \frac {e \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac {e \log \left (x\right )}{a^{2}} \]
-3/2*c*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2) - 1/2*(3*c*d*x^2 - a*e*x + 2*a*d)/(a^2*c*x^3 + a^3*x) - 1/2*e*log(c*x^2 + a)/a^2 + e*log(x)/a^2
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx=-\frac {3 \, c d \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a^{2}} - \frac {e \log \left (c x^{2} + a\right )}{2 \, a^{2}} + \frac {e \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {3 \, c d x^{2} - a e x + 2 \, a d}{2 \, {\left (c x^{3} + a x\right )} a^{2}} \]
-3/2*c*d*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2) - 1/2*e*log(c*x^2 + a)/a^2 + e*log(abs(x))/a^2 - 1/2*(3*c*d*x^2 - a*e*x + 2*a*d)/((c*x^3 + a*x)*a^2)
Time = 10.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.90 \[ \int \frac {d+e x}{x^2 \left (a+c x^2\right )^2} \, dx=\frac {e\,\ln \left (x\right )}{a^2}-\frac {\ln \left (2\,a^3\,e+d\,\sqrt {-a^5\,c}-2\,e\,x\,\sqrt {-a^5\,c}+a^2\,c\,d\,x\right )\,\left (2\,a^3\,e+3\,d\,\sqrt {-a^5\,c}\right )}{4\,a^5}-\frac {\ln \left (2\,a^3\,e-d\,\sqrt {-a^5\,c}+2\,e\,x\,\sqrt {-a^5\,c}+a^2\,c\,d\,x\right )\,\left (2\,a^3\,e-3\,d\,\sqrt {-a^5\,c}\right )}{4\,a^5}-\frac {\frac {d}{a}-\frac {e\,x}{2\,a}+\frac {3\,c\,d\,x^2}{2\,a^2}}{c\,x^3+a\,x} \]
(e*log(x))/a^2 - (log(2*a^3*e + d*(-a^5*c)^(1/2) - 2*e*x*(-a^5*c)^(1/2) + a^2*c*d*x)*(2*a^3*e + 3*d*(-a^5*c)^(1/2)))/(4*a^5) - (log(2*a^3*e - d*(-a^ 5*c)^(1/2) + 2*e*x*(-a^5*c)^(1/2) + a^2*c*d*x)*(2*a^3*e - 3*d*(-a^5*c)^(1/ 2)))/(4*a^5) - (d/a - (e*x)/(2*a) + (3*c*d*x^2)/(2*a^2))/(a*x + c*x^3)