Integrand size = 18, antiderivative size = 73 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx=\frac {c+d x}{2 a \left (a+b x^2\right )}+\frac {d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {c \log (x)}{a^2}-\frac {c \log \left (a+b x^2\right )}{2 a^2} \] Output:
1/2*(d*x+c)/a/(b*x^2+a)+1/2*d*arctan(b^(1/2)*x/a^(1/2))/a^(3/2)/b^(1/2)+c* ln(x)/a^2-1/2*c*ln(b*x^2+a)/a^2
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx=\frac {\frac {a (c+d x)}{a+b x^2}+\frac {\sqrt {a} d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+2 c \log (x)-c \log \left (a+b x^2\right )}{2 a^2} \] Input:
Integrate[(c + d*x)/(x*(a + b*x^2)^2),x]
Output:
((a*(c + d*x))/(a + b*x^2) + (Sqrt[a]*d*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[ b] + 2*c*Log[x] - c*Log[a + b*x^2])/(2*a^2)
Time = 0.43 (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.
\(\displaystyle \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx\) |
\(\Big \downarrow \) 532 |
\(\displaystyle \frac {c+d x}{2 a \left (a+b x^2\right )}-\frac {\int -\frac {2 c+d x}{x \left (b x^2+a\right )}dx}{2 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {2 c+d x}{x \left (b x^2+a\right )}dx}{2 a}+\frac {c+d x}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \frac {\int \left (\frac {2 c}{a x}+\frac {a d-2 b c x}{a \left (b x^2+a\right )}\right )dx}{2 a}+\frac {c+d x}{2 a \left (a+b x^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {d \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}-\frac {c \log \left (a+b x^2\right )}{a}+\frac {2 c \log (x)}{a}}{2 a}+\frac {c+d x}{2 a \left (a+b x^2\right )}\) |
Input:
Int[(c + d*x)/(x*(a + b*x^2)^2),x]
Output:
(c + d*x)/(2*a*(a + b*x^2)) + ((d*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[a]*Sq rt[b]) + (2*c*Log[x])/a - (c*Log[a + b*x^2])/a)/(2*a)
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 = 0.21 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.86
method | result | size |
default | \(\frac {c \ln \left (x \right )}{a^{2}}+\frac {\frac {\frac {1}{2} a d x +\frac {1}{2} a c}{b \,x^{2}+a}-\frac {c \ln \left (b \,x^{2}+a \right )}{2}+\frac {a d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}}{a^{2}}\) | \(63\) |
risch | \(\frac {\frac {d x}{2 a}+\frac {c}{2 a}}{b \,x^{2}+a}+\frac {\ln \left (\left (-d a b -6 \sqrt {-a b}\, b c \right ) x -\sqrt {-a b}\, a d +6 a b c \right ) \sqrt {-a b}\, d}{4 b \,a^{2}}-\frac {\ln \left (\left (-d a b -6 \sqrt {-a b}\, b c \right ) x -\sqrt {-a b}\, a d +6 a b c \right ) c}{2 a^{2}}-\frac {\ln \left (\left (-d a b +6 \sqrt {-a b}\, b c \right ) x +\sqrt {-a b}\, a d +6 a b c \right ) \sqrt {-a b}\, d}{4 b \,a^{2}}-\frac {\ln \left (\left (-d a b +6 \sqrt {-a b}\, b c \right ) x +\sqrt {-a b}\, a d +6 a b c \right ) c}{2 a^{2}}+\frac {c \ln \left (x \right )}{a^{2}}\) | \(213\) |
Input:
int((d*x+c)/x/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
c*ln(x)/a^2+1/a^2*((1/2*a*d*x+1/2*a*c)/(b*x^2+a)-1/2*c*ln(b*x^2+a)+1/2*a*d /(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2)))
Time = 0.15 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.97 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx=\left [\frac {2 \, a b d x + 2 \, a b c - {\left (b d x^{2} + a d\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) - 2 \, {\left (b^{2} c x^{2} + a b c\right )} \log \left (b x^{2} + a\right ) + 4 \, {\left (b^{2} c x^{2} + a b c\right )} \log \left (x\right )}{4 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}, \frac {a b d x + a b c + {\left (b d x^{2} + a d\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) - {\left (b^{2} c x^{2} + a b c\right )} \log \left (b x^{2} + a\right ) + 2 \, {\left (b^{2} c x^{2} + a b c\right )} \log \left (x\right )}{2 \, {\left (a^{2} b^{2} x^{2} + a^{3} b\right )}}\right ] \] Input:
integrate((d*x+c)/x/(b*x^2+a)^2,x, algorithm="fricas")
Output:
[1/4*(2*a*b*d*x + 2*a*b*c - (b*d*x^2 + a*d)*sqrt(-a*b)*log((b*x^2 - 2*sqrt (-a*b)*x - a)/(b*x^2 + a)) - 2*(b^2*c*x^2 + a*b*c)*log(b*x^2 + a) + 4*(b^2 *c*x^2 + a*b*c)*log(x))/(a^2*b^2*x^2 + a^3*b), 1/2*(a*b*d*x + a*b*c + (b*d *x^2 + a*d)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) - (b^2*c*x^2 + a*b*c)*log(b*x^ 2 + a) + 2*(b^2*c*x^2 + a*b*c)*log(x))/(a^2*b^2*x^2 + a^3*b)]
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (66) = 132\).
Time = 0.82 (sec) , antiderivative size = 359, normalized size of antiderivative = 4.92 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx=\left (- \frac {c}{2 a^{2}} - \frac {d \sqrt {- a^{5} b}}{4 a^{4} b}\right ) \log {\left (x + \frac {- 96 a^{4} b c \left (- \frac {c}{2 a^{2}} - \frac {d \sqrt {- a^{5} b}}{4 a^{4} b}\right )^{2} + 4 a^{3} d^{2} \left (- \frac {c}{2 a^{2}} - \frac {d \sqrt {- a^{5} b}}{4 a^{4} b}\right ) + 48 a^{2} b c^{2} \left (- \frac {c}{2 a^{2}} - \frac {d \sqrt {- a^{5} b}}{4 a^{4} b}\right ) - 4 a c d^{2} + 48 b c^{3}}{a d^{3} + 36 b c^{2} d} \right )} + \left (- \frac {c}{2 a^{2}} + \frac {d \sqrt {- a^{5} b}}{4 a^{4} b}\right ) \log {\left (x + \frac {- 96 a^{4} b c \left (- \frac {c}{2 a^{2}} + \frac {d \sqrt {- a^{5} b}}{4 a^{4} b}\right )^{2} + 4 a^{3} d^{2} \left (- \frac {c}{2 a^{2}} + \frac {d \sqrt {- a^{5} b}}{4 a^{4} b}\right ) + 48 a^{2} b c^{2} \left (- \frac {c}{2 a^{2}} + \frac {d \sqrt {- a^{5} b}}{4 a^{4} b}\right ) - 4 a c d^{2} + 48 b c^{3}}{a d^{3} + 36 b c^{2} d} \right )} + \frac {c + d x}{2 a^{2} + 2 a b x^{2}} + \frac {c \log {\left (x \right )}}{a^{2}} \] Input:
integrate((d*x+c)/x/(b*x**2+a)**2,x)
Output:
(-c/(2*a**2) - d*sqrt(-a**5*b)/(4*a**4*b))*log(x + (-96*a**4*b*c*(-c/(2*a* *2) - d*sqrt(-a**5*b)/(4*a**4*b))**2 + 4*a**3*d**2*(-c/(2*a**2) - d*sqrt(- a**5*b)/(4*a**4*b)) + 48*a**2*b*c**2*(-c/(2*a**2) - d*sqrt(-a**5*b)/(4*a** 4*b)) - 4*a*c*d**2 + 48*b*c**3)/(a*d**3 + 36*b*c**2*d)) + (-c/(2*a**2) + d *sqrt(-a**5*b)/(4*a**4*b))*log(x + (-96*a**4*b*c*(-c/(2*a**2) + d*sqrt(-a* *5*b)/(4*a**4*b))**2 + 4*a**3*d**2*(-c/(2*a**2) + d*sqrt(-a**5*b)/(4*a**4* b)) + 48*a**2*b*c**2*(-c/(2*a**2) + d*sqrt(-a**5*b)/(4*a**4*b)) - 4*a*c*d* *2 + 48*b*c**3)/(a*d**3 + 36*b*c**2*d)) + (c + d*x)/(2*a**2 + 2*a*b*x**2) + c*log(x)/a**2
Time = 0.11 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.84 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx=\frac {d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} + \frac {d x + c}{2 \, {\left (a b x^{2} + a^{2}\right )}} - \frac {c \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {c \log \left (x\right )}{a^{2}} \] Input:
integrate((d*x+c)/x/(b*x^2+a)^2,x, algorithm="maxima")
Output:
1/2*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) + 1/2*(d*x + c)/(a*b*x^2 + a^2) - 1/2*c*log(b*x^2 + a)/a^2 + c*log(x)/a^2
Time = 0.12 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx=\frac {d \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a} - \frac {c \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac {c \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {a d x + a c}{2 \, {\left (b x^{2} + a\right )} a^{2}} \] Input:
integrate((d*x+c)/x/(b*x^2+a)^2,x, algorithm="giac")
Output:
1/2*d*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a) - 1/2*c*log(b*x^2 + a)/a^2 + c*l og(abs(x))/a^2 + 1/2*(a*d*x + a*c)/((b*x^2 + a)*a^2)
Time = 6.96 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.26 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx=\frac {\frac {c}{2\,a}+\frac {d\,x}{2\,a}}{b\,x^2+a}+\frac {c\,\ln \left (x\right )}{a^2}+\frac {\ln \left (a\,d\,\sqrt {-a^5\,b}-6\,a^3\,b\,c+a^3\,b\,d\,x+6\,b\,c\,x\,\sqrt {-a^5\,b}\right )\,\left (d\,\sqrt {-a^5\,b}-2\,a^2\,b\,c\right )}{4\,a^4\,b}-\frac {\ln \left (a\,d\,\sqrt {-a^5\,b}+6\,a^3\,b\,c-a^3\,b\,d\,x+6\,b\,c\,x\,\sqrt {-a^5\,b}\right )\,\left (d\,\sqrt {-a^5\,b}+2\,a^2\,b\,c\right )}{4\,a^4\,b} \] Input:
int((c + d*x)/(x*(a + b*x^2)^2),x)
Output:
(c/(2*a) + (d*x)/(2*a))/(a + b*x^2) + (c*log(x))/a^2 + (log(a*d*(-a^5*b)^( 1/2) - 6*a^3*b*c + a^3*b*d*x + 6*b*c*x*(-a^5*b)^(1/2))*(d*(-a^5*b)^(1/2) - 2*a^2*b*c))/(4*a^4*b) - (log(a*d*(-a^5*b)^(1/2) + 6*a^3*b*c - a^3*b*d*x + 6*b*c*x*(-a^5*b)^(1/2))*(d*(-a^5*b)^(1/2) + 2*a^2*b*c))/(4*a^4*b)
Time = 0.18 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.66 \[ \int \frac {c+d x}{x \left (a+b x^2\right )^2} \, dx=\frac {\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a d +\sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b d \,x^{2}-\mathrm {log}\left (b \,x^{2}+a \right ) a b c -\mathrm {log}\left (b \,x^{2}+a \right ) b^{2} c \,x^{2}+2 \,\mathrm {log}\left (x \right ) a b c +2 \,\mathrm {log}\left (x \right ) b^{2} c \,x^{2}+a b d x -b^{2} c \,x^{2}}{2 a^{2} b \left (b \,x^{2}+a \right )} \] Input:
int((d*x+c)/x/(b*x^2+a)^2,x)
Output:
(sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*d + sqrt(b)*sqrt(a)*atan( (b*x)/(sqrt(b)*sqrt(a)))*b*d*x**2 - log(a + b*x**2)*a*b*c - log(a + b*x**2 )*b**2*c*x**2 + 2*log(x)*a*b*c + 2*log(x)*b**2*c*x**2 + a*b*d*x - b**2*c*x **2)/(2*a**2*b*(a + b*x**2))