Integrand size = 20, antiderivative size = 118 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )} \, dx=-\frac {1}{a c x}-\frac {b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \left (b c^2+a d^2\right )}-\frac {d \log (x)}{a c^2}+\frac {d^3 \log (c+d x)}{c^2 \left (b c^2+a d^2\right )}+\frac {b d \log \left (a+b x^2\right )}{2 a \left (b c^2+a d^2\right )} \] Output:
-1/a/c/x-b^(3/2)*c*arctan(b^(1/2)*x/a^(1/2))/a^(3/2)/(a*d^2+b*c^2)-d*ln(x) /a/c^2+d^3*ln(d*x+c)/c^2/(a*d^2+b*c^2)+1/2*b*d*ln(b*x^2+a)/a/(a*d^2+b*c^2)
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )} \, dx=-\frac {1}{a c x}-\frac {b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \left (b c^2+a d^2\right )}-\frac {d \log (x)}{a c^2}+\frac {d^3 \log (c+d x)}{c^2 \left (b c^2+a d^2\right )}+\frac {b d \log \left (a+b x^2\right )}{2 a \left (b c^2+a d^2\right )} \] Input:
Integrate[1/(x^2*(c + d*x)*(a + b*x^2)),x]
Output:
-(1/(a*c*x)) - (b^(3/2)*c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(b*c^2 + a *d^2)) - (d*Log[x])/(a*c^2) + (d^3*Log[c + d*x])/(c^2*(b*c^2 + a*d^2)) + ( b*d*Log[a + b*x^2])/(2*a*(b*c^2 + a*d^2))
Time = 0.47 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {615, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^2\right ) (c+d x)} \, dx\) |
\(\Big \downarrow \) 615 |
\(\displaystyle \int \left (-\frac {b^2 (c-d x)}{a \left (a+b x^2\right ) \left (a d^2+b c^2\right )}+\frac {d^4}{c^2 (c+d x) \left (a d^2+b c^2\right )}-\frac {d}{a c^2 x}+\frac {1}{a c x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^{3/2} c \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{a^{3/2} \left (a d^2+b c^2\right )}+\frac {b d \log \left (a+b x^2\right )}{2 a \left (a d^2+b c^2\right )}+\frac {d^3 \log (c+d x)}{c^2 \left (a d^2+b c^2\right )}-\frac {d \log (x)}{a c^2}-\frac {1}{a c x}\) |
Input:
Int[1/(x^2*(c + d*x)*(a + b*x^2)),x]
Output:
-(1/(a*c*x)) - (b^(3/2)*c*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*(b*c^2 + a *d^2)) - (d*Log[x])/(a*c^2) + (d^3*Log[c + d*x])/(c^2*(b*c^2 + a*d^2)) + ( b*d*Log[a + b*x^2])/(2*a*(b*c^2 + a*d^2))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Time = 0.33 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.86
method | result | size |
default | \(-\frac {1}{a c x}-\frac {d \ln \left (x \right )}{a \,c^{2}}-\frac {b^{2} \left (-\frac {d \ln \left (b \,x^{2}+a \right )}{2 b}+\frac {c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) a}+\frac {d^{3} \ln \left (d x +c \right )}{c^{2} \left (a \,d^{2}+b \,c^{2}\right )}\) | \(102\) |
risch | \(-\frac {1}{a c x}+\frac {d^{3} \ln \left (-d x -c \right )}{c^{2} \left (a \,d^{2}+b \,c^{2}\right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{4} d^{2}+a^{3} b \,c^{2}\right ) \textit {\_Z}^{2}-2 a^{2} b d \textit {\_Z} +b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{5} c^{2} d^{4}-3 a^{4} b \,c^{4} d^{2}-3 a^{3} b^{2} c^{6}\right ) \textit {\_R}^{3}+\left (12 a^{3} b \,c^{2} d^{3}+a^{2} b^{2} c^{4} d \right ) \textit {\_R}^{2}+\left (-4 a^{2} b \,d^{4}-4 a \,b^{2} c^{2} d^{2}-2 b^{3} c^{4}\right ) \textit {\_R} +4 b^{2} d^{3}\right ) x +\left (-2 a^{5} c^{3} d^{3}+2 a^{4} b \,c^{5} d \right ) \textit {\_R}^{3}+\left (-4 a^{4} c \,d^{4}+4 a^{3} b \,c^{3} d^{2}-a^{2} b^{2} c^{5}\right ) \textit {\_R}^{2}-2 a \,b^{2} c^{3} d \textit {\_R} \right )\right )}{2}-\frac {d \ln \left (x \right )}{a \,c^{2}}\) | \(279\) |
Input:
int(1/x^2/(d*x+c)/(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
-1/a/c/x-d*ln(x)/a/c^2-b^2/(a*d^2+b*c^2)/a*(-1/2*d*ln(b*x^2+a)/b+c/(a*b)^( 1/2)*arctan(b*x/(a*b)^(1/2)))+d^3*ln(d*x+c)/c^2/(a*d^2+b*c^2)
Time = 7.02 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.02 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )} \, dx=\left [\frac {b c^{3} x \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + b c^{2} d x \log \left (b x^{2} + a\right ) + 2 \, a d^{3} x \log \left (d x + c\right ) - 2 \, b c^{3} - 2 \, a c d^{2} - 2 \, {\left (b c^{2} d + a d^{3}\right )} x \log \left (x\right )}{2 \, {\left (a b c^{4} + a^{2} c^{2} d^{2}\right )} x}, -\frac {2 \, b c^{3} x \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) - b c^{2} d x \log \left (b x^{2} + a\right ) - 2 \, a d^{3} x \log \left (d x + c\right ) + 2 \, b c^{3} + 2 \, a c d^{2} + 2 \, {\left (b c^{2} d + a d^{3}\right )} x \log \left (x\right )}{2 \, {\left (a b c^{4} + a^{2} c^{2} d^{2}\right )} x}\right ] \] Input:
integrate(1/x^2/(d*x+c)/(b*x^2+a),x, algorithm="fricas")
Output:
[1/2*(b*c^3*x*sqrt(-b/a)*log((b*x^2 - 2*a*x*sqrt(-b/a) - a)/(b*x^2 + a)) + b*c^2*d*x*log(b*x^2 + a) + 2*a*d^3*x*log(d*x + c) - 2*b*c^3 - 2*a*c*d^2 - 2*(b*c^2*d + a*d^3)*x*log(x))/((a*b*c^4 + a^2*c^2*d^2)*x), -1/2*(2*b*c^3* x*sqrt(b/a)*arctan(x*sqrt(b/a)) - b*c^2*d*x*log(b*x^2 + a) - 2*a*d^3*x*log (d*x + c) + 2*b*c^3 + 2*a*c*d^2 + 2*(b*c^2*d + a*d^3)*x*log(x))/((a*b*c^4 + a^2*c^2*d^2)*x)]
Timed out. \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**2/(d*x+c)/(b*x**2+a),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )} \, dx=\frac {d^{3} \log \left (d x + c\right )}{b c^{4} + a c^{2} d^{2}} - \frac {b^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a b c^{2} + a^{2} d^{2}\right )} \sqrt {a b}} + \frac {b d \log \left (b x^{2} + a\right )}{2 \, {\left (a b c^{2} + a^{2} d^{2}\right )}} - \frac {d \log \left (x\right )}{a c^{2}} - \frac {1}{a c x} \] Input:
integrate(1/x^2/(d*x+c)/(b*x^2+a),x, algorithm="maxima")
Output:
d^3*log(d*x + c)/(b*c^4 + a*c^2*d^2) - b^2*c*arctan(b*x/sqrt(a*b))/((a*b*c ^2 + a^2*d^2)*sqrt(a*b)) + 1/2*b*d*log(b*x^2 + a)/(a*b*c^2 + a^2*d^2) - d* log(x)/(a*c^2) - 1/(a*c*x)
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.98 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )} \, dx=\frac {d^{4} \log \left ({\left | d x + c \right |}\right )}{b c^{4} d + a c^{2} d^{3}} - \frac {b^{2} c \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (a b c^{2} + a^{2} d^{2}\right )} \sqrt {a b}} + \frac {b d \log \left (b x^{2} + a\right )}{2 \, {\left (a b c^{2} + a^{2} d^{2}\right )}} - \frac {d \log \left ({\left | x \right |}\right )}{a c^{2}} - \frac {1}{a c x} \] Input:
integrate(1/x^2/(d*x+c)/(b*x^2+a),x, algorithm="giac")
Output:
d^4*log(abs(d*x + c))/(b*c^4*d + a*c^2*d^3) - b^2*c*arctan(b*x/sqrt(a*b))/ ((a*b*c^2 + a^2*d^2)*sqrt(a*b)) + 1/2*b*d*log(b*x^2 + a)/(a*b*c^2 + a^2*d^ 2) - d*log(abs(x))/(a*c^2) - 1/(a*c*x)
Time = 9.41 (sec) , antiderivative size = 688, normalized size of antiderivative = 5.83 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )} \, dx=\frac {d^3\,\ln \left (c+d\,x\right )}{b\,c^4+a\,c^2\,d^2}-\frac {\ln \left (4\,a^9\,b^4\,d^8-a^5\,b^8\,c^8+12\,a^6\,b^7\,c^6\,d^2-7\,a^7\,b^6\,c^4\,d^4+8\,a^4\,c\,d^7\,{\left (-a^3\,b^3\right )}^{3/2}+4\,a^4\,d^8\,x\,{\left (-a^3\,b^3\right )}^{3/2}-b^4\,c^8\,x\,{\left (-a^3\,b^3\right )}^{3/2}+8\,a^3\,b\,c^3\,d^5\,{\left (-a^3\,b^3\right )}^{3/2}-6\,a^5\,b^8\,c^7\,d\,x-8\,a^8\,b^5\,c\,d^7\,x-10\,a^2\,b^2\,c^5\,d^3\,{\left (-a^3\,b^3\right )}^{3/2}+10\,a^6\,b^7\,c^5\,d^3\,x-8\,a^7\,b^6\,c^3\,d^5\,x+6\,a\,b^3\,c^7\,d\,{\left (-a^3\,b^3\right )}^{3/2}-7\,a^2\,b^2\,c^4\,d^4\,x\,{\left (-a^3\,b^3\right )}^{3/2}+12\,a\,b^3\,c^6\,d^2\,x\,{\left (-a^3\,b^3\right )}^{3/2}\right )\,\left (c\,\sqrt {-a^3\,b^3}-a^2\,b\,d\right )}{2\,\left (a^4\,d^2+b\,a^3\,c^2\right )}+\frac {\ln \left (a^5\,b^8\,c^8-4\,a^9\,b^4\,d^8-12\,a^6\,b^7\,c^6\,d^2+7\,a^7\,b^6\,c^4\,d^4+8\,a^4\,c\,d^7\,{\left (-a^3\,b^3\right )}^{3/2}+4\,a^4\,d^8\,x\,{\left (-a^3\,b^3\right )}^{3/2}-b^4\,c^8\,x\,{\left (-a^3\,b^3\right )}^{3/2}+8\,a^3\,b\,c^3\,d^5\,{\left (-a^3\,b^3\right )}^{3/2}+6\,a^5\,b^8\,c^7\,d\,x+8\,a^8\,b^5\,c\,d^7\,x-10\,a^2\,b^2\,c^5\,d^3\,{\left (-a^3\,b^3\right )}^{3/2}-10\,a^6\,b^7\,c^5\,d^3\,x+8\,a^7\,b^6\,c^3\,d^5\,x+6\,a\,b^3\,c^7\,d\,{\left (-a^3\,b^3\right )}^{3/2}-7\,a^2\,b^2\,c^4\,d^4\,x\,{\left (-a^3\,b^3\right )}^{3/2}+12\,a\,b^3\,c^6\,d^2\,x\,{\left (-a^3\,b^3\right )}^{3/2}\right )\,\left (c\,\sqrt {-a^3\,b^3}+a^2\,b\,d\right )}{2\,a^4\,d^2+2\,b\,a^3\,c^2}-\frac {1}{a\,c\,x}-\frac {d\,\ln \left (x\right )}{a\,c^2} \] Input:
int(1/(x^2*(a + b*x^2)*(c + d*x)),x)
Output:
(d^3*log(c + d*x))/(b*c^4 + a*c^2*d^2) - (log(4*a^9*b^4*d^8 - a^5*b^8*c^8 + 12*a^6*b^7*c^6*d^2 - 7*a^7*b^6*c^4*d^4 + 8*a^4*c*d^7*(-a^3*b^3)^(3/2) + 4*a^4*d^8*x*(-a^3*b^3)^(3/2) - b^4*c^8*x*(-a^3*b^3)^(3/2) + 8*a^3*b*c^3*d^ 5*(-a^3*b^3)^(3/2) - 6*a^5*b^8*c^7*d*x - 8*a^8*b^5*c*d^7*x - 10*a^2*b^2*c^ 5*d^3*(-a^3*b^3)^(3/2) + 10*a^6*b^7*c^5*d^3*x - 8*a^7*b^6*c^3*d^5*x + 6*a* b^3*c^7*d*(-a^3*b^3)^(3/2) - 7*a^2*b^2*c^4*d^4*x*(-a^3*b^3)^(3/2) + 12*a*b ^3*c^6*d^2*x*(-a^3*b^3)^(3/2))*(c*(-a^3*b^3)^(1/2) - a^2*b*d))/(2*(a^4*d^2 + a^3*b*c^2)) + (log(a^5*b^8*c^8 - 4*a^9*b^4*d^8 - 12*a^6*b^7*c^6*d^2 + 7 *a^7*b^6*c^4*d^4 + 8*a^4*c*d^7*(-a^3*b^3)^(3/2) + 4*a^4*d^8*x*(-a^3*b^3)^( 3/2) - b^4*c^8*x*(-a^3*b^3)^(3/2) + 8*a^3*b*c^3*d^5*(-a^3*b^3)^(3/2) + 6*a ^5*b^8*c^7*d*x + 8*a^8*b^5*c*d^7*x - 10*a^2*b^2*c^5*d^3*(-a^3*b^3)^(3/2) - 10*a^6*b^7*c^5*d^3*x + 8*a^7*b^6*c^3*d^5*x + 6*a*b^3*c^7*d*(-a^3*b^3)^(3/ 2) - 7*a^2*b^2*c^4*d^4*x*(-a^3*b^3)^(3/2) + 12*a*b^3*c^6*d^2*x*(-a^3*b^3)^ (3/2))*(c*(-a^3*b^3)^(1/2) + a^2*b*d))/(2*a^4*d^2 + 2*a^3*b*c^2) - 1/(a*c* x) - (d*log(x))/(a*c^2)
Time = 0.25 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^2 (c+d x) \left (a+b x^2\right )} \, dx=\frac {-2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,c^{3} x +\mathrm {log}\left (b \,x^{2}+a \right ) a b \,c^{2} d x +2 \,\mathrm {log}\left (d x +c \right ) a^{2} d^{3} x -2 \,\mathrm {log}\left (x \right ) a^{2} d^{3} x -2 \,\mathrm {log}\left (x \right ) a b \,c^{2} d x -2 a^{2} c \,d^{2}-2 a b \,c^{3}}{2 a^{2} c^{2} x \left (a \,d^{2}+b \,c^{2}\right )} \] Input:
int(1/x^2/(d*x+c)/(b*x^2+a),x)
Output:
( - 2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b*c**3*x + log(a + b*x **2)*a*b*c**2*d*x + 2*log(c + d*x)*a**2*d**3*x - 2*log(x)*a**2*d**3*x - 2* log(x)*a*b*c**2*d*x - 2*a**2*c*d**2 - 2*a*b*c**3)/(2*a**2*c**2*x*(a*d**2 + b*c**2))