Integrand size = 25, antiderivative size = 193 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx=-\frac {1}{c d x}+\frac {\left (2 a^2 c d+b^3 e-a b (b d+3 c e)\right ) \text {arctanh}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{c^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}-\frac {(b d+c e) \log (x)}{c^2 d^2}+\frac {e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}+\frac {\left (a b d-b^2 e+a c e\right ) \log \left (c+b x+a x^2\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )} \] Output:
-1/c/d/x+(2*a^2*c*d+b^3*e-a*b*(b*d+3*c*e))*arctanh((2*a*x+b)/(-4*a*c+b^2)^ (1/2))/c^2/(-4*a*c+b^2)^(1/2)/(a*d^2-e*(b*d-c*e))-(b*d+c*e)*ln(x)/c^2/d^2+ e^3*ln(e*x+d)/d^2/(a*d^2-e*(b*d-c*e))+1/2*(a*b*d+a*c*e-b^2*e)*ln(a*x^2+b*x +c)/c^2/(a*d^2-e*(b*d-c*e))
Time = 0.17 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx=-\frac {1}{c d x}+\frac {\left (2 a^2 c d+b^3 e-a b (b d+3 c e)\right ) \arctan \left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{c^2 \sqrt {-b^2+4 a c} \left (-a d^2+e (b d-c e)\right )}-\frac {(b d+c e) \log (x)}{c^2 d^2}+\frac {e^3 \log (d+e x)}{a d^4+d^2 e (-b d+c e)}+\frac {\left (a b d-b^2 e+a c e\right ) \log (c+x (b+a x))}{2 c^2 \left (a d^2+e (-b d+c e)\right )} \] Input:
Integrate[1/((a + c/x^2 + b/x)*x^4*(d + e*x)),x]
Output:
-(1/(c*d*x)) + ((2*a^2*c*d + b^3*e - a*b*(b*d + 3*c*e))*ArcTan[(b + 2*a*x) /Sqrt[-b^2 + 4*a*c]])/(c^2*Sqrt[-b^2 + 4*a*c]*(-(a*d^2) + e*(b*d - c*e))) - ((b*d + c*e)*Log[x])/(c^2*d^2) + (e^3*Log[d + e*x])/(a*d^4 + d^2*e*(-(b* d) + c*e)) + ((a*b*d - b^2*e + a*c*e)*Log[c + x*(b + a*x)])/(2*c^2*(a*d^2 + e*(-(b*d) + c*e)))
Time = 0.52 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {1893, 1200, 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.
\(\displaystyle \int \frac {1}{x^4 (d+e x) \left (a+\frac {b}{x}+\frac {c}{x^2}\right )} \, dx\) |
\(\Big \downarrow \) 1893 |
\(\displaystyle \int \frac {1}{x^2 (d+e x) \left (a x^2+b x+c\right )}dx\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \int \left (\frac {-a^2 c d+a x \left (a b d+a c e+b^2 (-e)\right )+a b (b d+2 c e)+b^3 (-e)}{c^2 \left (a x^2+b x+c\right ) \left (a d^2-e (b d-c e)\right )}+\frac {e^4}{d^2 (d+e x) \left (a d^2-e (b d-c e)\right )}+\frac {-b d-c e}{c^2 d^2 x}+\frac {1}{c d x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\text {arctanh}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right ) \left (2 a^2 c d-a b (b d+3 c e)+b^3 e\right )}{c^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )}+\frac {\left (a b d+a c e+b^2 (-e)\right ) \log \left (a x^2+b x+c\right )}{2 c^2 \left (a d^2-e (b d-c e)\right )}+\frac {e^3 \log (d+e x)}{d^2 \left (a d^2-e (b d-c e)\right )}-\frac {\log (x) (b d+c e)}{c^2 d^2}-\frac {1}{c d x}\) |
Input:
Int[1/((a + c/x^2 + b/x)*x^4*(d + e*x)),x]
Output:
-(1/(c*d*x)) + ((2*a^2*c*d + b^3*e - a*b*(b*d + 3*c*e))*ArcTanh[(b + 2*a*x )/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))) - (( b*d + c*e)*Log[x])/(c^2*d^2) + (e^3*Log[d + e*x])/(d^2*(a*d^2 - e*(b*d - c *e))) + ((a*b*d - b^2*e + a*c*e)*Log[c + b*x + a*x^2])/(2*c^2*(a*d^2 - e*( b*d - c*e)))
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* (x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In tegersQ[n]
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^(mn_.) + (c_.)*(x_)^(mn2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Int[x^(m - 2*n*p)*(d + e*x^n)^q*(c + b*x^n + a*x^(2*n))^p, x] /; FreeQ[{a, b, c, d, e, m, n, q}, x] && EqQ[mn , -n] && EqQ[mn2, 2*mn] && IntegerQ[p]
Time = 0.22 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.07
method | result | size |
default | \(\frac {\frac {\left (a^{2} b d +a^{2} c e -a \,b^{2} e \right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (-a^{2} c d +a \,b^{2} d +2 a b c e -b^{3} e -\frac {\left (a^{2} b d +a^{2} c e -a \,b^{2} e \right ) b}{2 a}\right ) \arctan \left (\frac {2 x a +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-b d e +c \,e^{2}\right ) c^{2}}+\frac {e^{3} \ln \left (e x +d \right )}{d^{2} \left (a \,d^{2}-b d e +c \,e^{2}\right )}-\frac {1}{c d x}+\frac {\left (-b d -c e \right ) \ln \left (x \right )}{c^{2} d^{2}}\) | \(207\) |
risch | \(\text {Expression too large to display}\) | \(101857\) |
Input:
int(1/(a+c/x^2+b/x)/x^4/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/(a*d^2-b*d*e+c*e^2)/c^2*(1/2*(a^2*b*d+a^2*c*e-a*b^2*e)/a*ln(a*x^2+b*x+c) +2*(-a^2*c*d+a*b^2*d+2*a*b*c*e-b^3*e-1/2*(a^2*b*d+a^2*c*e-a*b^2*e)*b/a)/(4 *a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2)))+e^3/d^2/(a*d^2-b*d*e+ c*e^2)*ln(e*x+d)-1/c/d/x+1/c^2/d^2*(-b*d-c*e)*ln(x)
Timed out. \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx=\text {Timed out} \] Input:
integrate(1/(a+c/x^2+b/x)/x^4/(e*x+d),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx=\text {Timed out} \] Input:
integrate(1/(a+c/x**2+b/x)/x**4/(e*x+d),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(a+c/x^2+b/x)/x^4/(e*x+d),x, algorithm="maxima")
Output:
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*a*c-b^2>0)', see `assume?` for more deta
Time = 0.13 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx=\frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{a d^{4} e - b d^{3} e^{2} + c d^{2} e^{3}} + \frac {{\left (a b d - b^{2} e + a c e\right )} \log \left (a x^{2} + b x + c\right )}{2 \, {\left (a c^{2} d^{2} - b c^{2} d e + c^{3} e^{2}\right )}} + \frac {{\left (a b^{2} d - 2 \, a^{2} c d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac {2 \, a x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (a c^{2} d^{2} - b c^{2} d e + c^{3} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (b d + c e\right )} \log \left ({\left | x \right |}\right )}{c^{2} d^{2}} - \frac {1}{c d x} \] Input:
integrate(1/(a+c/x^2+b/x)/x^4/(e*x+d),x, algorithm="giac")
Output:
e^4*log(abs(e*x + d))/(a*d^4*e - b*d^3*e^2 + c*d^2*e^3) + 1/2*(a*b*d - b^2 *e + a*c*e)*log(a*x^2 + b*x + c)/(a*c^2*d^2 - b*c^2*d*e + c^3*e^2) + (a*b^ 2*d - 2*a^2*c*d - b^3*e + 3*a*b*c*e)*arctan((2*a*x + b)/sqrt(-b^2 + 4*a*c) )/((a*c^2*d^2 - b*c^2*d*e + c^3*e^2)*sqrt(-b^2 + 4*a*c)) - (b*d + c*e)*log (abs(x))/(c^2*d^2) - 1/(c*d*x)
Time = 52.19 (sec) , antiderivative size = 2388, normalized size of antiderivative = 12.37 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx=\text {Too large to display} \] Input:
int(1/(x^4*(d + e*x)*(a + b/x + c/x^2)),x)
Output:
(e^3*log(d + e*x))/(a*d^4 + c*d^2*e^2 - b*d^3*e) + (log((a^4*e^4*x)/(c^2*d ^2) - (((a*e*x*(a^4*d^4 + b^4*e^4 + 2*a^2*c^2*e^4 + 2*a^3*c*d^2*e^2 - 4*a* b^2*c*e^4 + 2*a^2*b*c*d*e^3))/(c^2*d^2) - (((a*e*(a^2*b^2*d^4 - 4*a*c^3*e^ 4 - a^3*c*d^4 + b^2*c^2*e^4 + b^4*d^2*e^2 + 4*a^2*c^2*d^2*e^2 - 2*a*b^3*d^ 3*e + b^3*c*d*e^3 - 4*a*b*c^2*d*e^3 + 5*a^2*b*c*d^3*e - 5*a*b^2*c*d^2*e^2) )/(c*d) + (a*e*x*(2*a^3*b*d^4 + 2*b^3*c*e^4 + 2*b^4*d*e^3 - 2*a*b^3*d^2*e^ 2 - 2*a^2*b^2*d^3*e + 12*a^2*c^2*d*e^3 - 8*a*b*c^2*e^4 + a^3*c*d^3*e - 11* a*b^2*c*d*e^3 + 8*a^2*b*c*d^2*e^2))/(c*d) + (a*e*(b^4*e + b^3*e*(b^2 - 4*a *c)^(1/2) + 4*a^2*c^2*e - a*b^3*d + 4*a^2*b*c*d - 5*a*b^2*c*e - a*b^2*d*(b ^2 - 4*a*c)^(1/2) + 2*a^2*c*d*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e*(b^2 - 4*a*c )^(1/2))*(4*a^2*c^2*d^3*e + b^2*c^2*d*e^3 + b^3*c*d^2*e^2 + 2*a^2*b^2*d^4* x + 2*b^2*c^2*e^4*x + 2*b^4*d^2*e^2*x + a^2*b*c*d^4 - 4*a*c^3*d*e^3 - 6*a^ 3*c*d^4*x - 8*a*c^3*e^4*x - 2*a*b^2*c*d^3*e - 4*a*b^3*d^3*e*x - 2*b^3*c*d* e^3*x - 3*a*b*c^2*d^2*e^2 - 6*a^2*c^2*d^2*e^2*x + 8*a*b*c^2*d*e^3*x + 14*a ^2*b*c*d^3*e*x - 6*a*b^2*c*d^2*e^2*x))/(2*c^2*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)))*(b^4*e + b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - a*b^3*d + 4 *a^2*b*c*d - 5*a*b^2*c*e - a*b^2*d*(b^2 - 4*a*c)^(1/2) + 2*a^2*c*d*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e*(b^2 - 4*a*c)^(1/2)))/(2*c^2*(4*a*c - b^2)*(a*d^2 + c*e^2 - b*d*e)) + (a*e*(b*d + c*e)*(a^3*d^3 + b^3*e^3 - 3*a*b*c*e^3))/( c^2*d^2))*(b^4*e + b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - a*b^3*d + ...
Time = 0.15 (sec) , antiderivative size = 532, normalized size of antiderivative = 2.76 \[ \int \frac {1}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) x^4 (d+e x)} \, dx=\frac {-4 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right ) a^{2} c \,d^{3} x +2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right ) a \,b^{2} d^{3} x +6 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right ) a b c \,d^{2} e x -2 \sqrt {4 a c -b^{2}}\, \mathit {atan} \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right ) b^{3} d^{2} e x +4 \,\mathrm {log}\left (a \,x^{2}+b x +c \right ) a^{2} b c \,d^{3} x +4 \,\mathrm {log}\left (a \,x^{2}+b x +c \right ) a^{2} c^{2} d^{2} e x -\mathrm {log}\left (a \,x^{2}+b x +c \right ) a \,b^{3} d^{3} x -5 \,\mathrm {log}\left (a \,x^{2}+b x +c \right ) a \,b^{2} c \,d^{2} e x +\mathrm {log}\left (a \,x^{2}+b x +c \right ) b^{4} d^{2} e x +8 \,\mathrm {log}\left (e x +d \right ) a \,c^{3} e^{3} x -2 \,\mathrm {log}\left (e x +d \right ) b^{2} c^{2} e^{3} x -8 \,\mathrm {log}\left (x \right ) a^{2} b c \,d^{3} x -8 \,\mathrm {log}\left (x \right ) a^{2} c^{2} d^{2} e x +2 \,\mathrm {log}\left (x \right ) a \,b^{3} d^{3} x +10 \,\mathrm {log}\left (x \right ) a \,b^{2} c \,d^{2} e x -8 \,\mathrm {log}\left (x \right ) a \,c^{3} e^{3} x -2 \,\mathrm {log}\left (x \right ) b^{4} d^{2} e x +2 \,\mathrm {log}\left (x \right ) b^{2} c^{2} e^{3} x -8 a^{2} c^{2} d^{3}+2 a \,b^{2} c \,d^{3}+8 a b \,c^{2} d^{2} e -8 a \,c^{3} d \,e^{2}-2 b^{3} c \,d^{2} e +2 b^{2} c^{2} d \,e^{2}}{2 c^{2} d^{2} x \left (4 a^{2} c \,d^{2}-a \,b^{2} d^{2}-4 a b c d e +4 a \,c^{2} e^{2}+b^{3} d e -b^{2} c \,e^{2}\right )} \] Input:
int(1/(a+c/x^2+b/x)/x^4/(e*x+d),x)
Output:
( - 4*sqrt(4*a*c - b**2)*atan((2*a*x + b)/sqrt(4*a*c - b**2))*a**2*c*d**3* x + 2*sqrt(4*a*c - b**2)*atan((2*a*x + b)/sqrt(4*a*c - b**2))*a*b**2*d**3* x + 6*sqrt(4*a*c - b**2)*atan((2*a*x + b)/sqrt(4*a*c - b**2))*a*b*c*d**2*e *x - 2*sqrt(4*a*c - b**2)*atan((2*a*x + b)/sqrt(4*a*c - b**2))*b**3*d**2*e *x + 4*log(a*x**2 + b*x + c)*a**2*b*c*d**3*x + 4*log(a*x**2 + b*x + c)*a** 2*c**2*d**2*e*x - log(a*x**2 + b*x + c)*a*b**3*d**3*x - 5*log(a*x**2 + b*x + c)*a*b**2*c*d**2*e*x + log(a*x**2 + b*x + c)*b**4*d**2*e*x + 8*log(d + e*x)*a*c**3*e**3*x - 2*log(d + e*x)*b**2*c**2*e**3*x - 8*log(x)*a**2*b*c*d **3*x - 8*log(x)*a**2*c**2*d**2*e*x + 2*log(x)*a*b**3*d**3*x + 10*log(x)*a *b**2*c*d**2*e*x - 8*log(x)*a*c**3*e**3*x - 2*log(x)*b**4*d**2*e*x + 2*log (x)*b**2*c**2*e**3*x - 8*a**2*c**2*d**3 + 2*a*b**2*c*d**3 + 8*a*b*c**2*d** 2*e - 8*a*c**3*d*e**2 - 2*b**3*c*d**2*e + 2*b**2*c**2*d*e**2)/(2*c**2*d**2 *x*(4*a**2*c*d**2 - a*b**2*d**2 - 4*a*b*c*d*e + 4*a*c**2*e**2 + b**3*d*e - b**2*c*e**2))