Integrand size = 25, antiderivative size = 274 \[ \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\frac {x}{a e^2}-\frac {d^4}{e^3 \left (a d^2-e (b d-c e)\right ) (d+e x)}-\frac {\left (b^4 d^2-2 b^3 c d e+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )-b^2 c \left (4 a d^2-c e^2\right )\right ) \text {arctanh}\left (\frac {b+2 a x}{\sqrt {b^2-4 a c}}\right )}{a^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac {d^3 \left (2 a d^2-e (3 b d-4 c e)\right ) \log (d+e x)}{e^3 \left (a d^2-e (b d-c e)\right )^2}-\frac {(b d-c e) \left (b^2 d-2 a c d-b c e\right ) \log \left (c+b x+a x^2\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )^2} \]
x/a/e^2-d^4/e^3/(a*d^2-e*(b*d-c*e))/(e*x+d)-d^3*(2*a*d^2-e*(3*b*d-4*c*e))* ln(e*x+d)/e^3/(a*d^2-e*(b*d-c*e))^2-1/2*(b*d-c*e)*(-2*a*c*d+b^2*d-b*c*e)*l n(a*x^2+b*x+c)/a^2/(a*d^2-e*(b*d-c*e))^2-(b^4*d^2-2*b^3*c*d*e+6*a*b*c^2*d* e+2*a*c^2*(a*d^2-c*e^2)-b^2*c*(4*a*d^2-c*e^2))*arctanh((2*a*x+b)/(-4*a*c+b ^2)^(1/2))/a^2/(a*d^2-e*(b*d-c*e))^2/(-4*a*c+b^2)^(1/2)
Time = 0.19 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\frac {x}{a e^2}-\frac {d^4}{e^3 \left (a d^2+e (-b d+c e)\right ) (d+e x)}+\frac {\left (b^4 d^2-2 b^3 c d e+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )+b^2 c \left (-4 a d^2+c e^2\right )\right ) \arctan \left (\frac {b+2 a x}{\sqrt {-b^2+4 a c}}\right )}{a^2 \sqrt {-b^2+4 a c} \left (a d^2+e (-b d+c e)\right )^2}-\frac {\left (2 a d^5+d^3 e (-3 b d+4 c e)\right ) \log (d+e x)}{e^3 \left (a d^2+e (-b d+c e)\right )^2}+\frac {(b d-c e) \left (-b^2 d+2 a c d+b c e\right ) \log (c+x (b+a x))}{2 a^2 \left (a d^2+e (-b d+c e)\right )^2} \]
x/(a*e^2) - d^4/(e^3*(a*d^2 + e*(-(b*d) + c*e))*(d + e*x)) + ((b^4*d^2 - 2 *b^3*c*d*e + 6*a*b*c^2*d*e + 2*a*c^2*(a*d^2 - c*e^2) + b^2*c*(-4*a*d^2 + c *e^2))*ArcTan[(b + 2*a*x)/Sqrt[-b^2 + 4*a*c]])/(a^2*Sqrt[-b^2 + 4*a*c]*(a* d^2 + e*(-(b*d) + c*e))^2) - ((2*a*d^5 + d^3*e*(-3*b*d + 4*c*e))*Log[d + e *x])/(e^3*(a*d^2 + e*(-(b*d) + c*e))^2) + ((b*d - c*e)*(-(b^2*d) + 2*a*c*d + b*c*e)*Log[c + x*(b + a*x)])/(2*a^2*(a*d^2 + e*(-(b*d) + c*e))^2)
Time = 0.66 (sec) , antiderivative size = 274, 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.
\(\displaystyle \int \frac {x^2}{(d+e x)^2 \left (a+\frac {b}{x}+\frac {c}{x^2}\right )} \, dx\) |
\(\Big \downarrow \) 1893 |
\(\displaystyle \int \frac {x^4}{(d+e x)^2 \left (a x^2+b x+c\right )}dx\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \int \left (\frac {-c \left (-c \left (a d^2-c e^2\right )+b^2 d^2-2 b c d e\right )-x (b d-c e) \left (-2 a c d+b^2 d-b c e\right )}{a \left (a x^2+b x+c\right ) \left (a d^2-e (b d-c e)\right )^2}+\frac {d^4}{e^2 (d+e x)^2 \left (a d^2-e (b d-c e)\right )}+\frac {d^3 \left (e (3 b d-4 c e)-2 a d^2\right )}{e^2 (d+e x) \left (a d^2-e (b d-c e)\right )^2}+\frac {1}{a e^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {2 a x+b}{\sqrt {b^2-4 a c}}\right ) \left (-b^2 c \left (4 a d^2-c e^2\right )+6 a b c^2 d e+2 a c^2 \left (a d^2-c e^2\right )+b^4 d^2-2 b^3 c d e\right )}{a^2 \sqrt {b^2-4 a c} \left (a d^2-e (b d-c e)\right )^2}-\frac {(b d-c e) \left (-2 a c d+b^2 d-b c e\right ) \log \left (a x^2+b x+c\right )}{2 a^2 \left (a d^2-e (b d-c e)\right )^2}-\frac {d^4}{e^3 (d+e x) \left (a d^2-e (b d-c e)\right )}-\frac {d^3 \log (d+e x) \left (2 a d^2-e (3 b d-4 c e)\right )}{e^3 \left (a d^2-e (b d-c e)\right )^2}+\frac {x}{a e^2}\) |
x/(a*e^2) - d^4/(e^3*(a*d^2 - e*(b*d - c*e))*(d + e*x)) - ((b^4*d^2 - 2*b^ 3*c*d*e + 6*a*b*c^2*d*e + 2*a*c^2*(a*d^2 - c*e^2) - b^2*c*(4*a*d^2 - c*e^2 ))*ArcTanh[(b + 2*a*x)/Sqrt[b^2 - 4*a*c]])/(a^2*Sqrt[b^2 - 4*a*c]*(a*d^2 - e*(b*d - c*e))^2) - (d^3*(2*a*d^2 - e*(3*b*d - 4*c*e))*Log[d + e*x])/(e^3 *(a*d^2 - e*(b*d - c*e))^2) - ((b*d - c*e)*(b^2*d - 2*a*c*d - b*c*e)*Log[c + b*x + a*x^2])/(2*a^2*(a*d^2 - e*(b*d - c*e))^2)
3.1.71.3.1 Defintions of rubi rules used
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 = 0.74 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.06
method | result | size |
default | \(\frac {x}{a \,e^{2}}+\frac {\frac {\left (2 b \,d^{2} c a -2 a \,c^{2} d e -b^{3} d^{2}+2 b^{2} c d e -b \,c^{2} e^{2}\right ) \ln \left (a \,x^{2}+b x +c \right )}{2 a}+\frac {2 \left (a \,c^{2} d^{2}-b^{2} c \,d^{2}+2 e d \,c^{2} b -c^{3} e^{2}-\frac {\left (2 b \,d^{2} c a -2 a \,c^{2} d e -b^{3} d^{2}+2 b^{2} c d e -b \,c^{2} e^{2}\right ) b}{2 a}\right ) \arctan \left (\frac {2 a x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,d^{2}-b d e +c \,e^{2}\right )^{2} a}-\frac {d^{4}}{e^{3} \left (a \,d^{2}-b d e +c \,e^{2}\right ) \left (e x +d \right )}-\frac {d^{3} \left (2 a \,d^{2}-3 b d e +4 c \,e^{2}\right ) \ln \left (e x +d \right )}{e^{3} \left (a \,d^{2}-b d e +c \,e^{2}\right )^{2}}\) | \(290\) |
risch | \(\text {Expression too large to display}\) | \(1434\) |
x/a/e^2+1/(a*d^2-b*d*e+c*e^2)^2/a*(1/2*(2*a*b*c*d^2-2*a*c^2*d*e-b^3*d^2+2* b^2*c*d*e-b*c^2*e^2)/a*ln(a*x^2+b*x+c)+2*(a*c^2*d^2-b^2*c*d^2+2*e*d*c^2*b- c^3*e^2-1/2*(2*a*b*c*d^2-2*a*c^2*d*e-b^3*d^2+2*b^2*c*d*e-b*c^2*e^2)*b/a)/( 4*a*c-b^2)^(1/2)*arctan((2*a*x+b)/(4*a*c-b^2)^(1/2)))-1/e^3*d^4/(a*d^2-b*d *e+c*e^2)/(e*x+d)-1/e^3*d^3*(2*a*d^2-3*b*d*e+4*c*e^2)/(a*d^2-b*d*e+c*e^2)^ 2*ln(e*x+d)
Leaf count of result is larger than twice the leaf count of optimal. 1060 vs. \(2 (268) = 536\).
Time = 26.47 (sec) , antiderivative size = 2139, normalized size of antiderivative = 7.81 \[ \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\text {Too large to display} \]
[-1/2*(2*(a^3*b^2 - 4*a^4*c)*d^6 - 2*(a^2*b^3 - 4*a^3*b*c)*d^5*e + 2*(a^2* b^2*c - 4*a^3*c^2)*d^4*e^2 - 2*((a^3*b^2 - 4*a^4*c)*d^4*e^2 - 2*(a^2*b^3 - 4*a^3*b*c)*d^3*e^3 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^2*e^4 - 2*(a*b^3 *c - 4*a^2*b*c^2)*d*e^5 + (a*b^2*c^2 - 4*a^2*c^3)*e^6)*x^2 + ((b^4 - 4*a*b ^2*c + 2*a^2*c^2)*d^3*e^3 - 2*(b^3*c - 3*a*b*c^2)*d^2*e^4 + (b^2*c^2 - 2*a *c^3)*d*e^5 + ((b^4 - 4*a*b^2*c + 2*a^2*c^2)*d^2*e^4 - 2*(b^3*c - 3*a*b*c^ 2)*d*e^5 + (b^2*c^2 - 2*a*c^3)*e^6)*x)*sqrt(b^2 - 4*a*c)*log((2*a^2*x^2 + 2*a*b*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*a*x + b))/(a*x^2 + b*x + c)) - 2*((a^3*b^2 - 4*a^4*c)*d^5*e - 2*(a^2*b^3 - 4*a^3*b*c)*d^4*e^2 + (a*b^4 - 2*a^2*b^2*c - 8*a^3*c^2)*d^3*e^3 - 2*(a*b^3*c - 4*a^2*b*c^2)*d^2*e^4 + ( a*b^2*c^2 - 4*a^2*c^3)*d*e^5)*x + ((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d^3*e^3 - 2*(b^4*c - 5*a*b^2*c^2 + 4*a^2*c^3)*d^2*e^4 + (b^3*c^2 - 4*a*b*c^3)*d*e ^5 + ((b^5 - 6*a*b^3*c + 8*a^2*b*c^2)*d^2*e^4 - 2*(b^4*c - 5*a*b^2*c^2 + 4 *a^2*c^3)*d*e^5 + (b^3*c^2 - 4*a*b*c^3)*e^6)*x)*log(a*x^2 + b*x + c) + 2*( 2*(a^3*b^2 - 4*a^4*c)*d^6 - 3*(a^2*b^3 - 4*a^3*b*c)*d^5*e + 4*(a^2*b^2*c - 4*a^3*c^2)*d^4*e^2 + (2*(a^3*b^2 - 4*a^4*c)*d^5*e - 3*(a^2*b^3 - 4*a^3*b* c)*d^4*e^2 + 4*(a^2*b^2*c - 4*a^3*c^2)*d^3*e^3)*x)*log(e*x + d))/((a^4*b^2 - 4*a^5*c)*d^5*e^3 - 2*(a^3*b^3 - 4*a^4*b*c)*d^4*e^4 + (a^2*b^4 - 2*a^3*b ^2*c - 8*a^4*c^2)*d^3*e^5 - 2*(a^2*b^3*c - 4*a^3*b*c^2)*d^2*e^6 + (a^2*b^2 *c^2 - 4*a^3*c^3)*d*e^7 + ((a^4*b^2 - 4*a^5*c)*d^4*e^4 - 2*(a^3*b^3 - 4...
Timed out. \[ \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\text {Exception raised: ValueError} \]
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 = 0.32 (sec) , antiderivative size = 483, normalized size of antiderivative = 1.76 \[ \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=-\frac {d^{4} e^{3}}{{\left (a d^{2} e^{6} - b d e^{7} + c e^{8}\right )} {\left (e x + d\right )}} - \frac {{\left (b^{3} d^{2} - 2 \, a b c d^{2} - 2 \, b^{2} c d e + 2 \, a c^{2} d e + b c^{2} e^{2}\right )} \log \left (-a + \frac {2 \, a d}{e x + d} - \frac {a d^{2}}{{\left (e x + d\right )}^{2}} - \frac {b e}{e x + d} + \frac {b d e}{{\left (e x + d\right )}^{2}} - \frac {c e^{2}}{{\left (e x + d\right )}^{2}}\right )}{2 \, {\left (a^{4} d^{4} - 2 \, a^{3} b d^{3} e + a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} c d^{2} e^{2} - 2 \, a^{2} b c d e^{3} + a^{2} c^{2} e^{4}\right )}} - \frac {{\left (b^{4} d^{2} e^{2} - 4 \, a b^{2} c d^{2} e^{2} + 2 \, a^{2} c^{2} d^{2} e^{2} - 2 \, b^{3} c d e^{3} + 6 \, a b c^{2} d e^{3} + b^{2} c^{2} e^{4} - 2 \, a c^{3} e^{4}\right )} \arctan \left (-\frac {2 \, a d - \frac {2 \, a d^{2}}{e x + d} - b e + \frac {2 \, b d e}{e x + d} - \frac {2 \, c e^{2}}{e x + d}}{\sqrt {-b^{2} + 4 \, a c} e}\right )}{{\left (a^{4} d^{4} - 2 \, a^{3} b d^{3} e + a^{2} b^{2} d^{2} e^{2} + 2 \, a^{3} c d^{2} e^{2} - 2 \, a^{2} b c d e^{3} + a^{2} c^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c} e^{2}} + \frac {e x + d}{a e^{3}} + \frac {{\left (2 \, a d + b e\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{a^{2} e^{3}} \]
-d^4*e^3/((a*d^2*e^6 - b*d*e^7 + c*e^8)*(e*x + d)) - 1/2*(b^3*d^2 - 2*a*b* c*d^2 - 2*b^2*c*d*e + 2*a*c^2*d*e + b*c^2*e^2)*log(-a + 2*a*d/(e*x + d) - a*d^2/(e*x + d)^2 - b*e/(e*x + d) + b*d*e/(e*x + d)^2 - c*e^2/(e*x + d)^2) /(a^4*d^4 - 2*a^3*b*d^3*e + a^2*b^2*d^2*e^2 + 2*a^3*c*d^2*e^2 - 2*a^2*b*c* d*e^3 + a^2*c^2*e^4) - (b^4*d^2*e^2 - 4*a*b^2*c*d^2*e^2 + 2*a^2*c^2*d^2*e^ 2 - 2*b^3*c*d*e^3 + 6*a*b*c^2*d*e^3 + b^2*c^2*e^4 - 2*a*c^3*e^4)*arctan(-( 2*a*d - 2*a*d^2/(e*x + d) - b*e + 2*b*d*e/(e*x + d) - 2*c*e^2/(e*x + d))/( sqrt(-b^2 + 4*a*c)*e))/((a^4*d^4 - 2*a^3*b*d^3*e + a^2*b^2*d^2*e^2 + 2*a^3 *c*d^2*e^2 - 2*a^2*b*c*d*e^3 + a^2*c^2*e^4)*sqrt(-b^2 + 4*a*c)*e^2) + (e*x + d)/(a*e^3) + (2*a*d + b*e)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/(a^2* e^3)
Time = 11.20 (sec) , antiderivative size = 2495, normalized size of antiderivative = 9.11 \[ \int \frac {x^2}{\left (a+\frac {c}{x^2}+\frac {b}{x}\right ) (d+e x)^2} \, dx=\text {Too large to display} \]
x/(a*e^2) - (log(d + e*x)*(2*a*d^5 + 4*c*d^3*e^2 - 3*b*d^4*e))/(c^2*e^7 + a^2*d^4*e^3 + b^2*d^2*e^5 - 2*b*c*d*e^6 - 2*a*b*d^3*e^4 + 2*a*c*d^2*e^5) + (log(8*a^4*c*d^7 + b*c^4*e^7 + c^4*e^7*(b^2 - 4*a*c)^(1/2) - 2*a^3*b^2*d^ 7 + b^5*d^4*e^3 + 3*a^2*b^3*d^6*e - 4*b^2*c^3*d*e^6 - 4*b^4*c*d^3*e^4 + b^ 4*d^4*e^3*(b^2 - 4*a*c)^(1/2) - 24*a^2*c^3*d^3*e^4 + 8*a^3*c^2*d^5*e^2 + 6 *b^3*c^2*d^2*e^5 + 8*a*c^4*d*e^6 + 2*a*c^4*e^7*x - 2*a^3*b*d^7*(b^2 - 4*a* c)^(1/2) - 4*a^4*d^7*x*(b^2 - 4*a*c)^(1/2) - 12*a^3*b*c*d^6*e + 17*a^2*c^2 *d^4*e^3*(b^2 - 4*a*c)^(1/2) + 6*b^2*c^2*d^2*e^5*(b^2 - 4*a*c)^(1/2) + 16* a^4*c*d^6*e*x + 8*a^3*c*d^6*e*(b^2 - 4*a*c)^(1/2) - 4*b*c^3*d*e^6*(b^2 - 4 *a*c)^(1/2) - 18*a*b*c^3*d^2*e^5 - 8*a*b^3*c*d^4*e^3 - 2*a*b^4*d^4*e^3*x - 4*a^3*b^2*d^6*e*x + 3*a^2*b^2*d^6*e*(b^2 - 4*a*c)^(1/2) - 6*a*c^3*d^2*e^5 *(b^2 - 4*a*c)^(1/2) - 4*b^3*c*d^3*e^4*(b^2 - 4*a*c)^(1/2) + 20*a*b^2*c^2* d^3*e^4 + 17*a^2*b*c^2*d^4*e^3 - 2*a^2*b^2*c*d^5*e^2 + 8*a^2*b^3*d^5*e^2*x - 12*a^2*c^3*d^2*e^5*x + 34*a^3*c^2*d^4*e^3*x + 4*a*b*c^2*d^3*e^4*(b^2 - 4*a*c)^(1/2) - 18*a^2*b*c*d^5*e^2*(b^2 - 4*a*c)^(1/2) + 4*a*b^3*d^4*e^3*x* (b^2 - 4*a*c)^(1/2) - 4*a^3*c*d^5*e^2*x*(b^2 - 4*a*c)^(1/2) + 6*a*b^2*c^2* d^2*e^5*x - 4*a^2*b*c^2*d^3*e^4*x - 8*a^2*b^2*d^5*e^2*x*(b^2 - 4*a*c)^(1/2 ) - 4*a*b*c^3*d*e^6*x + 12*a^2*c^2*d^3*e^4*x*(b^2 - 4*a*c)^(1/2) + 10*a^3* b*d^6*e*x*(b^2 - 4*a*c)^(1/2) - 4*a*c^3*d*e^6*x*(b^2 - 4*a*c)^(1/2) - 32*a ^3*b*c*d^5*e^2*x + 6*a*b*c^2*d^2*e^5*x*(b^2 - 4*a*c)^(1/2) - 8*a*b^2*c*...