Integrand size = 20, antiderivative size = 186 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=-\frac {e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}+\frac {e (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}-\frac {e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2} \] Output:
-e/(a*e^2-b*d*e+c*d^2)/(e*x+d)-(2*c^2*d^2+b^2*e^2-2*c*e*(a*e+b*d))*arctanh ((2*c*x+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2)/(a*e^2-b*d*e+c*d^2)^2+e* (-b*e+2*c*d)*ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^2-1/2*e*(-b*e+2*c*d)*ln(c*x^2+b *x+a)/(a*e^2-b*d*e+c*d^2)^2
Time = 0.14 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\frac {-\frac {2 e \left (c d^2+e (-b d+a e)\right )}{d+e x}+\frac {2 \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-2 e (-2 c d+b e) \log (d+e x)+e (-2 c d+b e) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^2} \] Input:
Integrate[1/((d + e*x)^2*(a + b*x + c*x^2)),x]
Output:
((-2*e*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x) + (2*(2*c^2*d^2 + b^2*e^2 - 2 *c*e*(b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a* c] - 2*e*(-2*c*d + b*e)*Log[d + e*x] + e*(-2*c*d + b*e)*Log[a + x*(b + c*x )])/(2*(c*d^2 + e*(-(b*d) + a*e))^2)
Time = 0.47 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1145, 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.
\(\displaystyle \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx\) |
\(\Big \downarrow \) 1145 |
\(\displaystyle \frac {\int \frac {c d-b e-c e x}{(d+e x) \left (c x^2+b x+a\right )}dx}{a e^2-b d e+c d^2}-\frac {e}{(d+e x) \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \frac {\int \left (\frac {c^2 d^2+b^2 e^2-c e (2 b d+a e)-c e (2 c d-b e) x}{\left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )}-\frac {e^2 (b e-2 c d)}{\left (c d^2-b e d+a e^2\right ) (d+e x)}\right )dx}{a e^2-b d e+c d^2}-\frac {e}{(d+e x) \left (a e^2-b d e+c d^2\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {e (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}+\frac {e (2 c d-b e) \log (d+e x)}{a e^2-b d e+c d^2}}{a e^2-b d e+c d^2}-\frac {e}{(d+e x) \left (a e^2-b d e+c d^2\right )}\) |
Input:
Int[1/((d + e*x)^2*(a + b*x + c*x^2)),x]
Output:
-(e/((c*d^2 - b*d*e + a*e^2)*(d + e*x))) + (-(((2*c^2*d^2 + b^2*e^2 - 2*c* e*(b*d + a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^2 - 4*a*c]* (c*d^2 - b*d*e + a*e^2))) + (e*(2*c*d - b*e)*Log[d + e*x])/(c*d^2 - b*d*e + a*e^2) - (e*(2*c*d - b*e)*Log[a + b*x + c*x^2])/(2*(c*d^2 - b*d*e + a*e^ 2)))/(c*d^2 - b*d*e + a*e^2)
Int[((d_.) + (e_.)*(x_))^(m_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp [1/(c*d^2 - b*d*e + a*e^2) Int[(d + e*x)^(m + 1)*(Simp[c*d - b*e - c*e*x, x]/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[m, -1]
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]
Time = 1.07 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {e}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )}-\frac {e \left (b e -2 c d \right ) \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {\frac {\left (b c \,e^{2}-2 c^{2} d e \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a c \,e^{2}+b^{2} e^{2}-2 b c d e +c^{2} d^{2}-\frac {\left (b c \,e^{2}-2 c^{2} d e \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}\) | \(197\) |
risch | \(\text {Expression too large to display}\) | \(12892\) |
Input:
int(1/(e*x+d)^2/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
-e/(a*e^2-b*d*e+c*d^2)/(e*x+d)-e*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+ d)+1/(a*e^2-b*d*e+c*d^2)^2*(1/2*(b*c*e^2-2*c^2*d*e)/c*ln(c*x^2+b*x+a)+2*(- a*c*e^2+b^2*e^2-2*b*c*d*e+c^2*d^2-1/2*(b*c*e^2-2*c^2*d*e)*b/c)/(4*a*c-b^2) ^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 530 vs. \(2 (180) = 360\).
Time = 1.64 (sec) , antiderivative size = 1079, normalized size of antiderivative = 5.80 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[-1/2*(2*(b^2*c - 4*a*c^2)*d^2*e - 2*(b^3 - 4*a*b*c)*d*e^2 + 2*(a*b^2 - 4* a^2*c)*e^3 + (2*c^2*d^3 - 2*b*c*d^2*e + (b^2 - 2*a*c)*d*e^2 + (2*c^2*d^2*e - 2*b*c*d*e^2 + (b^2 - 2*a*c)*e^3)*x)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + (2*(b^2*c - 4*a*c^2)*d^2*e - (b^3 - 4*a*b*c)*d*e^2 + (2*(b^2*c - 4*a*c^2 )*d*e^2 - (b^3 - 4*a*b*c)*e^3)*x)*log(c*x^2 + b*x + a) - 2*(2*(b^2*c - 4*a *c^2)*d^2*e - (b^3 - 4*a*b*c)*d*e^2 + (2*(b^2*c - 4*a*c^2)*d*e^2 - (b^3 - 4*a*b*c)*e^3)*x)*log(e*x + d))/((b^2*c^2 - 4*a*c^3)*d^5 - 2*(b^3*c - 4*a*b *c^2)*d^4*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^3*e^2 - 2*(a*b^3 - 4*a^2*b*c )*d^2*e^3 + (a^2*b^2 - 4*a^3*c)*d*e^4 + ((b^2*c^2 - 4*a*c^3)*d^4*e - 2*(b^ 3*c - 4*a*b*c^2)*d^3*e^2 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^2*e^3 - 2*(a*b^ 3 - 4*a^2*b*c)*d*e^4 + (a^2*b^2 - 4*a^3*c)*e^5)*x), -1/2*(2*(b^2*c - 4*a*c ^2)*d^2*e - 2*(b^3 - 4*a*b*c)*d*e^2 + 2*(a*b^2 - 4*a^2*c)*e^3 + 2*(2*c^2*d ^3 - 2*b*c*d^2*e + (b^2 - 2*a*c)*d*e^2 + (2*c^2*d^2*e - 2*b*c*d*e^2 + (b^2 - 2*a*c)*e^3)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b )/(b^2 - 4*a*c)) + (2*(b^2*c - 4*a*c^2)*d^2*e - (b^3 - 4*a*b*c)*d*e^2 + (2 *(b^2*c - 4*a*c^2)*d*e^2 - (b^3 - 4*a*b*c)*e^3)*x)*log(c*x^2 + b*x + a) - 2*(2*(b^2*c - 4*a*c^2)*d^2*e - (b^3 - 4*a*b*c)*d*e^2 + (2*(b^2*c - 4*a*c^2 )*d*e^2 - (b^3 - 4*a*b*c)*e^3)*x)*log(e*x + d))/((b^2*c^2 - 4*a*c^3)*d^5 - 2*(b^3*c - 4*a*b*c^2)*d^4*e + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*d^3*e^2 - ...
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+d)**2/(c*x**2+b*x+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x+a),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 = 0.38 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=-\frac {e^{3}}{{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} {\left (e x + d\right )}} - \frac {{\left (2 \, c d e - b e^{2}\right )} \log \left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}{2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} + \frac {{\left (2 \, c^{2} d^{2} e^{2} - 2 \, b c d e^{3} + b^{2} e^{4} - 2 \, a c e^{4}\right )} \arctan \left (\frac {2 \, c d - \frac {2 \, c d^{2}}{e x + d} - b e + \frac {2 \, b d e}{e x + d} - \frac {2 \, a e^{2}}{e x + d}}{\sqrt {-b^{2} + 4 \, a c} e}\right )}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c} e^{2}} \] Input:
integrate(1/(e*x+d)^2/(c*x^2+b*x+a),x, algorithm="giac")
Output:
-e^3/((c*d^2*e^2 - b*d*e^3 + a*e^4)*(e*x + d)) - 1/2*(2*c*d*e - b*e^2)*log (c - 2*c*d/(e*x + d) + c*d^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d) ^2 + a*e^2/(e*x + d)^2)/(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e ^2 - 2*a*b*d*e^3 + a^2*e^4) + (2*c^2*d^2*e^2 - 2*b*c*d*e^3 + b^2*e^4 - 2*a *c*e^4)*arctan((2*c*d - 2*c*d^2/(e*x + d) - b*e + 2*b*d*e/(e*x + d) - 2*a* e^2/(e*x + d))/(sqrt(-b^2 + 4*a*c)*e))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e ^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-b^2 + 4*a*c)*e^2)
Time = 9.81 (sec) , antiderivative size = 1786, normalized size of antiderivative = 9.60 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
int(1/((d + e*x)^2*(a + b*x + c*x^2)),x)
Output:
(log(a*e^4*(b^2 - 4*a*c)^(5/2) + 8*a*b^5*e^4 + 8*b^6*e^4*x - 4*c^3*d^4*(b^ 2 - 4*a*c)^(3/2) + 4*b^3*c^3*d^4 + 4*b^3*e^4*x*(b^2 - 4*a*c)^(3/2) - 60*a^ 2*b^3*c*e^4 + 112*a^3*b*c^2*e^4 + 256*a^2*c^4*d^3*e - 256*a^3*c^3*d*e^3 + 8*b^4*c^2*d^3*e - 4*b^5*c*d^2*e^2 - 32*a^3*c^3*e^4*x + 8*b^2*c^4*d^4*x + 1 0*b*d*e^3*(b^2 - 4*a*c)^(5/2) + 4*b*e^4*x*(b^2 - 4*a*c)^(5/2) - 16*a*b*c^4 *d^4 - 32*a*c^5*d^4*x + 7*a*b^2*e^4*(b^2 - 4*a*c)^(3/2) - 10*b^3*d*e^3*(b^ 2 - 4*a*c)^(3/2) - 14*c*d^2*e^2*(b^2 - 4*a*c)^(5/2) - 8*c*d*e^3*x*(b^2 - 4 *a*c)^(5/2) - 24*a*b^4*c*d*e^3 - 64*a*b^4*c*e^4*x - 32*b^5*c*d*e^3*x - 8*b *c^2*d^3*e*(b^2 - 4*a*c)^(3/2) - 32*c^3*d^3*e*x*(b^2 - 4*a*c)^(3/2) - 96*a *b^2*c^3*d^3*e - 16*b^3*c^3*d^3*e*x + 18*b^2*c*d^2*e^2*(b^2 - 4*a*c)^(3/2) + 56*a*b^3*c^2*d^2*e^2 - 160*a^2*b*c^3*d^2*e^2 + 160*a^2*b^2*c^2*d*e^3 + 136*a^2*b^2*c^2*e^4*x + 448*a^2*c^4*d^2*e^2*x + 40*b^4*c^2*d^2*e^2*x + 48* b*c^2*d^2*e^2*x*(b^2 - 4*a*c)^(3/2) - 272*a*b^2*c^3*d^2*e^2*x + 64*a*b*c^4 *d^3*e*x - 24*b^2*c*d*e^3*x*(b^2 - 4*a*c)^(3/2) + 240*a*b^3*c^2*d*e^3*x - 448*a^2*b*c^3*d*e^3*x)*(a*(e^2*(2*b*c + c*(b^2 - 4*a*c)^(1/2)) - 4*c^2*d*e ) + e*(b^2*c*d + b*c*d*(b^2 - 4*a*c)^(1/2)) - e^2*(b^3/2 + (b^2*(b^2 - 4*a *c)^(1/2))/2) - c^2*d^2*(b^2 - 4*a*c)^(1/2)))/(4*a*c^3*d^4 + 4*a^3*c*e^4 - a^2*b^2*e^4 - b^2*c^2*d^4 - b^4*d^2*e^2 + 8*a^2*c^2*d^2*e^2 + 2*a*b^3*d*e ^3 + 2*b^3*c*d^3*e - 8*a*b*c^2*d^3*e - 8*a^2*b*c*d*e^3 + 2*a*b^2*c*d^2*e^2 ) + (log(a*e^4*(b^2 - 4*a*c)^(5/2) - 8*a*b^5*e^4 - 8*b^6*e^4*x - 4*c^3*...
Time = 0.20 (sec) , antiderivative size = 983, normalized size of antiderivative = 5.28 \[ \int \frac {1}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:
int(1/(e*x+d)^2/(c*x^2+b*x+a),x)
Output:
( - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c*d**2*e** 2 - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c*d*e**3*x + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*d**2*e** 2 + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b**2*d*e**3* x - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c*d**3*e - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b*c*d**2*e**2*x + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*c**2*d**4 + 4 *sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*c**2*d**3*e*x + 4 *log(a + b*x + c*x**2)*a*b*c*d**2*e**2 + 4*log(a + b*x + c*x**2)*a*b*c*d*e **3*x - 8*log(a + b*x + c*x**2)*a*c**2*d**3*e - 8*log(a + b*x + c*x**2)*a* c**2*d**2*e**2*x - log(a + b*x + c*x**2)*b**3*d**2*e**2 - log(a + b*x + c* x**2)*b**3*d*e**3*x + 2*log(a + b*x + c*x**2)*b**2*c*d**3*e + 2*log(a + b* x + c*x**2)*b**2*c*d**2*e**2*x - 8*log(d + e*x)*a*b*c*d**2*e**2 - 8*log(d + e*x)*a*b*c*d*e**3*x + 16*log(d + e*x)*a*c**2*d**3*e + 16*log(d + e*x)*a* c**2*d**2*e**2*x + 2*log(d + e*x)*b**3*d**2*e**2 + 2*log(d + e*x)*b**3*d*e **3*x - 4*log(d + e*x)*b**2*c*d**3*e - 4*log(d + e*x)*b**2*c*d**2*e**2*x + 8*a**2*c*e**4*x - 2*a*b**2*e**4*x - 8*a*b*c*d*e**3*x + 8*a*c**2*d**2*e**2 *x + 2*b**3*d*e**3*x - 2*b**2*c*d**2*e**2*x)/(2*d*(4*a**3*c*d*e**4 + 4*a** 3*c*e**5*x - a**2*b**2*d*e**4 - a**2*b**2*e**5*x - 8*a**2*b*c*d**2*e**3 - 8*a**2*b*c*d*e**4*x + 8*a**2*c**2*d**3*e**2 + 8*a**2*c**2*d**2*e**3*x +...