Integrand size = 25, antiderivative size = 249 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=-\frac {e f-d g}{\left (c d^2-b d e+a e^2\right ) (d+e x)}-\frac {\left (2 c^2 d^2 f+b e^2 (b f-a g)-c (2 a e (e f-2 d g)+b d (2 e f+d g))\right ) \text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2}-\frac {\left (e^2 (b f-a g)-c d (2 e f-d g)\right ) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^2}+\frac {\left (e^2 (b f-a g)-c d (2 e f-d g)\right ) \log \left (a+b x+c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )^2} \] Output:
-(-d*g+e*f)/(a*e^2-b*d*e+c*d^2)/(e*x+d)-(2*c^2*d^2*f+b*e^2*(-a*g+b*f)-c*(2 *a*e*(-2*d*g+e*f)+b*d*(d*g+2*e*f)))*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^2*(-a*g+b*f)-c*d*(-d*g+2*e*f)) *ln(e*x+d)/(a*e^2-b*d*e+c*d^2)^2+1/2*(e^2*(-a*g+b*f)-c*d*(-d*g+2*e*f))*ln( c*x^2+b*x+a)/(a*e^2-b*d*e+c*d^2)^2
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.85 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\frac {\frac {2 \left (c d^2+e (-b d+a e)\right ) (-e f+d g)}{d+e x}+\frac {2 \left (2 c^2 d^2 f+b e^2 (b f-a g)-c (2 a e (e f-2 d g)+b d (2 e f+d g))\right ) \arctan \left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}-2 \left (e^2 (b f-a g)+c d (-2 e f+d g)\right ) \log (d+e x)+\left (e^2 (b f-a g)+c d (-2 e f+d g)\right ) \log (a+x (b+c x))}{2 \left (c d^2+e (-b d+a e)\right )^2} \] Input:
Integrate[(f + g*x)/((d + e*x)^2*(a + b*x + c*x^2)),x]
Output:
((2*(c*d^2 + e*(-(b*d) + a*e))*(-(e*f) + d*g))/(d + e*x) + (2*(2*c^2*d^2*f + b*e^2*(b*f - a*g) - c*(2*a*e*(e*f - 2*d*g) + b*d*(2*e*f + d*g)))*ArcTan [(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] - 2*(e^2*(b*f - a*g) + c*d*(-2*e*f + d*g))*Log[d + e*x] + (e^2*(b*f - a*g) + c*d*(-2*e*f + d*g) )*Log[a + x*(b + c*x)])/(2*(c*d^2 + e*(-(b*d) + a*e))^2)
Time = 1.10 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {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.
\(\displaystyle \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx\) |
\(\Big \downarrow \) 1200 |
\(\displaystyle \int \left (\frac {c x \left (e^2 (b f-a g)-c d (2 e f-d g)\right )-c e (-2 a d g+a e f+2 b d f)+b e^2 (b f-a g)+c^2 d^2 f}{\left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {e (e f-d g)}{(d+e x)^2 \left (a e^2-b d e+c d^2\right )}+\frac {e \left (c d (2 e f-d g)-e^2 (b f-a g)\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\text {arctanh}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right ) \left (-c (2 a e (e f-2 d g)+b d (d g+2 e f))+b e^2 (b f-a g)+2 c^2 d^2 f\right )}{\sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )^2}+\frac {\log \left (a+b x+c x^2\right ) \left (e^2 (b f-a g)-c d (2 e f-d g)\right )}{2 \left (a e^2-b d e+c d^2\right )^2}-\frac {e f-d g}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {\log (d+e x) \left (e^2 (b f-a g)-c d (2 e f-d g)\right )}{\left (a e^2-b d e+c d^2\right )^2}\) |
Input:
Int[(f + g*x)/((d + e*x)^2*(a + b*x + c*x^2)),x]
Output:
-((e*f - d*g)/((c*d^2 - b*d*e + a*e^2)*(d + e*x))) - ((2*c^2*d^2*f + b*e^2 *(b*f - a*g) - c*(2*a*e*(e*f - 2*d*g) + b*d*(2*e*f + d*g)))*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)^2) - ((e^2*(b*f - a*g) - c*d*(2*e*f - d*g))*Log[d + e*x])/(c*d^2 - b*d*e + a*e^ 2)^2 + ((e^2*(b*f - a*g) - c*d*(2*e*f - d*g))*Log[a + b*x + c*x^2])/(2*(c* d^2 - b*d*e + a*e^2)^2)
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 = 2.07 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\left (a \,e^{2} g -b \,e^{2} f -c \,d^{2} g +2 c d e f \right ) \ln \left (e x +d \right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {d g -e f}{\left (a \,e^{2}-b d e +c \,d^{2}\right ) \left (e x +d \right )}+\frac {\frac {\left (-a c \,e^{2} g +b c \,e^{2} f +c^{2} d^{2} g -2 c^{2} d e f \right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-a b \,e^{2} g +2 a c d e g -a c \,e^{2} f +b^{2} e^{2} f -2 b c d e f +c^{2} d^{2} f -\frac {\left (-a c \,e^{2} g +b c \,e^{2} f +c^{2} d^{2} g -2 c^{2} d e f \right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}\) | \(275\) |
risch | \(\text {Expression too large to display}\) | \(63095\) |
Input:
int((g*x+f)/(e*x+d)^2/(c*x^2+b*x+a),x,method=_RETURNVERBOSE)
Output:
(a*e^2*g-b*e^2*f-c*d^2*g+2*c*d*e*f)/(a*e^2-b*d*e+c*d^2)^2*ln(e*x+d)+(d*g-e *f)/(a*e^2-b*d*e+c*d^2)/(e*x+d)+1/(a*e^2-b*d*e+c*d^2)^2*(1/2*(-a*c*e^2*g+b *c*e^2*f+c^2*d^2*g-2*c^2*d*e*f)/c*ln(c*x^2+b*x+a)+2*(-a*b*e^2*g+2*a*c*d*e* g-a*c*e^2*f+b^2*e^2*f-2*b*c*d*e*f+c^2*d^2*f-1/2*(-a*c*e^2*g+b*c*e^2*f+c^2* d^2*g-2*c^2*d*e*f)*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. 804 vs. \(2 (242) = 484\).
Time = 51.21 (sec) , antiderivative size = 1627, normalized size of antiderivative = 6.53 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
integrate((g*x+f)/(e*x+d)^2/(c*x^2+b*x+a),x, algorithm="fricas")
Output:
[-1/2*(sqrt(b^2 - 4*a*c)*((2*c^2*d^3 - 2*b*c*d^2*e + (b^2 - 2*a*c)*d*e^2)* f - (b*c*d^3 - 4*a*c*d^2*e + a*b*d*e^2)*g + ((2*c^2*d^2*e - 2*b*c*d*e^2 + (b^2 - 2*a*c)*e^3)*f - (b*c*d^2*e - 4*a*c*d*e^2 + a*b*e^3)*g)*x)*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 + (a*b^2 - 4* a^2*c)*e^3)*f - 2*((b^2*c - 4*a*c^2)*d^3 - (b^3 - 4*a*b*c)*d^2*e + (a*b^2 - 4*a^2*c)*d*e^2)*g + ((2*(b^2*c - 4*a*c^2)*d^2*e - (b^3 - 4*a*b*c)*d*e^2) *f - ((b^2*c - 4*a*c^2)*d^3 - (a*b^2 - 4*a^2*c)*d*e^2)*g + ((2*(b^2*c - 4* a*c^2)*d*e^2 - (b^3 - 4*a*b*c)*e^3)*f - ((b^2*c - 4*a*c^2)*d^2*e - (a*b^2 - 4*a^2*c)*e^3)*g)*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)*f - ((b^2*c - 4*a*c^2)*d^3 - (a*b^2 - 4*a^2*c)*d *e^2)*g + ((2*(b^2*c - 4*a*c^2)*d*e^2 - (b^3 - 4*a*b*c)*e^3)*f - ((b^2*c - 4*a*c^2)*d^2*e - (a*b^2 - 4*a^2*c)*e^3)*g)*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*sqrt(-b^2 + 4*a*c)*((2*c^2*d^3 - 2*b*c*d^2*e + (b^2 - 2*a* c)*d*e^2)*f - (b*c*d^3 - 4*a*c*d^2*e + a*b*d*e^2)*g + ((2*c^2*d^2*e - 2*b* c*d*e^2 + (b^2 - 2*a*c)*e^3)*f - (b*c*d^2*e - 4*a*c*d*e^2 + a*b*e^3)*g)...
Timed out. \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\text {Timed out} \] Input:
integrate((g*x+f)/(e*x+d)**2/(c*x**2+b*x+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((g*x+f)/(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 = 0.21 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.58 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=-\frac {{\left (2 \, c d e f - b e^{2} f - c d^{2} g + a e^{2} g\right )} \log \left (c - \frac {2 \, c d}{e x + d} + \frac {c d^{2}}{{\left (e x + d\right )}^{2}} + \frac {b e}{e x + d} - \frac {b d e}{{\left (e x + d\right )}^{2}} + \frac {a e^{2}}{{\left (e x + d\right )}^{2}}\right )}{2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} - \frac {\frac {e^{3} f}{e x + d} - \frac {d e^{2} g}{e x + d}}{c d^{2} e^{2} - b d e^{3} + a e^{4}} + \frac {{\left (2 \, c^{2} d^{2} e^{2} f - 2 \, b c d e^{3} f + b^{2} e^{4} f - 2 \, a c e^{4} f - b c d^{2} e^{2} g + 4 \, a c d e^{3} g - a b e^{4} g\right )} \arctan \left (\frac {2 \, c d - \frac {2 \, c d^{2}}{e x + d} - b e + \frac {2 \, b d e}{e x + d} - \frac {2 \, a e^{2}}{e x + d}}{\sqrt {-b^{2} + 4 \, a c} e}\right )}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c} e^{2}} \] Input:
integrate((g*x+f)/(e*x+d)^2/(c*x^2+b*x+a),x, algorithm="giac")
Output:
-1/2*(2*c*d*e*f - b*e^2*f - c*d^2*g + a*e^2*g)*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) - (e^3*f/(e*x + d) - d*e^2*g/(e*x + d))/(c*d^2*e^2 - b*d*e^3 + a*e^4) + (2*c^2*d^2*e^2*f - 2*b*c*d*e^3*f + b^2*e^4*f - 2*a*c*e^4*f - b*c*d^2*e^2 *g + 4*a*c*d*e^3*g - a*b*e^4*g)*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 = 16.95 (sec) , antiderivative size = 2649, normalized size of antiderivative = 10.64 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:
int((f + g*x)/((d + e*x)^2*(a + b*x + c*x^2)),x)
Output:
(log(d + e*x)*(e^2*(a*g - b*f) - c*d^2*g + 2*c*d*e*f))/(a^2*e^4 + c^2*d^4 + b^2*d^2*e^2 - 2*a*b*d*e^3 - 2*b*c*d^3*e + 2*a*c*d^2*e^2) - (log(2*a*b^3* e^4*f - 2*b^2*c^2*d^4*g - 2*a^2*b^2*e^4*g + 6*a*c^3*d^4*g + b*c^3*d^4*f + 6*a^3*c*e^4*g + 2*b^4*e^4*f*x + 2*c^4*d^4*f*x + c^3*d^4*f*(b^2 - 4*a*c)^(1 /2) - 7*a^2*b*c*e^4*f - 16*a*c^3*d^3*e*f - 2*a*b^3*e^4*g*x - b*c^3*d^4*g*x - 2*a*b^2*e^4*f*(b^2 - 4*a*c)^(1/2) + 2*a^2*b*e^4*g*(b^2 - 4*a*c)^(1/2) + a^2*c*e^4*f*(b^2 - 4*a*c)^(1/2) - 2*b*c^2*d^4*g*(b^2 - 4*a*c)^(1/2) - 2*b ^3*e^4*f*x*(b^2 - 4*a*c)^(1/2) - 3*c^3*d^4*g*x*(b^2 - 4*a*c)^(1/2) + 16*a^ 2*c^2*d*e^3*f + 2*b^2*c^2*d^3*e*f - b^3*c*d^2*e^2*f + 2*a^2*c^2*e^4*f*x - 20*a^2*c^2*d^2*e^2*g - 14*a*c^2*d^2*e^2*f*(b^2 - 4*a*c)^(1/2) - b^2*c*d^2* e^2*f*(b^2 - 4*a*c)^(1/2) + 10*b^2*c^2*d^2*e^2*f*x - 6*a*b^2*c*d*e^3*f + 4 *a*b*c^2*d^3*e*g + 4*a^2*b*c*d*e^3*g - 8*a*b^2*c*e^4*f*x + 7*a^2*b*c*e^4*g *x + 16*a*c^3*d^3*e*g*x - 4*b*c^3*d^3*e*f*x - 8*b^3*c*d*e^3*f*x + 8*a*c^2* d^3*e*g*(b^2 - 4*a*c)^(1/2) + 2*b*c^2*d^3*e*f*(b^2 - 4*a*c)^(1/2) - 8*a^2* c*d*e^3*g*(b^2 - 4*a*c)^(1/2) + 2*a*b^2*e^4*g*x*(b^2 - 4*a*c)^(1/2) - 3*a^ 2*c*e^4*g*x*(b^2 - 4*a*c)^(1/2) + 8*c^3*d^3*e*f*x*(b^2 - 4*a*c)^(1/2) + 10 *a*b*c^2*d^2*e^2*f + 2*a*b^2*c*d^2*e^2*g - 28*a*c^3*d^2*e^2*f*x - 16*a^2*c ^2*d*e^3*g*x - 2*b^2*c^2*d^3*e*g*x + b^3*c*d^2*e^2*g*x - 8*a*c^2*d*e^3*f*x *(b^2 - 4*a*c)^(1/2) + 8*b^2*c*d*e^3*f*x*(b^2 - 4*a*c)^(1/2) + 2*b*c^2*d^3 *e*g*x*(b^2 - 4*a*c)^(1/2) - 10*a*b*c^2*d^2*e^2*g*x + 10*a*c^2*d^2*e^2*...
Time = 0.26 (sec) , antiderivative size = 1661, normalized size of antiderivative = 6.67 \[ \int \frac {f+g x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:
int((g*x+f)/(e*x+d)^2/(c*x^2+b*x+a),x)
Output:
( - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*d**2*e** 2*g - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*b*d*e**3 *g*x + 8*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c*d**3* e*g - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c*d**2*e **2*f + 8*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c*d**2 *e**2*g*x - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*a*c* d*e**3*f*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*f + 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))* b**2*d*e**3*f*x - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2) )*b*c*d**4*g - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b *c*d**3*e*f - 2*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b* c*d**3*e*g*x - 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2))*b *c*d**2*e**2*f*x + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2 ))*c**2*d**4*f + 4*sqrt(4*a*c - b**2)*atan((b + 2*c*x)/sqrt(4*a*c - b**2)) *c**2*d**3*e*f*x - 4*log(a + b*x + c*x**2)*a**2*c*d**2*e**2*g - 4*log(a + b*x + c*x**2)*a**2*c*d*e**3*g*x + log(a + b*x + c*x**2)*a*b**2*d**2*e**2*g + log(a + b*x + c*x**2)*a*b**2*d*e**3*g*x + 4*log(a + b*x + c*x**2)*a*b*c *d**2*e**2*f + 4*log(a + b*x + c*x**2)*a*b*c*d*e**3*f*x + 4*log(a + b*x + c*x**2)*a*c**2*d**4*g - 8*log(a + b*x + c*x**2)*a*c**2*d**3*e*f + 4*log(a + b*x + c*x**2)*a*c**2*d**3*e*g*x - 8*log(a + b*x + c*x**2)*a*c**2*d**2...