Integrand size = 29, antiderivative size = 105 \[ \int \frac {(e+f x)^3}{a c+(b c+a d) x+b d x^2} \, dx=\frac {f^2 (3 b d e-b c f-a d f) x}{b^2 d^2}+\frac {f^3 x^2}{2 b d}+\frac {(b e-a f)^3 \log (a+b x)}{b^3 (b c-a d)}-\frac {(d e-c f)^3 \log (c+d x)}{d^3 (b c-a d)} \] Output:
f^2*(-a*d*f-b*c*f+3*b*d*e)*x/b^2/d^2+1/2*f^3*x^2/b/d+(-a*f+b*e)^3*ln(b*x+a )/b^3/(-a*d+b*c)-(-c*f+d*e)^3*ln(d*x+c)/d^3/(-a*d+b*c)
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.94 \[ \int \frac {(e+f x)^3}{a c+(b c+a d) x+b d x^2} \, dx=\frac {b d (b c-a d) f^2 x (-2 a d f+b (6 d e-2 c f+d f x))+2 d^3 (b e-a f)^3 \log (a+b x)-2 b^3 (d e-c f)^3 \log (c+d x)}{2 b^3 d^3 (b c-a d)} \] Input:
Integrate[(e + f*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2),x]
Output:
(b*d*(b*c - a*d)*f^2*x*(-2*a*d*f + b*(6*d*e - 2*c*f + d*f*x)) + 2*d^3*(b*e - a*f)^3*Log[a + b*x] - 2*b^3*(d*e - c*f)^3*Log[c + d*x])/(2*b^3*d^3*(b*c - a*d))
Time = 0.32 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {1141, 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.
\(\displaystyle \int \frac {(e+f x)^3}{x (a d+b c)+a c+b d x^2} \, dx\) |
\(\Big \downarrow \) 1141 |
\(\displaystyle b d \int \left (\frac {x f^3}{b^2 d^2}+\frac {(3 b d e-b c f-a d f) f^2}{b^3 d^3}+\frac {(b e-a f)^3}{b^3 d (b c-a d) (a+b x)}-\frac {(d e-c f)^3}{b d^3 (b c-a d) (c+d x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle b d \left (\frac {(b e-a f)^3 \log (a+b x)}{b^4 d (b c-a d)}+\frac {f^2 x (-a d f-b c f+3 b d e)}{b^3 d^3}-\frac {(d e-c f)^3 \log (c+d x)}{b d^4 (b c-a d)}+\frac {f^3 x^2}{2 b^2 d^2}\right )\) |
Input:
Int[(e + f*x)^3/(a*c + (b*c + a*d)*x + b*d*x^2),x]
Output:
b*d*((f^2*(3*b*d*e - b*c*f - a*d*f)*x)/(b^3*d^3) + (f^3*x^2)/(2*b^2*d^2) + ((b*e - a*f)^3*Log[a + b*x])/(b^4*d*(b*c - a*d)) - ((d*e - c*f)^3*Log[c + d*x])/(b*d^4*(b*c - a*d)))
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_ Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[1/c^p Int[ExpandIntegrand[ (d + e*x)^m*(b/2 - q/2 + c*x)^p*(b/2 + q/2 + c*x)^p, x], x], x] /; EqQ[p, - 1] || !FractionalPowerFactorQ[q]] /; FreeQ[{a, b, c, d, e}, x] && ILtQ[p, 0] && IntegerQ[m] && NiceSqrtQ[b^2 - 4*a*c]
Time = 0.99 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.43
method | result | size |
default | \(-\frac {f^{2} \left (-\frac {1}{2} b d f \,x^{2}+a d f x +b c f x -3 b d x e \right )}{b^{2} d^{2}}+\frac {\left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )}+\frac {\left (-c^{3} f^{3}+3 c^{2} d e \,f^{2}-3 c \,d^{2} e^{2} f +d^{3} e^{3}\right ) \ln \left (d x +c \right )}{d^{3} \left (a d -b c \right )}\) | \(150\) |
norman | \(\frac {f^{3} x^{2}}{2 b d}-\frac {f^{2} \left (a d f +b c f -3 b d e \right ) x}{b^{2} d^{2}}+\frac {\left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \ln \left (b x +a \right )}{b^{3} \left (a d -b c \right )}-\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,d^{2} e^{2} f -d^{3} e^{3}\right ) \ln \left (d x +c \right )}{d^{3} \left (a d -b c \right )}\) | \(155\) |
parallelrisch | \(\frac {x^{2} a \,b^{2} d^{3} f^{3}-x^{2} b^{3} c \,d^{2} f^{3}+2 \ln \left (b x +a \right ) a^{3} d^{3} f^{3}-6 \ln \left (b x +a \right ) a^{2} b \,d^{3} e \,f^{2}+6 \ln \left (b x +a \right ) a \,b^{2} d^{3} e^{2} f -2 \ln \left (b x +a \right ) b^{3} d^{3} e^{3}-2 \ln \left (d x +c \right ) b^{3} c^{3} f^{3}+6 \ln \left (d x +c \right ) b^{3} c^{2} d e \,f^{2}-6 \ln \left (d x +c \right ) b^{3} c \,d^{2} e^{2} f +2 \ln \left (d x +c \right ) b^{3} d^{3} e^{3}-2 x \,a^{2} b \,d^{3} f^{3}+6 x a \,b^{2} d^{3} e \,f^{2}+2 x \,b^{3} c^{2} d \,f^{3}-6 x \,b^{3} c \,d^{2} e \,f^{2}}{2 b^{3} d^{3} \left (a d -b c \right )}\) | \(247\) |
risch | \(\frac {f^{3} x^{2}}{2 b d}-\frac {f^{3} a x}{b^{2} d}-\frac {f^{3} c x}{b \,d^{2}}+\frac {3 f^{2} x e}{b d}-\frac {\ln \left (d x +c \right ) c^{3} f^{3}}{d^{3} \left (a d -b c \right )}+\frac {3 \ln \left (d x +c \right ) c^{2} e \,f^{2}}{d^{2} \left (a d -b c \right )}-\frac {3 \ln \left (d x +c \right ) c \,e^{2} f}{d \left (a d -b c \right )}+\frac {\ln \left (d x +c \right ) e^{3}}{a d -b c}+\frac {\ln \left (-b x -a \right ) a^{3} f^{3}}{b^{3} \left (a d -b c \right )}-\frac {3 \ln \left (-b x -a \right ) a^{2} e \,f^{2}}{b^{2} \left (a d -b c \right )}+\frac {3 \ln \left (-b x -a \right ) a \,e^{2} f}{b \left (a d -b c \right )}-\frac {\ln \left (-b x -a \right ) e^{3}}{a d -b c}\) | \(269\) |
Input:
int((f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x,method=_RETURNVERBOSE)
Output:
-f^2/b^2/d^2*(-1/2*b*d*f*x^2+a*d*f*x+b*c*f*x-3*b*d*x*e)+1/b^3*(a^3*f^3-3*a ^2*b*e*f^2+3*a*b^2*e^2*f-b^3*e^3)/(a*d-b*c)*ln(b*x+a)+(-c^3*f^3+3*c^2*d*e* f^2-3*c*d^2*e^2*f+d^3*e^3)/d^3/(a*d-b*c)*ln(d*x+c)
Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (103) = 206\).
Time = 0.14 (sec) , antiderivative size = 211, normalized size of antiderivative = 2.01 \[ \int \frac {(e+f x)^3}{a c+(b c+a d) x+b d x^2} \, dx=\frac {{\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} f^{3} x^{2} + 2 \, {\left (3 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} e f^{2} - {\left (b^{3} c^{2} d - a^{2} b d^{3}\right )} f^{3}\right )} x + 2 \, {\left (b^{3} d^{3} e^{3} - 3 \, a b^{2} d^{3} e^{2} f + 3 \, a^{2} b d^{3} e f^{2} - a^{3} d^{3} f^{3}\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{3} d^{3} e^{3} - 3 \, b^{3} c d^{2} e^{2} f + 3 \, b^{3} c^{2} d e f^{2} - b^{3} c^{3} f^{3}\right )} \log \left (d x + c\right )}{2 \, {\left (b^{4} c d^{3} - a b^{3} d^{4}\right )}} \] Input:
integrate((f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="fricas")
Output:
1/2*((b^3*c*d^2 - a*b^2*d^3)*f^3*x^2 + 2*(3*(b^3*c*d^2 - a*b^2*d^3)*e*f^2 - (b^3*c^2*d - a^2*b*d^3)*f^3)*x + 2*(b^3*d^3*e^3 - 3*a*b^2*d^3*e^2*f + 3* a^2*b*d^3*e*f^2 - a^3*d^3*f^3)*log(b*x + a) - 2*(b^3*d^3*e^3 - 3*b^3*c*d^2 *e^2*f + 3*b^3*c^2*d*e*f^2 - b^3*c^3*f^3)*log(d*x + c))/(b^4*c*d^3 - a*b^3 *d^4)
Timed out. \[ \int \frac {(e+f x)^3}{a c+(b c+a d) x+b d x^2} \, dx=\text {Timed out} \] Input:
integrate((f*x+e)**3/(a*c+(a*d+b*c)*x+b*d*x**2),x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.52 \[ \int \frac {(e+f x)^3}{a c+(b c+a d) x+b d x^2} \, dx=\frac {{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \log \left (b x + a\right )}{b^{4} c - a b^{3} d} - \frac {{\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} \log \left (d x + c\right )}{b c d^{3} - a d^{4}} + \frac {b d f^{3} x^{2} + 2 \, {\left (3 \, b d e f^{2} - {\left (b c + a d\right )} f^{3}\right )} x}{2 \, b^{2} d^{2}} \] Input:
integrate((f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="maxima")
Output:
(b^3*e^3 - 3*a*b^2*e^2*f + 3*a^2*b*e*f^2 - a^3*f^3)*log(b*x + a)/(b^4*c - a*b^3*d) - (d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*log(d*x + c )/(b*c*d^3 - a*d^4) + 1/2*(b*d*f^3*x^2 + 2*(3*b*d*e*f^2 - (b*c + a*d)*f^3) *x)/(b^2*d^2)
Time = 0.94 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.55 \[ \int \frac {(e+f x)^3}{a c+(b c+a d) x+b d x^2} \, dx=\frac {{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \log \left ({\left | b x + a \right |}\right )}{b^{4} c - a b^{3} d} - \frac {{\left (d^{3} e^{3} - 3 \, c d^{2} e^{2} f + 3 \, c^{2} d e f^{2} - c^{3} f^{3}\right )} \log \left ({\left | d x + c \right |}\right )}{b c d^{3} - a d^{4}} + \frac {b d f^{3} x^{2} + 6 \, b d e f^{2} x - 2 \, b c f^{3} x - 2 \, a d f^{3} x}{2 \, b^{2} d^{2}} \] Input:
integrate((f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x, algorithm="giac")
Output:
(b^3*e^3 - 3*a*b^2*e^2*f + 3*a^2*b*e*f^2 - a^3*f^3)*log(abs(b*x + a))/(b^4 *c - a*b^3*d) - (d^3*e^3 - 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 - c^3*f^3)*log(ab s(d*x + c))/(b*c*d^3 - a*d^4) + 1/2*(b*d*f^3*x^2 + 6*b*d*e*f^2*x - 2*b*c*f ^3*x - 2*a*d*f^3*x)/(b^2*d^2)
Time = 6.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04 \[ \int \frac {(e+f x)^3}{a c+(b c+a d) x+b d x^2} \, dx=x\,\left (\frac {3\,e\,f^2}{b\,d}-\frac {f^3\,\left (a\,d+b\,c\right )}{b^2\,d^2}\right )+\frac {f^3\,x^2}{2\,b\,d}+\frac {\ln \left (a+b\,x\right )\,{\left (a\,f-b\,e\right )}^3}{b^3\,\left (a\,d-b\,c\right )}-\frac {\ln \left (c+d\,x\right )\,{\left (c\,f-d\,e\right )}^3}{d^3\,\left (a\,d-b\,c\right )} \] Input:
int((e + f*x)^3/(a*c + x*(a*d + b*c) + b*d*x^2),x)
Output:
x*((3*e*f^2)/(b*d) - (f^3*(a*d + b*c))/(b^2*d^2)) + (f^3*x^2)/(2*b*d) + (l og(a + b*x)*(a*f - b*e)^3)/(b^3*(a*d - b*c)) - (log(c + d*x)*(c*f - d*e)^3 )/(d^3*(a*d - b*c))
Time = 0.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.34 \[ \int \frac {(e+f x)^3}{a c+(b c+a d) x+b d x^2} \, dx=\frac {2 \,\mathrm {log}\left (b x +a \right ) a^{3} d^{3} f^{3}-6 \,\mathrm {log}\left (b x +a \right ) a^{2} b \,d^{3} e \,f^{2}+6 \,\mathrm {log}\left (b x +a \right ) a \,b^{2} d^{3} e^{2} f -2 \,\mathrm {log}\left (b x +a \right ) b^{3} d^{3} e^{3}-2 \,\mathrm {log}\left (d x +c \right ) b^{3} c^{3} f^{3}+6 \,\mathrm {log}\left (d x +c \right ) b^{3} c^{2} d e \,f^{2}-6 \,\mathrm {log}\left (d x +c \right ) b^{3} c \,d^{2} e^{2} f +2 \,\mathrm {log}\left (d x +c \right ) b^{3} d^{3} e^{3}-2 a^{2} b \,d^{3} f^{3} x +6 a \,b^{2} d^{3} e \,f^{2} x +a \,b^{2} d^{3} f^{3} x^{2}+2 b^{3} c^{2} d \,f^{3} x -6 b^{3} c \,d^{2} e \,f^{2} x -b^{3} c \,d^{2} f^{3} x^{2}}{2 b^{3} d^{3} \left (a d -b c \right )} \] Input:
int((f*x+e)^3/(a*c+(a*d+b*c)*x+b*d*x^2),x)
Output:
(2*log(a + b*x)*a**3*d**3*f**3 - 6*log(a + b*x)*a**2*b*d**3*e*f**2 + 6*log (a + b*x)*a*b**2*d**3*e**2*f - 2*log(a + b*x)*b**3*d**3*e**3 - 2*log(c + d *x)*b**3*c**3*f**3 + 6*log(c + d*x)*b**3*c**2*d*e*f**2 - 6*log(c + d*x)*b* *3*c*d**2*e**2*f + 2*log(c + d*x)*b**3*d**3*e**3 - 2*a**2*b*d**3*f**3*x + 6*a*b**2*d**3*e*f**2*x + a*b**2*d**3*f**3*x**2 + 2*b**3*c**2*d*f**3*x - 6* b**3*c*d**2*e*f**2*x - b**3*c*d**2*f**3*x**2)/(2*b**3*d**3*(a*d - b*c))