\(\int (f+g x)^2 (A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})) \, dx\) [264]

Optimal result

Integrand size = 29, antiderivative size = 152 \[ \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=-\frac {2 B (b c-a d) g (3 b d f-b c g-a d g) x}{3 b^2 d^2}-\frac {B (b c-a d) g^2 x^2}{3 b d}-\frac {2 B (b f-a g)^3 \log (a+b x)}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 g}+\frac {2 B (d f-c g)^3 \log (c+d x)}{3 d^3 g} \]

[Out]

-2/3*B*(-a*d+b*c)*g*(-a*d*g-b*c*g+3*b*d*f)*x/b^2/d^2-1/3*B*(-a*d+b*c)*g^2*x^2/b/d-2/3*B*(-a*g+b*f)^3*ln(b*x+a)
/b^3/g+1/3*(g*x+f)^3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/g+2/3*B*(-c*g+d*f)^3*ln(d*x+c)/d^3/g

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2548, 84} \[ \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {(f+g x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 g}-\frac {2 B (b f-a g)^3 \log (a+b x)}{3 b^3 g}-\frac {2 B g x (b c-a d) (-a d g-b c g+3 b d f)}{3 b^2 d^2}-\frac {B g^2 x^2 (b c-a d)}{3 b d}+\frac {2 B (d f-c g)^3 \log (c+d x)}{3 d^3 g} \]

[In]

Int[(f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]),x]

[Out]

(-2*B*(b*c - a*d)*g*(3*b*d*f - b*c*g - a*d*g)*x)/(3*b^2*d^2) - (B*(b*c - a*d)*g^2*x^2)/(3*b*d) - (2*B*(b*f - a
*g)^3*Log[a + b*x])/(3*b^3*g) + ((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*g) + (2*B*(d*f - c*g
)^3*Log[c + d*x])/(3*d^3*g)

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = \frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 g}-\frac {(2 B (b c-a d)) \int \frac {(f+g x)^3}{(a+b x) (c+d x)} \, dx}{3 g} \\ & = \frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 g}-\frac {(2 B (b c-a d)) \int \left (\frac {g^2 (3 b d f-b c g-a d g)}{b^2 d^2}+\frac {g^3 x}{b d}+\frac {(b f-a g)^3}{b^2 (b c-a d) (a+b x)}+\frac {(d f-c g)^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx}{3 g} \\ & = -\frac {2 B (b c-a d) g (3 b d f-b c g-a d g) x}{3 b^2 d^2}-\frac {B (b c-a d) g^2 x^2}{3 b d}-\frac {2 B (b f-a g)^3 \log (a+b x)}{3 b^3 g}+\frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 g}+\frac {2 B (d f-c g)^3 \log (c+d x)}{3 d^3 g} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.93 \[ \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {(f+g x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-\frac {B \left (2 b d (b c-a d) g^2 (3 b d f-b c g-a d g) x+b^2 d^2 (b c-a d) g^3 x^2+2 d^3 (b f-a g)^3 \log (a+b x)-2 b^3 (d f-c g)^3 \log (c+d x)\right )}{b^3 d^3}}{3 g} \]

[In]

Integrate[(f + g*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]),x]

[Out]

((f + g*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) - (B*(2*b*d*(b*c - a*d)*g^2*(3*b*d*f - b*c*g - a*d*g)*x
+ b^2*d^2*(b*c - a*d)*g^3*x^2 + 2*d^3*(b*f - a*g)^3*Log[a + b*x] - 2*b^3*(d*f - c*g)^3*Log[c + d*x]))/(b^3*d^3
))/(3*g)

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.78

method	result	size
risch	\(\frac {\left (g x +f \right )^{3} B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{3 g}+\frac {g^{2} A \,x^{3}}{3}+g A f \,x^{2}+\frac {g^{2} B a \,x^{2}}{3 b}-\frac {g^{2} B c \,x^{2}}{3 d}+A \,f^{2} x -\frac {2 g^{2} B \ln \left (d x +c \right ) c^{3}}{3 d^{3}}+\frac {2 g B \ln \left (d x +c \right ) c^{2} f}{d^{2}}-\frac {2 B \ln \left (d x +c \right ) c \,f^{2}}{d}+\frac {2 B \ln \left (d x +c \right ) f^{3}}{3 g}+\frac {2 g^{2} B \ln \left (-b x -a \right ) a^{3}}{3 b^{3}}-\frac {2 g B \ln \left (-b x -a \right ) a^{2} f}{b^{2}}+\frac {2 B \ln \left (-b x -a \right ) a \,f^{2}}{b}-\frac {2 B \ln \left (-b x -a \right ) f^{3}}{3 g}-\frac {2 g^{2} B \,a^{2} x}{3 b^{2}}+\frac {2 g B a f x}{b}+\frac {2 g^{2} B \,c^{2} x}{3 d^{2}}-\frac {2 g B c f x}{d}\)	\(270\)
parts	\(\frac {A \left (g x +f \right )^{3}}{3 g}-\frac {B \left (\left (-\left (d x +c \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (-2 a d +2 c b \right ) \left (\frac {\ln \left (\frac {1}{d x +c}\right )}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b \left (a d -c b \right )}\right )\right ) \left (g^{2} c^{2}-2 g f d c +f^{2} d^{2}\right )-2 \left (-\frac {\left (d x +c \right )^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2}-\left (-a d +c b \right ) \left (-\frac {d x +c}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{d x +c}\right )}{b^{2}}+\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{2}}\right )\right ) g \left (c g -d f \right )+\left (-\frac {\left (d x +c \right )^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{3}-\left (-\frac {2 a d}{3}+\frac {2 c b}{3}\right ) \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {1}{d x +c}\right )}{b^{3}}-\frac {\left (-a d +c b \right ) \left (d x +c \right )}{b^{2}}-\frac {\left (d x +c \right )^{2}}{2 b}-\frac {\left (a d -c b \right )^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{3}}\right )\right ) g^{2}\right )}{d^{3}}\)	\(441\)
derivativedivides	\(-\frac {\frac {A \left (-\left (g^{2} c^{2}-2 g f d c +f^{2} d^{2}\right ) \left (d x +c \right )+g \left (c g -d f \right ) \left (d x +c \right )^{2}-\frac {g^{2} \left (d x +c \right )^{3}}{3}\right )}{d^{2}}+\frac {B \left (\left (-\left (d x +c \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (-2 a d +2 c b \right ) \left (\frac {\ln \left (\frac {1}{d x +c}\right )}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b \left (a d -c b \right )}\right )\right ) \left (g^{2} c^{2}-2 g f d c +f^{2} d^{2}\right )-2 \left (-\frac {\left (d x +c \right )^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2}-\left (-a d +c b \right ) \left (-\frac {d x +c}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{d x +c}\right )}{b^{2}}+\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{2}}\right )\right ) g \left (c g -d f \right )+\left (-\frac {\left (d x +c \right )^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{3}-\left (-\frac {2 a d}{3}+\frac {2 c b}{3}\right ) \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {1}{d x +c}\right )}{b^{3}}-\frac {\left (-a d +c b \right ) \left (d x +c \right )}{b^{2}}-\frac {\left (d x +c \right )^{2}}{2 b}-\frac {\left (a d -c b \right )^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{3}}\right )\right ) g^{2}\right )}{d^{2}}}{d}\)	\(495\)
default	\(-\frac {\frac {A \left (-\left (g^{2} c^{2}-2 g f d c +f^{2} d^{2}\right ) \left (d x +c \right )+g \left (c g -d f \right ) \left (d x +c \right )^{2}-\frac {g^{2} \left (d x +c \right )^{3}}{3}\right )}{d^{2}}+\frac {B \left (\left (-\left (d x +c \right ) \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )-\left (-2 a d +2 c b \right ) \left (\frac {\ln \left (\frac {1}{d x +c}\right )}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b \left (a d -c b \right )}\right )\right ) \left (g^{2} c^{2}-2 g f d c +f^{2} d^{2}\right )-2 \left (-\frac {\left (d x +c \right )^{2} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{2}-\left (-a d +c b \right ) \left (-\frac {d x +c}{b}+\frac {\left (-a d +c b \right ) \ln \left (\frac {1}{d x +c}\right )}{b^{2}}+\frac {\left (a d -c b \right ) \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{2}}\right )\right ) g \left (c g -d f \right )+\left (-\frac {\left (d x +c \right )^{3} \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{3}-\left (-\frac {2 a d}{3}+\frac {2 c b}{3}\right ) \left (\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {1}{d x +c}\right )}{b^{3}}-\frac {\left (-a d +c b \right ) \left (d x +c \right )}{b^{2}}-\frac {\left (d x +c \right )^{2}}{2 b}-\frac {\left (a d -c b \right )^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{b^{3}}\right )\right ) g^{2}\right )}{d^{2}}}{d}\)	\(495\)
parallelrisch	\(\frac {2 B \,x^{2} a \,b^{2} d^{3} g^{2}-2 B \,x^{2} b^{3} c \,d^{2} g^{2}+12 B x a \,b^{2} d^{3} f g +6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{3} f g -4 B x \,a^{2} b \,d^{3} g^{2}-4 B \,c^{3} g^{2} b^{3}+4 B \,a^{3} d^{3} g^{2}-6 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c^{2} d f g -12 B x \,b^{3} c \,d^{2} f g -2 B a \,b^{2} c^{2} d \,g^{2}+12 B \,b^{3} c^{2} d f g +2 B \,a^{2} b c \,d^{2} g^{2}-6 A a \,b^{2} c \,d^{2} f g +12 B \ln \left (b x +a \right ) a \,b^{2} d^{3} f^{2}-12 B \ln \left (b x +a \right ) b^{3} c \,d^{2} f^{2}+2 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c^{3} g^{2}+4 B x \,b^{3} c^{2} d \,g^{2}-12 B \,a^{2} b \,d^{3} f g -12 B \ln \left (b x +a \right ) a^{2} b \,d^{3} f g +12 B \ln \left (b x +a \right ) b^{3} c^{2} d f g +2 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{3} g^{2}+6 B x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} d^{3} f^{2}+6 A \,x^{2} b^{3} d^{3} f g -6 A a \,b^{2} d^{3} f^{2}+6 B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right ) b^{3} c \,d^{2} f^{2}-6 A \,b^{3} c \,d^{2} f^{2}+2 A \,x^{3} b^{3} d^{3} g^{2}+4 B \ln \left (b x +a \right ) a^{3} d^{3} g^{2}-4 B \ln \left (b x +a \right ) b^{3} c^{3} g^{2}+6 A x \,b^{3} d^{3} f^{2}}{6 b^{3} d^{3}}\)	\(544\)

[In]

int((g*x+f)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2)),x,method=_RETURNVERBOSE)

[Out]

1/3*(g*x+f)^3*B/g*ln(e*(b*x+a)^2/(d*x+c)^2)+1/3*g^2*A*x^3+g*A*f*x^2+1/3/b*g^2*B*a*x^2-1/3/d*g^2*B*c*x^2+A*f^2*
x-2/3/d^3*g^2*B*ln(d*x+c)*c^3+2/d^2*g*B*ln(d*x+c)*c^2*f-2/d*B*ln(d*x+c)*c*f^2+2/3/g*B*ln(d*x+c)*f^3+2/3/b^3*g^
2*B*ln(-b*x-a)*a^3-2/b^2*g*B*ln(-b*x-a)*a^2*f+2/b*B*ln(-b*x-a)*a*f^2-2/3/g*B*ln(-b*x-a)*f^3-2/3/b^2*g^2*B*a^2*
x+2/b*g*B*a*f*x+2/3/d^2*g^2*B*c^2*x-2/d*g*B*c*f*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (142) = 284\).

Time = 0.33 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.98 \[ \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {A b^{3} d^{3} g^{2} x^{3} + {\left (3 \, A b^{3} d^{3} f g - {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} g^{2}\right )} x^{2} + {\left (3 \, A b^{3} d^{3} f^{2} - 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} f g + 2 \, {\left (B b^{3} c^{2} d - B a^{2} b d^{3}\right )} g^{2}\right )} x + 2 \, {\left (3 \, B a b^{2} d^{3} f^{2} - 3 \, B a^{2} b d^{3} f g + B a^{3} d^{3} g^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, B b^{3} c d^{2} f^{2} - 3 \, B b^{3} c^{2} d f g + B b^{3} c^{3} g^{2}\right )} \log \left (d x + c\right ) + {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B b^{3} d^{3} f g x^{2} + 3 \, B b^{3} d^{3} f^{2} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{3 \, b^{3} d^{3}} \]

[In]

integrate((g*x+f)^2*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="fricas")

[Out]

1/3*(A*b^3*d^3*g^2*x^3 + (3*A*b^3*d^3*f*g - (B*b^3*c*d^2 - B*a*b^2*d^3)*g^2)*x^2 + (3*A*b^3*d^3*f^2 - 6*(B*b^3
*c*d^2 - B*a*b^2*d^3)*f*g + 2*(B*b^3*c^2*d - B*a^2*b*d^3)*g^2)*x + 2*(3*B*a*b^2*d^3*f^2 - 3*B*a^2*b*d^3*f*g +
B*a^3*d^3*g^2)*log(b*x + a) - 2*(3*B*b^3*c*d^2*f^2 - 3*B*b^3*c^2*d*f*g + B*b^3*c^3*g^2)*log(d*x + c) + (B*b^3*
d^3*g^2*x^3 + 3*B*b^3*d^3*f*g*x^2 + 3*B*b^3*d^3*f^2*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x
+ c^2)))/(b^3*d^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (139) = 278\).

Time = 3.27 (sec) , antiderivative size = 692, normalized size of antiderivative = 4.55 \[ \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {A g^{2} x^{3}}{3} + \frac {2 B a \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right ) \log {\left (x + \frac {2 B a^{3} c d^{2} g^{2} - 6 B a^{2} b c d^{2} f g + \frac {2 B a^{2} d^{3} \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right )}{b} + 2 B a b^{2} c^{3} g^{2} - 6 B a b^{2} c^{2} d f g + 12 B a b^{2} c d^{2} f^{2} - 2 B a c d^{2} \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right )}{2 B a^{3} d^{3} g^{2} - 6 B a^{2} b d^{3} f g + 6 B a b^{2} d^{3} f^{2} + 2 B b^{3} c^{3} g^{2} - 6 B b^{3} c^{2} d f g + 6 B b^{3} c d^{2} f^{2}} \right )}}{3 b^{3}} - \frac {2 B c \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right ) \log {\left (x + \frac {2 B a^{3} c d^{2} g^{2} - 6 B a^{2} b c d^{2} f g + 2 B a b^{2} c^{3} g^{2} - 6 B a b^{2} c^{2} d f g + 12 B a b^{2} c d^{2} f^{2} - 2 B a b^{2} c \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right ) + \frac {2 B b^{3} c^{2} \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right )}{d}}{2 B a^{3} d^{3} g^{2} - 6 B a^{2} b d^{3} f g + 6 B a b^{2} d^{3} f^{2} + 2 B b^{3} c^{3} g^{2} - 6 B b^{3} c^{2} d f g + 6 B b^{3} c d^{2} f^{2}} \right )}}{3 d^{3}} + x^{2} \left (A f g + \frac {B a g^{2}}{3 b} - \frac {B c g^{2}}{3 d}\right ) + x \left (A f^{2} - \frac {2 B a^{2} g^{2}}{3 b^{2}} + \frac {2 B a f g}{b} + \frac {2 B c^{2} g^{2}}{3 d^{2}} - \frac {2 B c f g}{d}\right ) + \left (B f^{2} x + B f g x^{2} + \frac {B g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )} \]

[In]

integrate((g*x+f)**2*(A+B*ln(e*(b*x+a)**2/(d*x+c)**2)),x)

[Out]

A*g**2*x**3/3 + 2*B*a*(a**2*g**2 - 3*a*b*f*g + 3*b**2*f**2)*log(x + (2*B*a**3*c*d**2*g**2 - 6*B*a**2*b*c*d**2*
f*g + 2*B*a**2*d**3*(a**2*g**2 - 3*a*b*f*g + 3*b**2*f**2)/b + 2*B*a*b**2*c**3*g**2 - 6*B*a*b**2*c**2*d*f*g + 1
2*B*a*b**2*c*d**2*f**2 - 2*B*a*c*d**2*(a**2*g**2 - 3*a*b*f*g + 3*b**2*f**2))/(2*B*a**3*d**3*g**2 - 6*B*a**2*b*
d**3*f*g + 6*B*a*b**2*d**3*f**2 + 2*B*b**3*c**3*g**2 - 6*B*b**3*c**2*d*f*g + 6*B*b**3*c*d**2*f**2))/(3*b**3) -
2*B*c*(c**2*g**2 - 3*c*d*f*g + 3*d**2*f**2)*log(x + (2*B*a**3*c*d**2*g**2 - 6*B*a**2*b*c*d**2*f*g + 2*B*a*b**
2*c**3*g**2 - 6*B*a*b**2*c**2*d*f*g + 12*B*a*b**2*c*d**2*f**2 - 2*B*a*b**2*c*(c**2*g**2 - 3*c*d*f*g + 3*d**2*f
**2) + 2*B*b**3*c**2*(c**2*g**2 - 3*c*d*f*g + 3*d**2*f**2)/d)/(2*B*a**3*d**3*g**2 - 6*B*a**2*b*d**3*f*g + 6*B*
a*b**2*d**3*f**2 + 2*B*b**3*c**3*g**2 - 6*B*b**3*c**2*d*f*g + 6*B*b**3*c*d**2*f**2))/(3*d**3) + x**2*(A*f*g +
B*a*g**2/(3*b) - B*c*g**2/(3*d)) + x*(A*f**2 - 2*B*a**2*g**2/(3*b**2) + 2*B*a*f*g/b + 2*B*c**2*g**2/(3*d**2) -
2*B*c*f*g/d) + (B*f**2*x + B*f*g*x**2 + B*g**2*x**3/3)*log(e*(a + b*x)**2/(c + d*x)**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (142) = 284\).

Time = 0.21 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.76 \[ \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {1}{3} \, A g^{2} x^{3} + A f g x^{2} + {\left (x \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a \log \left (b x + a\right )}{b} - \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B f^{2} + {\left (x^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B f g + \frac {1}{3} \, {\left (x^{3} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B g^{2} + A f^{2} x \]

[In]

integrate((g*x+f)^2*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="maxima")

[Out]

1/3*A*g^2*x^3 + A*f*g*x^2 + (x*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) +
a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 2*a*log(b*x + a)/b - 2*c*log(d*x + c)/d)*B*f^2 + (x^2*log(b^2*e*x^2/(d^2*x
^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - 2*a^2*log(b*x +
a)/b^2 + 2*c^2*log(d*x + c)/d^2 - 2*(b*c - a*d)*x/(b*d))*B*f*g + 1/3*(x^3*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x +
c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) + 2*a^3*log(b*x + a)/b^3 - 2*c^3
*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2))*B*g^2 + A*f^2*x

Giac [A] (verification not implemented)

none

Time = 5.20 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.69 \[ \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\frac {1}{3} \, A g^{2} x^{3} + \frac {1}{3} \, {\left (B g^{2} x^{3} + 3 \, B f g x^{2} + 3 \, B f^{2} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {{\left (3 \, A b d f g - B b c g^{2} + B a d g^{2}\right )} x^{2}}{3 \, b d} + \frac {2 \, {\left (3 \, B a b^{2} f^{2} - 3 \, B a^{2} b f g + B a^{3} g^{2}\right )} \log \left (b x + a\right )}{3 \, b^{3}} - \frac {2 \, {\left (3 \, B c d^{2} f^{2} - 3 \, B c^{2} d f g + B c^{3} g^{2}\right )} \log \left (-d x - c\right )}{3 \, d^{3}} + \frac {{\left (3 \, A b^{2} d^{2} f^{2} - 6 \, B b^{2} c d f g + 6 \, B a b d^{2} f g + 2 \, B b^{2} c^{2} g^{2} - 2 \, B a^{2} d^{2} g^{2}\right )} x}{3 \, b^{2} d^{2}} \]

[In]

integrate((g*x+f)^2*(A+B*log(e*(b*x+a)^2/(d*x+c)^2)),x, algorithm="giac")

[Out]

1/3*A*g^2*x^3 + 1/3*(B*g^2*x^3 + 3*B*f*g*x^2 + 3*B*f^2*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d
*x + c^2)) + 1/3*(3*A*b*d*f*g - B*b*c*g^2 + B*a*d*g^2)*x^2/(b*d) + 2/3*(3*B*a*b^2*f^2 - 3*B*a^2*b*f*g + B*a^3*
g^2)*log(b*x + a)/b^3 - 2/3*(3*B*c*d^2*f^2 - 3*B*c^2*d*f*g + B*c^3*g^2)*log(-d*x - c)/d^3 + 1/3*(3*A*b^2*d^2*f
^2 - 6*B*b^2*c*d*f*g + 6*B*a*b*d^2*f*g + 2*B*b^2*c^2*g^2 - 2*B*a^2*d^2*g^2)*x/(b^2*d^2)

Mupad [B] (verification not implemented)

Time = 1.38 (sec) , antiderivative size = 362, normalized size of antiderivative = 2.38 \[ \int (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \, dx=\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (B\,f^2\,x+B\,f\,g\,x^2+\frac {B\,g^2\,x^3}{3}\right )+x^2\,\left (\frac {3\,A\,a\,d\,g^2+3\,A\,b\,c\,g^2+2\,B\,a\,d\,g^2-2\,B\,b\,c\,g^2+6\,A\,b\,d\,f\,g}{6\,b\,d}-\frac {A\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,b\,d}\right )-x\,\left (\frac {\left (\frac {3\,A\,a\,d\,g^2+3\,A\,b\,c\,g^2+2\,B\,a\,d\,g^2-2\,B\,b\,c\,g^2+6\,A\,b\,d\,f\,g}{3\,b\,d}-\frac {A\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,b\,d}\right )\,\left (3\,a\,d+3\,b\,c\right )}{3\,b\,d}-\frac {3\,A\,a\,c\,g^2+3\,A\,b\,d\,f^2+6\,A\,a\,d\,f\,g+6\,A\,b\,c\,f\,g+6\,B\,a\,d\,f\,g-6\,B\,b\,c\,f\,g}{3\,b\,d}+\frac {A\,a\,c\,g^2}{b\,d}\right )+\frac {\ln \left (a+b\,x\right )\,\left (2\,B\,a^3\,g^2-6\,B\,a^2\,b\,f\,g+6\,B\,a\,b^2\,f^2\right )}{3\,b^3}-\frac {\ln \left (c+d\,x\right )\,\left (2\,B\,c^3\,g^2-6\,B\,c^2\,d\,f\,g+6\,B\,c\,d^2\,f^2\right )}{3\,d^3}+\frac {A\,g^2\,x^3}{3} \]

[In]

int((f + g*x)^2*(A + B*log((e*(a + b*x)^2)/(c + d*x)^2)),x)

[Out]

log((e*(a + b*x)^2)/(c + d*x)^2)*((B*g^2*x^3)/3 + B*f^2*x + B*f*g*x^2) + x^2*((3*A*a*d*g^2 + 3*A*b*c*g^2 + 2*B
*a*d*g^2 - 2*B*b*c*g^2 + 6*A*b*d*f*g)/(6*b*d) - (A*g^2*(3*a*d + 3*b*c))/(6*b*d)) - x*((((3*A*a*d*g^2 + 3*A*b*c
*g^2 + 2*B*a*d*g^2 - 2*B*b*c*g^2 + 6*A*b*d*f*g)/(3*b*d) - (A*g^2*(3*a*d + 3*b*c))/(3*b*d))*(3*a*d + 3*b*c))/(3
*b*d) - (3*A*a*c*g^2 + 3*A*b*d*f^2 + 6*A*a*d*f*g + 6*A*b*c*f*g + 6*B*a*d*f*g - 6*B*b*c*f*g)/(3*b*d) + (A*a*c*g
^2)/(b*d)) + (log(a + b*x)*(2*B*a^3*g^2 + 6*B*a*b^2*f^2 - 6*B*a^2*b*f*g))/(3*b^3) - (log(c + d*x)*(2*B*c^3*g^2
+ 6*B*c*d^2*f^2 - 6*B*c^2*d*f*g))/(3*d^3) + (A*g^2*x^3)/3