\(\int (a g+b g x)^2 (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [90]

Optimal result

Integrand size = 30, antiderivative size = 118 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {B (b c-a d)^2 g^2 x}{3 d^2}-\frac {B (b c-a d) g^2 (a+b x)^2}{6 b d}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b}-\frac {B (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2548, 21, 45} \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}-\frac {B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac {B g^2 x (b c-a d)^2}{3 d^2}-\frac {B g^2 (a+b x)^2 (b c-a d)}{6 b d} \]

[In]

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

[Out]

(B*(b*c - a*d)^2*g^2*x)/(3*d^2) - (B*(b*c - a*d)*g^2*(a + b*x)^2)/(6*b*d) + (g^2*(a + b*x)^3*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]))/(3*b) - (B*(b*c - a*d)^3*g^2*Log[c + d*x])/(3*b*d^3)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b}-\frac {(B (b c-a d)) \int \frac {(a g+b g x)^3}{(a+b x) (c+d x)} \, dx}{3 b g} \\ & = \frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b}-\frac {\left (B (b c-a d) g^2\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{3 b} \\ & = \frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b}-\frac {\left (B (b c-a d) g^2\right ) \int \left (-\frac {b (b c-a d)}{d^2}+\frac {b (a+b x)}{d}+\frac {(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b} \\ & = \frac {B (b c-a d)^2 g^2 x}{3 d^2}-\frac {B (b c-a d) g^2 (a+b x)^2}{6 b d}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b}-\frac {B (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.84 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g^2 \left ((a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+\frac {B (-b c+a d) \left (d \left (a^2 d+4 a b d x+b^2 x (-2 c+d x)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{2 d^3}\right )}{3 b} \]

[In]

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

[Out]

(g^2*((a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + (B*(-(b*c) + a*d)*(d*(a^2*d + 4*a*b*d*x + b^2*x*(-2*c
+ d*x)) + 2*(b*c - a*d)^2*Log[c + d*x]))/(2*d^3)))/(3*b)

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.75

method	result	size
risch	\(\frac {\left (b x +a \right )^{3} g^{2} B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{3 b}+\frac {g^{2} b^{2} A \,x^{3}}{3}+g^{2} b A a \,x^{2}+\frac {g^{2} b B a \,x^{2}}{6}-\frac {g^{2} b^{2} B c \,x^{2}}{6 d}+g^{2} A \,a^{2} x +\frac {g^{2} B \ln \left (d x +c \right ) a^{3}}{3 b}-\frac {g^{2} B \ln \left (d x +c \right ) a^{2} c}{d}+\frac {g^{2} b B \ln \left (d x +c \right ) a \,c^{2}}{d^{2}}-\frac {g^{2} b^{2} B \ln \left (d x +c \right ) c^{3}}{3 d^{3}}+\frac {2 g^{2} B \,a^{2} x}{3}-\frac {g^{2} b B a c x}{d}+\frac {g^{2} b^{2} B \,c^{2} x}{3 d^{2}}\)	\(207\)
parallelrisch	\(\frac {2 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c \,d^{3} g^{2}+2 A \,x^{3} a \,b^{3} c \,d^{3} g^{2}+6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c \,d^{3} g^{2}+6 A \,x^{2} a^{2} b^{2} c \,d^{3} g^{2}+B \,x^{2} a^{2} b^{2} c \,d^{3} g^{2}-B \,x^{2} a \,b^{3} c^{2} d^{2} g^{2}+6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b c \,d^{3} g^{2}+6 A x \,a^{3} b c \,d^{3} g^{2}+2 B \ln \left (b x +a \right ) a^{4} c \,d^{3} g^{2}-6 B \ln \left (b x +a \right ) a^{3} b \,c^{2} d^{2} g^{2}+6 B \ln \left (b x +a \right ) a^{2} b^{2} c^{3} d \,g^{2}-2 B \ln \left (b x +a \right ) a \,b^{3} c^{4} g^{2}+4 B x \,a^{3} b c \,d^{3} g^{2}-6 B x \,a^{2} b^{2} c^{2} d^{2} g^{2}+2 B x a \,b^{3} c^{3} d \,g^{2}+6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{2} d^{2} g^{2}-6 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c^{3} d \,g^{2}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{3} c^{4} g^{2}}{6 a b c \,d^{3}}\)	\(421\)
parts	\(\frac {A \,g^{2} \left (b x +a \right )^{3}}{3 b}-\frac {B \,g^{2} \left (a d -c b \right ) e \left (2 b e \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {1}{2 e b d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}-\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{2 e^{2} b^{2} d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -2 b e \right )}{2 e^{2} b^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}\right )+\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right )+e^{2} b^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {1}{6 b e d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}+\frac {1}{3 b^{2} e^{2} d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}+\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{3 b^{3} e^{3} d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (3 e^{2} b^{2}-3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{3}}\right )\right )}{d^{2}}\)	\(798\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (A \,d^{2} g^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {b e}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {b^{2} e^{2}}{3 d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}+\frac {1}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )+B \,d^{2} g^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {2 b e \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}+\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )}{d^{2}}+\frac {b^{2} e^{2} \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{3 b^{3} e^{3} d}-\frac {1}{3 b^{2} e^{2} d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {1}{6 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (3 e^{2} b^{2}-3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}\right )}{d^{2}}+\frac {\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}}{d^{2}}\right )\right )}{d^{2}}\)	\(913\)
default	\(-\frac {e \left (a d -c b \right ) \left (A \,d^{2} g^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {b e}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {b^{2} e^{2}}{3 d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}+\frac {1}{d^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )+B \,d^{2} g^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {2 b e \left (-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} d}+\frac {1}{2 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (2 b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{2 b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}\right )}{d^{2}}+\frac {b^{2} e^{2} \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{3 b^{3} e^{3} d}-\frac {1}{3 b^{2} e^{2} d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {1}{6 b e d \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{2}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (3 e^{2} b^{2}-3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )^{3}}\right )}{d^{2}}+\frac {\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}}{d^{2}}\right )\right )}{d^{2}}\)	\(913\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (110) = 220\).

Time = 0.26 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.88 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {2 \, A b^{3} d^{3} g^{2} x^{3} + 2 \, B a^{3} d^{3} g^{2} \log \left (b x + a\right ) - {\left (B b^{3} c d^{2} - {\left (6 \, A + B\right )} a b^{2} d^{3}\right )} g^{2} x^{2} + 2 \, {\left (B b^{3} c^{2} d - 3 \, B a b^{2} c d^{2} + {\left (3 \, A + 2 \, B\right )} a^{2} b d^{3}\right )} g^{2} x - 2 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} \log \left (d x + c\right ) + 2 \, {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{6 \, b d^{3}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (100) = 200\).

Time = 1.31 (sec) , antiderivative size = 491, normalized size of antiderivative = 4.16 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A b^{2} g^{2} x^{3}}{3} + \frac {B a^{3} g^{2} \log {\left (x + \frac {\frac {B a^{4} d^{3} g^{2}}{b} + 3 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 b} - \frac {B c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {4 B a^{3} c d^{2} g^{2} - 3 B a^{2} b c^{2} d g^{2} + B a b^{2} c^{3} g^{2} - B a c g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac {B b c^{2} g^{2} \cdot \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d}}{B a^{3} d^{3} g^{2} + 3 B a^{2} b c d^{2} g^{2} - 3 B a b^{2} c^{2} d g^{2} + B b^{3} c^{3} g^{2}} \right )}}{3 d^{3}} + x^{2} \left (A a b g^{2} + \frac {B a b g^{2}}{6} - \frac {B b^{2} c g^{2}}{6 d}\right ) + x \left (A a^{2} g^{2} + \frac {2 B a^{2} g^{2}}{3} - \frac {B a b c g^{2}}{d} + \frac {B b^{2} c^{2} g^{2}}{3 d^{2}}\right ) + \left (B a^{2} g^{2} x + B a b g^{2} x^{2} + \frac {B b^{2} g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

[In]

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

[Out]

A*b**2*g**2*x**3/3 + B*a**3*g**2*log(x + (B*a**4*d**3*g**2/b + 3*B*a**3*c*d**2*g**2 - 3*B*a**2*b*c**2*d*g**2 +
B*a*b**2*c**3*g**2)/(B*a**3*d**3*g**2 + 3*B*a**2*b*c*d**2*g**2 - 3*B*a*b**2*c**2*d*g**2 + B*b**3*c**3*g**2))/
(3*b) - B*c*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)*log(x + (4*B*a**3*c*d**2*g**2 - 3*B*a**2*b*c**2*d*g**2
+ B*a*b**2*c**3*g**2 - B*a*c*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2) + B*b*c**2*g**2*(3*a**2*d**2 - 3*a*b*c
*d + b**2*c**2)/d)/(B*a**3*d**3*g**2 + 3*B*a**2*b*c*d**2*g**2 - 3*B*a*b**2*c**2*d*g**2 + B*b**3*c**3*g**2))/(3
*d**3) + x**2*(A*a*b*g**2 + B*a*b*g**2/6 - B*b**2*c*g**2/(6*d)) + x*(A*a**2*g**2 + 2*B*a**2*g**2/3 - B*a*b*c*g
**2/d + B*b**2*c**2*g**2/(3*d**2)) + (B*a**2*g**2*x + B*a*b*g**2*x**2 + B*b**2*g**2*x**3/3)*log(e*(a + b*x)/(c
+ d*x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (110) = 220\).

Time = 0.21 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.37 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{3} \, A b^{2} g^{2} x^{3} + A a b g^{2} x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a^{2} g^{2} + {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a b g^{2} + \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} g^{2} + A a^{2} g^{2} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (110) = 220\).

Time = 0.42 (sec) , antiderivative size = 1742, normalized size of antiderivative = 14.76 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/6*(2*(B*b^6*c^4*e^4*g^2 - 4*B*a*b^5*c^3*d*e^4*g^2 + 6*B*a^2*b^4*c^2*d^2*e^4*g^2 - 4*B*a^3*b^3*c*d^3*e^4*g^2
+ B*a^4*b^2*d^4*e^4*g^2 - 3*(b*e*x + a*e)*B*b^5*c^4*d*e^3*g^2/(d*x + c) + 12*(b*e*x + a*e)*B*a*b^4*c^3*d^2*e^3
*g^2/(d*x + c) - 18*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3*g^2/(d*x + c) + 12*(b*e*x + a*e)*B*a^3*b^2*c*d^4*e^3*g
^2/(d*x + c) - 3*(b*e*x + a*e)*B*a^4*b*d^5*e^3*g^2/(d*x + c) + 3*(b*e*x + a*e)^2*B*b^4*c^4*d^2*e^2*g^2/(d*x +
c)^2 - 12*(b*e*x + a*e)^2*B*a*b^3*c^3*d^3*e^2*g^2/(d*x + c)^2 + 18*(b*e*x + a*e)^2*B*a^2*b^2*c^2*d^4*e^2*g^2/(
d*x + c)^2 - 12*(b*e*x + a*e)^2*B*a^3*b*c*d^5*e^2*g^2/(d*x + c)^2 + 3*(b*e*x + a*e)^2*B*a^4*d^6*e^2*g^2/(d*x +
c)^2)*log((b*e*x + a*e)/(d*x + c))/(b^3*d^3*e^3 - 3*(b*e*x + a*e)*b^2*d^4*e^2/(d*x + c) + 3*(b*e*x + a*e)^2*b
*d^5*e/(d*x + c)^2 - (b*e*x + a*e)^3*d^6/(d*x + c)^3) + (2*A*b^6*c^4*e^4*g^2 + 3*B*b^6*c^4*e^4*g^2 - 8*A*a*b^5
*c^3*d*e^4*g^2 - 12*B*a*b^5*c^3*d*e^4*g^2 + 12*A*a^2*b^4*c^2*d^2*e^4*g^2 + 18*B*a^2*b^4*c^2*d^2*e^4*g^2 - 8*A*
a^3*b^3*c*d^3*e^4*g^2 - 12*B*a^3*b^3*c*d^3*e^4*g^2 + 2*A*a^4*b^2*d^4*e^4*g^2 + 3*B*a^4*b^2*d^4*e^4*g^2 - 6*(b*
e*x + a*e)*A*b^5*c^4*d*e^3*g^2/(d*x + c) - 7*(b*e*x + a*e)*B*b^5*c^4*d*e^3*g^2/(d*x + c) + 24*(b*e*x + a*e)*A*
a*b^4*c^3*d^2*e^3*g^2/(d*x + c) + 28*(b*e*x + a*e)*B*a*b^4*c^3*d^2*e^3*g^2/(d*x + c) - 36*(b*e*x + a*e)*A*a^2*
b^3*c^2*d^3*e^3*g^2/(d*x + c) - 42*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3*g^2/(d*x + c) + 24*(b*e*x + a*e)*A*a^3*
b^2*c*d^4*e^3*g^2/(d*x + c) + 28*(b*e*x + a*e)*B*a^3*b^2*c*d^4*e^3*g^2/(d*x + c) - 6*(b*e*x + a*e)*A*a^4*b*d^5
*e^3*g^2/(d*x + c) - 7*(b*e*x + a*e)*B*a^4*b*d^5*e^3*g^2/(d*x + c) + 6*(b*e*x + a*e)^2*A*b^4*c^4*d^2*e^2*g^2/(
d*x + c)^2 + 4*(b*e*x + a*e)^2*B*b^4*c^4*d^2*e^2*g^2/(d*x + c)^2 - 24*(b*e*x + a*e)^2*A*a*b^3*c^3*d^3*e^2*g^2/
(d*x + c)^2 - 16*(b*e*x + a*e)^2*B*a*b^3*c^3*d^3*e^2*g^2/(d*x + c)^2 + 36*(b*e*x + a*e)^2*A*a^2*b^2*c^2*d^4*e^
2*g^2/(d*x + c)^2 + 24*(b*e*x + a*e)^2*B*a^2*b^2*c^2*d^4*e^2*g^2/(d*x + c)^2 - 24*(b*e*x + a*e)^2*A*a^3*b*c*d^
5*e^2*g^2/(d*x + c)^2 - 16*(b*e*x + a*e)^2*B*a^3*b*c*d^5*e^2*g^2/(d*x + c)^2 + 6*(b*e*x + a*e)^2*A*a^4*d^6*e^2
*g^2/(d*x + c)^2 + 4*(b*e*x + a*e)^2*B*a^4*d^6*e^2*g^2/(d*x + c)^2)/(b^3*d^3*e^3 - 3*(b*e*x + a*e)*b^2*d^4*e^2
/(d*x + c) + 3*(b*e*x + a*e)^2*b*d^5*e/(d*x + c)^2 - (b*e*x + a*e)^3*d^6/(d*x + c)^3) + 2*(B*b^4*c^4*e*g^2 - 4
*B*a*b^3*c^3*d*e*g^2 + 6*B*a^2*b^2*c^2*d^2*e*g^2 - 4*B*a^3*b*c*d^3*e*g^2 + B*a^4*d^4*e*g^2)*log(-b*e + (b*e*x
+ a*e)*d/(d*x + c))/(b*d^3) - 2*(B*b^4*c^4*e*g^2 - 4*B*a*b^3*c^3*d*e*g^2 + 6*B*a^2*b^2*c^2*d^2*e*g^2 - 4*B*a^3
*b*c*d^3*e*g^2 + B*a^4*d^4*e*g^2)*log((b*e*x + a*e)/(d*x + c))/(b*d^3))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a
*d/((b*c*e - a*d*e)*(b*c - a*d)))

Mupad [B] (verification not implemented)

Time = 1.13 (sec) , antiderivative size = 290, normalized size of antiderivative = 2.46 \[ \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=x^2\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{6\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{3\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )}{3\,b\,d}-\frac {a\,g^2\,\left (3\,A\,a\,d+3\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{d}+\frac {A\,a\,b\,c\,g^2}{d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (B\,a^2\,g^2\,x+B\,a\,b\,g^2\,x^2+\frac {B\,b^2\,g^2\,x^3}{3}\right )-\frac {\ln \left (c+d\,x\right )\,\left (3\,B\,a^2\,c\,d^2\,g^2-3\,B\,a\,b\,c^2\,d\,g^2+B\,b^2\,c^3\,g^2\right )}{3\,d^3}+\frac {A\,b^2\,g^2\,x^3}{3}+\frac {B\,a^3\,g^2\,\ln \left (a+b\,x\right )}{3\,b} \]

[In]

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

[Out]

x^2*((b*g^2*(9*A*a*d + 3*A*b*c + B*a*d - B*b*c))/(6*d) - (A*b*g^2*(3*a*d + 3*b*c))/(6*d)) - x*(((3*a*d + 3*b*c
)*((b*g^2*(9*A*a*d + 3*A*b*c + B*a*d - B*b*c))/(3*d) - (A*b*g^2*(3*a*d + 3*b*c))/(3*d)))/(3*b*d) - (a*g^2*(3*A
*a*d + 3*A*b*c + B*a*d - B*b*c))/d + (A*a*b*c*g^2)/d) + log((e*(a + b*x))/(c + d*x))*((B*b^2*g^2*x^3)/3 + B*a^
2*g^2*x + B*a*b*g^2*x^2) - (log(c + d*x)*(B*b^2*c^3*g^2 + 3*B*a^2*c*d^2*g^2 - 3*B*a*b*c^2*d*g^2))/(3*d^3) + (A
*b^2*g^2*x^3)/3 + (B*a^3*g^2*log(a + b*x))/(3*b)