\(\int (a g+b g x) (c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x})) \, dx\) [3]

Optimal result

Integrand size = 36, antiderivative size = 140 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=-\frac {B (b c-a d)^2 g i x}{6 b d}+\frac {g i (a+b x)^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b}+\frac {(b c-a d) g i (a+b x)^2 \left (A-B+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 b^2}+\frac {B (b c-a d)^3 g i \log (c+d x)}{6 b^2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2560, 2548, 21, 45} \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g i (a+b x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A-B\right )}{6 b^2}+\frac {g i (a+b x)^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b}+\frac {B g i (b c-a d)^3 \log (c+d x)}{6 b^2 d^2}-\frac {B g i x (b c-a d)^2}{6 b d} \]

[In]

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

[Out]

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

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])

Rule 2560

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

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.29 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {g i \left (-a^2 B d^2 (3 b c+a d) \log (a+b x)+b \left (d x \left (a^2 B d^2-b^2 B c (c+d x)+A b^2 d x (3 c+2 d x)+a b d (6 A c+3 A d x+B d x)\right )+B d^2 \left (6 a^2 c+3 a b x (2 c+d x)+b^2 x^2 (3 c+2 d x)\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+B c \left (b^2 c^2-3 a b c d+6 a^2 d^2\right ) \log (c+d x)\right )\right )}{6 b^2 d^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.47

method	result	size
risch	\(\frac {g i B x \left (2 b d \,x^{2}+3 x a d +3 b c x +6 c a \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6}+\frac {i g b d A \,x^{3}}{3}+\frac {i g d A a \,x^{2}}{2}+\frac {i g b A c \,x^{2}}{2}+\frac {i g d B a \,x^{2}}{6}-\frac {i g b B c \,x^{2}}{6}+i g A a c x -\frac {i g d B \ln \left (b x +a \right ) a^{3}}{6 b^{2}}+\frac {i g B \ln \left (b x +a \right ) a^{2} c}{2 b}-\frac {i g B \ln \left (-d x -c \right ) a \,c^{2}}{2 d}+\frac {i g b B \ln \left (-d x -c \right ) c^{3}}{6 d^{2}}+\frac {i g d B \,a^{2} x}{6 b}-\frac {i g b B \,c^{2} x}{6 d}\)	\(206\)
parallelrisch	\(\frac {-B \ln \left (b x +a \right ) a^{3} d^{3} g i +B \ln \left (b x +a \right ) b^{3} c^{3} g i +2 A \,x^{3} b^{3} d^{3} g i -B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c^{3} g i +3 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c \,d^{2} g i +6 A x a \,b^{2} c \,d^{2} g i +3 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} c^{2} d g i +3 B \ln \left (b x +a \right ) a^{2} b c \,d^{2} g i -3 B \ln \left (b x +a \right ) a \,b^{2} c^{2} d g i -9 A \,a^{2} b c \,d^{2} g i -9 A a \,b^{2} c^{2} d g i +3 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{3} g i -B x \,b^{3} c^{2} d g i -B \,a^{3} d^{3} g i +B \,b^{3} c^{3} g i +2 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{3} g i +3 A \,x^{2} a \,b^{2} d^{3} g i +3 A \,x^{2} b^{3} c \,d^{2} g i +B \,x^{2} a \,b^{2} d^{3} g i -B \,x^{2} b^{3} c \,d^{2} g i +B x \,a^{2} b \,d^{3} g i -2 B \,a^{2} b c \,d^{2} g i +2 B a \,b^{2} c^{2} d g i +6 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} c \,d^{2} g i}{6 b^{2} d^{2}}\)	\(438\)
parts	\(A g i \left (x c a +\frac {\left (a d +c b \right ) x^{2}}{2}+\frac {b d \,x^{3}}{3}\right )-\frac {B g i \left (a d -c b \right )^{2} e^{2} \left (d^{2} \left (a d -c b \right ) \left (-\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{2 e^{2} b^{2} d}-\frac {1}{2 e b d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -2 b e \right )}{2 e^{2} b^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}\right )+b \,d^{2} e \left (a d -c b \right ) \left (-\frac {1}{6 b e d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{2}}+\frac {1}{3 b^{2} e^{2} d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}+\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{3 b^{3} e^{3} d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (3 e^{2} b^{2}-3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d b e +d^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}\right )}{3 b^{3} e^{3} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )^{3}}\right )\right )}{d^{3}}\)	\(619\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (A \,d^{2} e g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {1}{2 d^{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}}+\frac {b e}{3 d^{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 )^{3}}\right )+B \,d^{2} e g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\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 )}{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}}}{d}+\frac {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 )}{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}\right )\right )}{d^{2}}\)	\(712\)
default	\(-\frac {e \left (a d -c b \right ) \left (A \,d^{2} e g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {1}{2 d^{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}}+\frac {b e}{3 d^{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 )^{3}}\right )+B \,d^{2} e g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\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 )}{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}}}{d}+\frac {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 )}{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}\right )\right )}{d^{2}}\)	\(712\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.61 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {2 \, A b^{3} d^{3} g i x^{3} + {\left ({\left (3 \, A - B\right )} b^{3} c d^{2} + {\left (3 \, A + B\right )} a b^{2} d^{3}\right )} g i x^{2} - {\left (B b^{3} c^{2} d - 6 \, A a b^{2} c d^{2} - B a^{2} b d^{3}\right )} g i x + {\left (3 \, B a^{2} b c d^{2} - B a^{3} d^{3}\right )} g i \log \left (b x + a\right ) + {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} g i \log \left (d x + c\right ) + {\left (2 \, B b^{3} d^{3} g i x^{3} + 6 \, B a b^{2} c d^{2} g i x + 3 \, {\left (B b^{3} c d^{2} + B a b^{2} d^{3}\right )} g i x^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{6 \, b^{2} d^{2}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (126) = 252\).

Time = 1.44 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.56 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {A b d g i x^{3}}{3} - \frac {B a^{2} g i \left (a d - 3 b c\right ) \log {\left (x + \frac {B a^{3} c d^{2} g i + \frac {B a^{3} d^{2} g i \left (a d - 3 b c\right )}{b} - 6 B a^{2} b c^{2} d g i - B a^{2} c d g i \left (a d - 3 b c\right ) + B a b^{2} c^{3} g i}{B a^{3} d^{3} g i - 3 B a^{2} b c d^{2} g i - 3 B a b^{2} c^{2} d g i + B b^{3} c^{3} g i} \right )}}{6 b^{2}} - \frac {B c^{2} g i \left (3 a d - b c\right ) \log {\left (x + \frac {B a^{3} c d^{2} g i - 6 B a^{2} b c^{2} d g i + B a b^{2} c^{3} g i + B a b c^{2} g i \left (3 a d - b c\right ) - \frac {B b^{2} c^{3} g i \left (3 a d - b c\right )}{d}}{B a^{3} d^{3} g i - 3 B a^{2} b c d^{2} g i - 3 B a b^{2} c^{2} d g i + B b^{3} c^{3} g i} \right )}}{6 d^{2}} + x^{2} \left (\frac {A a d g i}{2} + \frac {A b c g i}{2} + \frac {B a d g i}{6} - \frac {B b c g i}{6}\right ) + x \left (A a c g i + \frac {B a^{2} d g i}{6 b} - \frac {B b c^{2} g i}{6 d}\right ) + \left (B a c g i x + \frac {B a d g i x^{2}}{2} + \frac {B b c g i x^{2}}{2} + \frac {B b d g i x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )} \]

[In]

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

[Out]

A*b*d*g*i*x**3/3 - B*a**2*g*i*(a*d - 3*b*c)*log(x + (B*a**3*c*d**2*g*i + B*a**3*d**2*g*i*(a*d - 3*b*c)/b - 6*B
*a**2*b*c**2*d*g*i - B*a**2*c*d*g*i*(a*d - 3*b*c) + B*a*b**2*c**3*g*i)/(B*a**3*d**3*g*i - 3*B*a**2*b*c*d**2*g*
i - 3*B*a*b**2*c**2*d*g*i + B*b**3*c**3*g*i))/(6*b**2) - B*c**2*g*i*(3*a*d - b*c)*log(x + (B*a**3*c*d**2*g*i -
6*B*a**2*b*c**2*d*g*i + B*a*b**2*c**3*g*i + B*a*b*c**2*g*i*(3*a*d - b*c) - B*b**2*c**3*g*i*(3*a*d - b*c)/d)/(
B*a**3*d**3*g*i - 3*B*a**2*b*c*d**2*g*i - 3*B*a*b**2*c**2*d*g*i + B*b**3*c**3*g*i))/(6*d**2) + x**2*(A*a*d*g*i
/2 + A*b*c*g*i/2 + B*a*d*g*i/6 - B*b*c*g*i/6) + x*(A*a*c*g*i + B*a**2*d*g*i/(6*b) - B*b*c**2*g*i/(6*d)) + (B*a
*c*g*i*x + B*a*d*g*i*x**2/2 + B*b*c*g*i*x**2/2 + B*b*d*g*i*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. 361 vs. \(2 (132) = 264\).

Time = 0.21 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.58 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\frac {1}{3} \, A b d g i x^{3} + \frac {1}{2} \, A b c g i x^{2} + \frac {1}{2} \, A a d g i x^{2} + {\left (x \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} B a c g i + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B b c g i + \frac {1}{2} \, {\left (x^{2} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {{\left (b c - a d\right )} x}{b d}\right )} B a d g i + \frac {1}{6} \, {\left (2 \, x^{3} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b d g i + A a c g i x \]

[In]

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

[Out]

1/3*A*b*d*g*i*x^3 + 1/2*A*b*c*g*i*x^2 + 1/2*A*a*d*g*i*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*c*g*i + 1/2*(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*b*c*g*i + 1/2*(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*d*g*i + 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*d*g*i + A*a*c*g*i*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1254 vs. \(2 (132) = 264\).

Time = 0.50 (sec) , antiderivative size = 1254, normalized size of antiderivative = 8.96 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/6*((B*b^5*c^4*e^4*g*i - 4*B*a*b^4*c^3*d*e^4*g*i + 6*B*a^2*b^3*c^2*d^2*e^4*g*i - 4*B*a^3*b^2*c*d^3*e^4*g*i +
B*a^4*b*d^4*e^4*g*i - 3*(b*e*x + a*e)*B*b^4*c^4*d*e^3*g*i/(d*x + c) + 12*(b*e*x + a*e)*B*a*b^3*c^3*d^2*e^3*g*
i/(d*x + c) - 18*(b*e*x + a*e)*B*a^2*b^2*c^2*d^3*e^3*g*i/(d*x + c) + 12*(b*e*x + a*e)*B*a^3*b*c*d^4*e^3*g*i/(d
*x + c) - 3*(b*e*x + a*e)*B*a^4*d^5*e^3*g*i/(d*x + c))*log((b*e*x + a*e)/(d*x + c))/(b^3*d^2*e^3 - 3*(b*e*x +
a*e)*b^2*d^3*e^2/(d*x + c) + 3*(b*e*x + a*e)^2*b*d^4*e/(d*x + c)^2 - (b*e*x + a*e)^3*d^5/(d*x + c)^3) + (A*b^6
*c^4*e^4*g*i - 4*A*a*b^5*c^3*d*e^4*g*i + 6*A*a^2*b^4*c^2*d^2*e^4*g*i - 4*A*a^3*b^3*c*d^3*e^4*g*i + A*a^4*b^2*d
^4*e^4*g*i - 3*(b*e*x + a*e)*A*b^5*c^4*d*e^3*g*i/(d*x + c) + (b*e*x + a*e)*B*b^5*c^4*d*e^3*g*i/(d*x + c) + 12*
(b*e*x + a*e)*A*a*b^4*c^3*d^2*e^3*g*i/(d*x + c) - 4*(b*e*x + a*e)*B*a*b^4*c^3*d^2*e^3*g*i/(d*x + c) - 18*(b*e*
x + a*e)*A*a^2*b^3*c^2*d^3*e^3*g*i/(d*x + c) + 6*(b*e*x + a*e)*B*a^2*b^3*c^2*d^3*e^3*g*i/(d*x + c) + 12*(b*e*x
+ a*e)*A*a^3*b^2*c*d^4*e^3*g*i/(d*x + c) - 4*(b*e*x + a*e)*B*a^3*b^2*c*d^4*e^3*g*i/(d*x + c) - 3*(b*e*x + a*e
)*A*a^4*b*d^5*e^3*g*i/(d*x + c) + (b*e*x + a*e)*B*a^4*b*d^5*e^3*g*i/(d*x + c) - (b*e*x + a*e)^2*B*b^4*c^4*d^2*
e^2*g*i/(d*x + c)^2 + 4*(b*e*x + a*e)^2*B*a*b^3*c^3*d^3*e^2*g*i/(d*x + c)^2 - 6*(b*e*x + a*e)^2*B*a^2*b^2*c^2*
d^4*e^2*g*i/(d*x + c)^2 + 4*(b*e*x + a*e)^2*B*a^3*b*c*d^5*e^2*g*i/(d*x + c)^2 - (b*e*x + a*e)^2*B*a^4*d^6*e^2*
g*i/(d*x + c)^2)/(b^4*d^2*e^3 - 3*(b*e*x + a*e)*b^3*d^3*e^2/(d*x + c) + 3*(b*e*x + a*e)^2*b^2*d^4*e/(d*x + c)^
2 - (b*e*x + a*e)^3*b*d^5/(d*x + c)^3) + (B*b^4*c^4*e*g*i - 4*B*a*b^3*c^3*d*e*g*i + 6*B*a^2*b^2*c^2*d^2*e*g*i
- 4*B*a^3*b*c*d^3*e*g*i + B*a^4*d^4*e*g*i)*log(-b*e + (b*e*x + a*e)*d/(d*x + c))/(b^2*d^2) - (B*b^4*c^4*e*g*i
- 4*B*a*b^3*c^3*d*e*g*i + 6*B*a^2*b^2*c^2*d^2*e*g*i - 4*B*a^3*b*c*d^3*e*g*i + B*a^4*d^4*e*g*i)*log((b*e*x + a*
e)/(d*x + c))/(b^2*d^2))*(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 = 1.31 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.01 \[ \int (a g+b g x) (c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx=x^2\,\left (\frac {g\,i\,\left (6\,A\,a\,d+6\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{6}-\frac {A\,g\,i\,\left (6\,a\,d+6\,b\,c\right )}{12}\right )-x\,\left (\frac {\left (\frac {g\,i\,\left (6\,A\,a\,d+6\,A\,b\,c+B\,a\,d-B\,b\,c\right )}{3}-\frac {A\,g\,i\,\left (6\,a\,d+6\,b\,c\right )}{6}\right )\,\left (6\,a\,d+6\,b\,c\right )}{6\,b\,d}+A\,a\,c\,g\,i-\frac {g\,i\,\left (2\,A\,a^2\,d^2+2\,A\,b^2\,c^2+B\,a^2\,d^2-B\,b^2\,c^2+8\,A\,a\,b\,c\,d\right )}{2\,b\,d}\right )+\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,b\,d\,g\,i\,x^3}{3}+\frac {B\,g\,i\,\left (a\,d+b\,c\right )\,x^2}{2}+B\,a\,c\,g\,i\,x\right )-\frac {\ln \left (a+b\,x\right )\,\left (B\,a^3\,d\,g\,i-3\,B\,a^2\,b\,c\,g\,i\right )}{6\,b^2}+\frac {\ln \left (c+d\,x\right )\,\left (B\,b\,c^3\,g\,i-3\,B\,a\,c^2\,d\,g\,i\right )}{6\,d^2}+\frac {A\,b\,d\,g\,i\,x^3}{3} \]

[In]

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

[Out]

x^2*((g*i*(6*A*a*d + 6*A*b*c + B*a*d - B*b*c))/6 - (A*g*i*(6*a*d + 6*b*c))/12) - x*((((g*i*(6*A*a*d + 6*A*b*c
+ B*a*d - B*b*c))/3 - (A*g*i*(6*a*d + 6*b*c))/6)*(6*a*d + 6*b*c))/(6*b*d) + A*a*c*g*i - (g*i*(2*A*a^2*d^2 + 2*
A*b^2*c^2 + B*a^2*d^2 - B*b^2*c^2 + 8*A*a*b*c*d))/(2*b*d)) + log((e*(a + b*x))/(c + d*x))*((B*g*i*x^2*(a*d + b
*c))/2 + (B*b*d*g*i*x^3)/3 + B*a*c*g*i*x) - (log(a + b*x)*(B*a^3*d*g*i - 3*B*a^2*b*c*g*i))/(6*b^2) + (log(c +
d*x)*(B*b*c^3*g*i - 3*B*a*c^2*d*g*i))/(6*d^2) + (A*b*d*g*i*x^3)/3