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

Optimal. Leaf size=140 \[ \frac{g i (a+b x)^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A-B\right )}{6 b^2}+\frac{g i (a+b x)^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b}+\frac{B g i (b c-a d)^3 \log (c+d x)}{6 b^2 d^2}-\frac{B g i x (b c-a d)^2}{6 b d} \]

[Out]

-(B*(b*c - a*d)^2*g*i*x)/(6*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)

________________________________________________________________________________________

Rubi [B]  time = 0.344012, antiderivative size = 294, normalized size of antiderivative = 2.1, number of steps used = 13, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2528, 2486, 31, 2525, 12, 72} \[ -\frac{1}{3} b B d g i x \left (\frac{a^2}{b^2}-\frac{c^2}{d^2}\right )-\frac{a^2 B g i (a d+b c) \log (a+b x)}{2 b^2}+\frac{a^3 B d g i \log (a+b x)}{3 b^2}+\frac{1}{3} b d g i x^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{1}{2} g i x^2 (a d+b c) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+a A c g i x+\frac{B c^2 g i (a d+b c) \log (c+d x)}{2 d^2}+\frac{a B c g i (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}-\frac{1}{6} B g i x^2 (b c-a d)-\frac{B g i x (b c-a d) (a d+b c)}{2 b d}-\frac{a B c g i (b c-a d) \log (c+d x)}{b d}-\frac{b B c^3 g i \log (c+d x)}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 72

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]

Rubi steps

\begin{align*} \int (3 c+3 d x) (a g+b g x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx &=\int \left (3 a c g \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+3 (b c+a d) g x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+3 b d g x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )\right ) \, dx\\ &=(3 a c g) \int \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx+(3 b d g) \int x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx+(3 (b c+a d) g) \int x \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \, dx\\ &=3 a A c g x+\frac{3}{2} (b c+a d) g x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+b d g x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+(3 a B c g) \int \log \left (\frac{e (a+b x)}{c+d x}\right ) \, dx-(b B d g) \int \frac{(b c-a d) x^3}{(a+b x) (c+d x)} \, dx-\frac{1}{2} (3 B (b c+a d) g) \int \frac{(b c-a d) x^2}{(a+b x) (c+d x)} \, dx\\ &=3 a A c g x+\frac{3 a B c g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}+\frac{3}{2} (b c+a d) g x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+b d g x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-\frac{(3 a B c (b c-a d) g) \int \frac{1}{c+d x} \, dx}{b}-(b B d (b c-a d) g) \int \frac{x^3}{(a+b x) (c+d x)} \, dx-\frac{1}{2} (3 B (b c-a d) (b c+a d) g) \int \frac{x^2}{(a+b x) (c+d x)} \, dx\\ &=3 a A c g x+\frac{3 a B c g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}+\frac{3}{2} (b c+a d) g x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+b d g x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-\frac{3 a B c (b c-a d) g \log (c+d x)}{b d}-(b B d (b c-a d) g) \int \left (\frac{-b c-a d}{b^2 d^2}+\frac{x}{b d}-\frac{a^3}{b^2 (b c-a d) (a+b x)}-\frac{c^3}{d^2 (-b c+a d) (c+d x)}\right ) \, dx-\frac{1}{2} (3 B (b c-a d) (b c+a d) g) \int \left (\frac{1}{b d}+\frac{a^2}{b (b c-a d) (a+b x)}+\frac{c^2}{d (-b c+a d) (c+d x)}\right ) \, dx\\ &=3 a A c g x-\frac{B (b c-a d) (b c+a d) g x}{2 b d}-\frac{1}{2} B (b c-a d) g x^2+\frac{a^3 B d g \log (a+b x)}{b^2}-\frac{3 a^2 B (b c+a d) g \log (a+b x)}{2 b^2}+\frac{3 a B c g (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{b}+\frac{3}{2} (b c+a d) g x^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+b d g x^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-\frac{b B c^3 g \log (c+d x)}{d^2}-\frac{3 a B c (b c-a d) g \log (c+d x)}{b d}+\frac{3 B c^2 (b c+a d) g \log (c+d x)}{2 d^2}\\ \end{align*}

Mathematica [A]  time = 0.238177, size = 181, normalized size = 1.29 \[ \frac{g i \left (b \left (d x \left (a^2 B d^2+a b d (6 A c+3 A d x+B d x)+A b^2 d x (3 c+2 d x)+b^2 (-B) c (c+d x)\right )+B c \left (6 a^2 d^2-3 a b c d+b^2 c^2\right ) \log (c+d x)+B d^2 \left (6 a^2 c+3 a b x (2 c+d x)+b^2 x^2 (3 c+2 d x)\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-a^2 B d^2 (a d+3 b c) \log (a+b x)\right )}{6 b^2 d^2} \]

Antiderivative was successfully verified.

[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 [B]  time = 0.177, size = 2407, normalized size = 17.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*e^3*d*B*g*i*b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3-3/2*e^2*B*g*i*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*b*c-e^3*B*g*i*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d
*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*c+1/2*e/d*B*g*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a*c^2*b+1/2*e^2/d*B*g*i/(d*e/(
d*x+c)*a-e/(d*x+c)*b*c)^2*b^2*c^2*a+3/2*e^2/d*A*g*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^2*c^2*a+1/2/d*B*g*i*ln(d
*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^2*a-1/6/d^2*B*g*i*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^3*b+1/2*e^2*
d*A*g*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3+1/6*e^2*d*B*g*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3-1/2*B*g*i/b*ln
(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^2*c+1/6*d*B*g*i/b^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a^3-1/2*e*
B*g*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*c+e^3/d*A*g*i*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^2*a-1/2*e^2/d^2*B*
g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^3-1/3*e^3/d^2*B*g*i*ln(b*e/d+(a*d-b*
c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^3*b^4+1/3*e^3*d*A*g*i*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^3-
1/6*e/d^2*B*g*i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*c^3-1/2*e^2/d^2*A*g*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*b^3*c^
3-1/3*e^3/d^2*A*g*i*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^3-1/6*e^2/d^2*B*g*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*
b^3*c^3+1/2*e^2*d*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3+1/6*e*d*B*g*i/b/(d
*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3-e^3*A*g*i*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*c-1/2*e^2*B*g*i/(d*e/(d*x+c)
*a-e/(d*x+c)*b*c)^2*a^2*b*c-3/2*e^2*A*g*i/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*b*c-20/3*e^3*d*B*g*i*ln(b*e/d+(a
*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^3/(d*x+c)^3*a^3*b-2*e^3/d*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/
(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^5/(d*x+c)^3*b^3*a+5/2*e^2*d^2*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))
/b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^4/(d*x+c)^2*c-2*e^3*d^3*B*g*i/b*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x
+c)*a-e/(d*x+c)*b*c)^3*a^5/(d*x+c)^3*c-5/2*e^2/d*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c
)*b*c)^2*c^4/(d*x+c)^2*a*b^2+5*e^3*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^4/(
d*x+c)^3*a^2*b^2+5*e^2*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2/(d*x+c)^2*c^3
*b+1/3*e^3/d^2*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^6/(d*x+c)^3*b^4+1/2*e^2
/d^2*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^5/(d*x+c)^2*b^3+5*e^3*d^2*B*g*i*l
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^2/(d*x+c)^3*a^4-1/2*e^2*d^3*B*g*i*ln(b*e/d+(a
*d-b*c)*e/d/(d*x+c))/b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^5/(d*x+c)^2+1/3*e^3*d^4*B*g*i/b^2*ln(b*e/d+(a*d-b*c
)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^6/(d*x+c)^3-5*e^2*d*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*
e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3/(d*x+c)^2*c^2+3/2*e^2/d*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2/(d*e/(d*x+c
)*a-e/(d*x+c)*b*c)^2*c^2*a+e^3/d*B*g*i*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^2*b^3
*a

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Maxima [B]  time = 1.31585, size = 487, normalized size = 3.48 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.11211, size = 486, normalized size = 3.47 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 4.38369, size = 505, normalized size = 3.61 \begin{align*} \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 ) + \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 )} + \frac{x \left (6 A a b c d g i + B a^{2} d^{2} g i - B b^{2} c^{2} g i\right )}{6 b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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) + (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)) + x*(6*A*a*b*c*d*g*i + B*a**2*d**2*g*i - B*b**2*c**2*g*i)/(6*b*d)

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Giac [A]  time = 2.54362, size = 296, normalized size = 2.11 \begin{align*} \frac{1}{3} \,{\left (A b d g i + B b d g i\right )} x^{3} + \frac{1}{6} \,{\left (3 \, A b c g i + 2 \, B b c g i + 3 \, A a d g i + 4 \, B a d g i\right )} x^{2} + \frac{1}{6} \,{\left (2 \, B b d g i x^{3} + 6 \, B a c g i x + 3 \,{\left (B b c g i + B a d g i\right )} x^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right ) - \frac{{\left (B b^{2} c^{2} g i - 6 \, A a b c d g i - 6 \, B a b c d g i - B a^{2} d^{2} g i\right )} x}{6 \, b d} + \frac{{\left (B b c^{3} g i - 3 \, B a c^{2} d g i\right )} \log \left (d i x + c i\right )}{6 \, d^{2}} + \frac{{\left (3 \, B a^{2} b c g i - B a^{3} d g i\right )} \log \left (b x + a\right )}{6 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

1/3*(A*b*d*g*i + B*b*d*g*i)*x^3 + 1/6*(3*A*b*c*g*i + 2*B*b*c*g*i + 3*A*a*d*g*i + 4*B*a*d*g*i)*x^2 + 1/6*(2*B*b
*d*g*i*x^3 + 6*B*a*c*g*i*x + 3*(B*b*c*g*i + B*a*d*g*i)*x^2)*log((b*x + a)/(d*x + c)) - 1/6*(B*b^2*c^2*g*i - 6*
A*a*b*c*d*g*i - 6*B*a*b*c*d*g*i - B*a^2*d^2*g*i)*x/(b*d) + 1/6*(B*b*c^3*g*i - 3*B*a*c^2*d*g*i)*log(d*i*x + c*i
)/d^2 + 1/6*(3*B*a^2*b*c*g*i - B*a^3*d*g*i)*log(b*x + a)/b^2