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

Optimal. Leaf size=150 \[ \frac{(f+g x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g}-\frac{B g x (b c-a d) (-a d g-b c g+3 b d f)}{3 b^2 d^2}-\frac{B (b f-a g)^3 \log (a+b x)}{3 b^3 g}-\frac{B g^2 x^2 (b c-a d)}{6 b d}+\frac{B (d f-c g)^3 \log (c+d x)}{3 d^3 g} \]

[Out]

-(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)/(6*b*d) - (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))/(c + d*x)]))/(3*g) + (B*(d*f - c*g)^3*Log[c
+ d*x])/(3*d^3*g)

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Rubi [A]  time = 0.169498, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2525, 12, 72} \[ \frac{(f+g x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g}-\frac{B g x (b c-a d) (-a d g-b c g+3 b d f)}{3 b^2 d^2}-\frac{B (b f-a g)^3 \log (a+b x)}{3 b^3 g}-\frac{B g^2 x^2 (b c-a d)}{6 b d}+\frac{B (d f-c g)^3 \log (c+d x)}{3 d^3 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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)/(6*b*d) - (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))/(c + d*x)]))/(3*g) + (B*(d*f - c*g)^3*Log[c
+ d*x])/(3*d^3*g)

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

Mathematica [A]  time = 0.131928, size = 142, normalized size = 0.95 \[ \frac{(f+g x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{B \left (b^2 d^2 g^3 x^2 (b c-a d)+2 b d g^2 x (b c-a d) (-a d g-b c g+3 b d f)+2 d^3 (b f-a g)^3 \log (a+b x)-2 b^3 (d f-c g)^3 \log (c+d x)\right )}{2 b^3 d^3}}{3 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.184, size = 4406, normalized size = 29.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.1324, size = 354, normalized size = 2.36 \begin{align*} \frac{1}{3} \, A g^{2} x^{3} + A f g 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 f^{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 f g + \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 g^{2} + A f^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*A*g^2*x^3 + A*f*g*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*f
^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*f*g + 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*g^2 + A*f^2*x

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Fricas [A]  time = 1.31593, size = 581, normalized size = 3.87 \begin{align*} \frac{2 \, A b^{3} d^{3} g^{2} x^{3} +{\left (6 \, 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} + 2 \,{\left (3 \, A b^{3} d^{3} f^{2} - 3 \,{\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} f g +{\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 ) + 2 \,{\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 e x + a e}{d x + c}\right )}{6 \, b^{3} d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [B]  time = 10.6912, size = 688, normalized size = 4.59 \begin{align*} \frac{A g^{2} x^{3}}{3} + \frac{B a \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right ) \log{\left (x + \frac{B a^{3} c d^{2} g^{2} - 3 B a^{2} b c d^{2} f g + \frac{B a^{2} d^{3} \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right )}{b} + B a b^{2} c^{3} g^{2} - 3 B a b^{2} c^{2} d f g + 6 B a b^{2} c d^{2} f^{2} - B a c d^{2} \left (a^{2} g^{2} - 3 a b f g + 3 b^{2} f^{2}\right )}{B a^{3} d^{3} g^{2} - 3 B a^{2} b d^{3} f g + 3 B a b^{2} d^{3} f^{2} + B b^{3} c^{3} g^{2} - 3 B b^{3} c^{2} d f g + 3 B b^{3} c d^{2} f^{2}} \right )}}{3 b^{3}} - \frac{B c \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right ) \log{\left (x + \frac{B a^{3} c d^{2} g^{2} - 3 B a^{2} b c d^{2} f g + B a b^{2} c^{3} g^{2} - 3 B a b^{2} c^{2} d f g + 6 B a b^{2} c d^{2} f^{2} - B a b^{2} c \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right ) + \frac{B b^{3} c^{2} \left (c^{2} g^{2} - 3 c d f g + 3 d^{2} f^{2}\right )}{d}}{B a^{3} d^{3} g^{2} - 3 B a^{2} b d^{3} f g + 3 B a b^{2} d^{3} f^{2} + B b^{3} c^{3} g^{2} - 3 B b^{3} c^{2} d f g + 3 B b^{3} c d^{2} f^{2}} \right )}}{3 d^{3}} + \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 )}{c + d x} \right )} + \frac{x^{2} \left (6 A b d f g + B a d g^{2} - B b c g^{2}\right )}{6 b d} - \frac{x \left (- 3 A b^{2} d^{2} f^{2} + B a^{2} d^{2} g^{2} - 3 B a b d^{2} f g - B b^{2} c^{2} g^{2} + 3 B b^{2} c d f g\right )}{3 b^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 15.9632, size = 346, normalized size = 2.31 \begin{align*} \frac{1}{3} \,{\left (A g^{2} + B g^{2}\right )} 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 x + a}{d x + c}\right ) + \frac{{\left (6 \, A b d f g + 6 \, B b d f g - B b c g^{2} + B a d g^{2}\right )} x^{2}}{6 \, b d} + \frac{{\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{{\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} + 3 \, B b^{2} d^{2} f^{2} - 3 \, B b^{2} c d f g + 3 \, B a b d^{2} f g + B b^{2} c^{2} g^{2} - B a^{2} d^{2} g^{2}\right )} x}{3 \, b^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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