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

Optimal. Leaf size=118 \[ \frac{g^2 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b}+\frac{B g^2 x (b c-a d)^2}{3 d^2}-\frac{B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}-\frac{B g^2 (a+b x)^2 (b c-a d)}{6 b d} \]

[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)

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Rubi [A]  time = 0.0792376, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2525, 12, 43} \[ \frac{g^2 (a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b}+\frac{B g^2 x (b c-a d)^2}{3 d^2}-\frac{B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}-\frac{B g^2 (a+b x)^2 (b c-a d)}{6 b d} \]

Antiderivative was successfully verified.

[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 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 43

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])

Rubi steps

\begin{align*} \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 (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b}-\frac{B \int \frac{(b c-a d) g^3 (a+b x)^2}{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]  time = 0.0537513, size = 99, normalized size = 0.84 \[ \frac{g^2 \left (\frac{B (a d-b c) \left (d \left (a^2 d+4 a b d x+b^2 x (d x-2 c)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{2 d^3}+(a+b x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.162, size = 3283, normalized size = 27.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*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)*a^3+1/d*B*g^2*ln(d*(b*e/d+(a*d-b*c)*e/d/
(d*x+c))-b*e)*a^2*c+1/3/d^3*B*g^2*b^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*c^3+1/3*e^3*A*g^2*b^2/(d*e/(d*x+
c)*a-e/(d*x+c)*b*c)^3*a^3+e^2*A*g^2*b/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^3+1/6*e^2*B*g^2*b/(d*e/(d*x+c)*a-e/(d*
x+c)*b*c)^2*a^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^4/(d*x+c)^3*c^
2*b+5*e^3/d*B*g^2*b^3*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/(d*x+c)^3*c^4-5*e^2/
d^2*B*g^2*b^3*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+10*e^2/d*B*g^2*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)^2*c^3/(d*x+c)^2*a^2+6*e/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)*a^2/(d*x+c)*c^2*b-2*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^5/(d*x+c)^3*b^4-4*e/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)*c^3/(d*x+c)*a-3*e/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)*a^2*c*b+e/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)*c^4/(d*x+c)+e*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)*a^4/(d*x+c)+5*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^4/(d*x+c)^2*c+e^3/d^2*B*g^2*b^4*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*a-2*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^5/(d*x+c)^3*c+1/3*e^3/d^3*B*g^2*b^5*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-e^2*d^2*B*g^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)^2*a^5/(d*x+c)^2
+e^2/d^3*B*g^2*b^4*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+1/3*e^3*d^3*B
*g^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)^3*a^6/(d*x+c)^3-3*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*b^2*c-e^3/d*B*g^2*b^3*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+3*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*b^3*c^2+3*e/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)*c^2*a-20/3*
e^3*B*g^2*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^3/(d*x+c)^3*c^3-10*e^2*B*g^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)^2*c^2/(d*x+c)^2*a^3+2/3*e*B*g^2/(d*e/(d*x+c)*a-
e/(d*x+c)*b*c)*a^3+e*A*g^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^3-1/3*B*g^2/b*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*
e)*a^3-3*e^2/d*A*g^2*b^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a^2*c+2*e/d^2*B*g^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a*b
^2*c^2+e^3/d^2*A*g^2*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a*c^2+3*e^2/d^2*A*g^2*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*
c)^2*a*c^2-3*e/d*A*g^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*b*c+3*e/d^2*A*g^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*b^2*c
^2*a-2*e/d*B*g^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*a^2*b*c+1/2*e^2/d^2*B*g^2*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*a
*c^2-e^3/d*A*g^2*b^3/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*a^2*c-4*e*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)*a^3/(d*x+c)*c-1/3*e^3/d^3*B*g^2*b^5*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-e^2/d^3*B*g^2*b^4*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-e/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)*c^3-1/2*e^2/d*B*g^2*b^2/(d*e/(d*x+c)*a-e/
(d*x+c)*b*c)^2*a^2*c-e/d^3*A*g^2/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)*b^3*c^3-2/3*e/d^3*B*g^2/(d*e/(d*x+c)*a-e/(d*x+c
)*b*c)*b^3*c^3-1/3*e^3/d^3*A*g^2*b^5/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^3*c^3-e^2/d^3*A*g^2*b^4/(d*e/(d*x+c)*a-e/(d
*x+c)*b*c)^2*c^3-1/6*e^2/d^3*B*g^2*b^4/(d*e/(d*x+c)*a-e/(d*x+c)*b*c)^2*c^3+e^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^3*b-1/d^2*B*g^2*ln(d*(b*e/d+(a*d-b*c)*e/d/(d*x+c))-b*e)*a*c^2*b+1/3*e
^3*B*g^2*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^3

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Maxima [B]  time = 1.17322, size = 378, normalized size = 3.2 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.09247, size = 467, normalized size = 3.96 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [B]  time = 5.21757, size = 503, normalized size = 4.26 \begin{align*} \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} \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} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac{B b c^{2} g^{2} \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}} + \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 )} + \frac{x^{2} \left (6 A a b d g^{2} + B a b d g^{2} - B b^{2} c g^{2}\right )}{6 d} + \frac{x \left (3 A a^{2} d^{2} g^{2} + 2 B a^{2} d^{2} g^{2} - 3 B a b c d g^{2} + B b^{2} c^{2} g^{2}\right )}{3 d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Giac [B]  time = 4.75747, size = 309, normalized size = 2.62 \begin{align*} \frac{B a^{3} g^{2} \log \left (b x + a\right )}{3 \, b} + \frac{1}{3} \,{\left (A b^{2} g^{2} + B b^{2} g^{2}\right )} x^{3} - \frac{{\left (B b^{2} c g^{2} - 6 \, A a b d g^{2} - 7 \, B a b d g^{2}\right )} x^{2}}{6 \, d} + \frac{1}{3} \,{\left (B b^{2} g^{2} x^{3} + 3 \, B a b g^{2} x^{2} + 3 \, B a^{2} g^{2} x\right )} \log \left (\frac{b x + a}{d x + c}\right ) + \frac{{\left (B b^{2} c^{2} g^{2} - 3 \, B a b c d g^{2} + 3 \, A a^{2} d^{2} g^{2} + 5 \, B a^{2} d^{2} g^{2}\right )} x}{3 \, d^{2}} - \frac{{\left (B b^{2} c^{3} g^{2} - 3 \, B a b c^{2} d g^{2} + 3 \, B a^{2} c d^{2} g^{2}\right )} \log \left (-d x - c\right )}{3 \, d^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*B*a^3*g^2*log(b*x + a)/b + 1/3*(A*b^2*g^2 + B*b^2*g^2)*x^3 - 1/6*(B*b^2*c*g^2 - 6*A*a*b*d*g^2 - 7*B*a*b*d*
g^2)*x^2/d + 1/3*(B*b^2*g^2*x^3 + 3*B*a*b*g^2*x^2 + 3*B*a^2*g^2*x)*log((b*x + a)/(d*x + c)) + 1/3*(B*b^2*c^2*g
^2 - 3*B*a*b*c*d*g^2 + 3*A*a^2*d^2*g^2 + 5*B*a^2*d^2*g^2)*x/d^2 - 1/3*(B*b^2*c^3*g^2 - 3*B*a*b*c^2*d*g^2 + 3*B
*a^2*c*d^2*g^2)*log(-d*x - c)/d^3