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

Optimal. Leaf size=120 \[ \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b}-\frac {2 B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}+\frac {2 B g^2 x (b c-a d)^2}{3 d^2}-\frac {B g^2 (a+b x)^2 (b c-a d)}{3 b d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2525, 12, 43} \[ \frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 b}+\frac {2 B g^2 x (b c-a d)^2}{3 d^2}-\frac {2 B g^2 (b c-a d)^3 \log (c+d x)}{3 b d^3}-\frac {B g^2 (a+b x)^2 (b c-a d)}{3 b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.06, size = 98, normalized size = 0.82 \[ \frac {g^2 \left (\frac {B (a d-b c) \left (d \left (a^2 d+4 a b d x+b^2 x (d x-2 c)\right )+2 (b c-a d)^2 \log (c+d x)\right )}{d^3}+(a+b x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )\right )}{3 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.75, size = 243, normalized size = 2.02 \[ \frac {A b^{3} d^{3} g^{2} x^{3} + 2 \, B a^{3} d^{3} g^{2} \log \left (b x + a\right ) - {\left (B b^{3} c d^{2} - {\left (3 \, A + B\right )} a b^{2} d^{3}\right )} g^{2} x^{2} + {\left (2 \, B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + {\left (3 \, A + 4 \, B\right )} a^{2} b d^{3}\right )} g^{2} x - 2 \, {\left (B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} g^{2} \log \left (d x + c\right ) + {\left (B b^{3} d^{3} g^{2} x^{3} + 3 \, B a b^{2} d^{3} g^{2} x^{2} + 3 \, B a^{2} b d^{3} g^{2} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{3 \, b d^{3}} \]

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)^2/(d*x+c)^2)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 3.43, size = 252, normalized size = 2.10 \[ \frac {2 \, B a^{3} g^{2} \log \left (b x + a\right )}{3 \, b} + \frac {1}{3} \, {\left (A b^{2} g^{2} + B b^{2} g^{2}\right )} x^{3} - \frac {{\left (B b^{2} c g^{2} - 3 \, A a b d g^{2} - 4 \, B a b d g^{2}\right )} x^{2}}{3 \, d} + \frac {1}{3} \, {\left (B b^{2} g^{2} x^{3} + 3 \, B a b g^{2} x^{2} + 3 \, B a^{2} g^{2} x\right )} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {{\left (2 \, B b^{2} c^{2} g^{2} - 6 \, B a b c d g^{2} + 3 \, A a^{2} d^{2} g^{2} + 7 \, B a^{2} d^{2} g^{2}\right )} x}{3 \, d^{2}} - \frac {2 \, {\left (B b^{2} c^{3} g^{2} - 3 \, B a b c^{2} d g^{2} + 3 \, B a^{2} c d^{2} g^{2}\right )} \log \left (-d x - c\right )}{3 \, d^{3}} \]

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)^2/(d*x+c)^2)),x, algorithm="giac")

[Out]

2/3*B*a^3*g^2*log(b*x + a)/b + 1/3*(A*b^2*g^2 + B*b^2*g^2)*x^3 - 1/3*(B*b^2*c*g^2 - 3*A*a*b*d*g^2 - 4*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^2*x^2 + 2*a*b*x + a^2)/(d^2*x^2 + 2*
c*d*x + c^2)) + 1/3*(2*B*b^2*c^2*g^2 - 6*B*a*b*c*d*g^2 + 3*A*a^2*d^2*g^2 + 7*B*a^2*d^2*g^2)*x/d^2 - 2/3*(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

________________________________________________________________________________________

maple [B]  time = 0.08, size = 915, normalized size = 7.62 \[ \frac {B \,b^{2} g^{2} x^{3} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{3}+\frac {A \,b^{2} g^{2} x^{3}}{3}+\frac {2 B \,a^{4} d \,g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{\left (a d -b c \right ) b}-\frac {8 B \,a^{3} c \,g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{a d -b c}+\frac {12 B \,a^{2} b \,c^{2} g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{\left (a d -b c \right ) d}-\frac {8 B a \,b^{2} c^{3} g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{\left (a d -b c \right ) d^{2}}+B a b \,g^{2} x^{2} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )+\frac {2 B \,b^{3} c^{4} g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{\left (a d -b c \right ) d^{3}}+A a b \,g^{2} x^{2}+B \,a^{2} g^{2} x \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )+\frac {B a b \,g^{2} x^{2}}{3}-\frac {B \,b^{2} c \,g^{2} x^{2}}{3 d}+A \,a^{2} g^{2} x -\frac {2 B \,a^{3} g^{2} \ln \left (\frac {1}{d x +c}\right )}{3 b}-\frac {4 B \,a^{3} g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{3 b}+\frac {2 B \,a^{2} c \,g^{2} \ln \left (\frac {1}{d x +c}\right )}{d}+\frac {B \,a^{2} c \,g^{2} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{d}+\frac {4 B \,a^{2} c \,g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{d}+\frac {4 B \,a^{2} g^{2} x}{3}-\frac {2 B a b \,c^{2} g^{2} \ln \left (\frac {1}{d x +c}\right )}{d^{2}}-\frac {B a b \,c^{2} g^{2} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{d^{2}}-\frac {4 B a b \,c^{2} g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{d^{2}}-\frac {2 B a b c \,g^{2} x}{d}+\frac {2 B \,b^{2} c^{3} g^{2} \ln \left (\frac {1}{d x +c}\right )}{3 d^{3}}+\frac {B \,b^{2} c^{3} g^{2} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{3 d^{3}}+\frac {4 B \,b^{2} c^{3} g^{2} \ln \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}{3 d^{3}}+\frac {2 B \,b^{2} c^{2} g^{2} x}{3 d^{2}}+\frac {A \,a^{2} c \,g^{2}}{d}-\frac {A a b \,c^{2} g^{2}}{d^{2}}+\frac {A \,b^{2} c^{3} g^{2}}{3 d^{3}}+\frac {4 B \,a^{2} c \,g^{2}}{3 d}-\frac {7 B a b \,c^{2} g^{2}}{3 d^{2}}+\frac {B \,b^{2} c^{3} g^{2}}{d^{3}} \]

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)^2/(d*x+c)^2)),x)

[Out]

1/3*B*g^2*b*a*x^2+B*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*x*a^2*g^2+1/d*A*g^2*a^2*c+1/3/d^3*A*g^2*b^2*c^
3+1/d^3*B*g^2*c^3*b^2+4/3/d*B*g^2*a^2*c-4/3*B*g^2/b*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^3-2/3*B*g^2/b*ln(1/(d*
x+c))*a^3+1/3*B*g^2*b^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*x^3+A*g^2*x^2*a*b+4/3*B*x*a^2*g^2+A*g^2*x*
a^2+1/3*A*g^2*x^3*b^2-1/d^2*B*g^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*a*b*c^2-2/d*B*g^2*a*b*c*x+2*d*B*
g^2/b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^4-2/d^2*B*g^2*b*ln(1/(d*x+c))*a*c^2-4/d^2*B*g^2*b*ln(1/(d*
x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^2+4/d*B*g^2*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*c+2/d*B*g^2*ln(1/(d*x+c))*a^2*
c+1/d*B*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*a^2*c*g^2+4/3/d^3*B*g^2*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*
c^3*b^2-1/d^2*A*g^2*a*b*c^2-8/d^2*B*g^2/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a*c^3*b^2-7/3/d^2*B*g^2*b*
a*c^2+12/d*B*g^2*b/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*a^2*c^2+2/3/d^3*B*g^2*ln(1/(d*x+c))*c^3*b^2+1/3
/d^3*B*g^2*b^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*c^3-8*B*g^2/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*
c+b)*a^3*c+B*g^2*b*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)*a*x^2+2/3/d^2*B*g^2*c^2*b^2*x-1/3/d*B*g^2*b^2*c
*x^2+2/d^3*B*g^2/(a*d-b*c)*ln(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)*c^4*b^3

________________________________________________________________________________________

maxima [B]  time = 1.41, size = 437, normalized size = 3.64 \[ \frac {1}{3} \, A b^{2} g^{2} x^{3} + A a b g^{2} x^{2} + {\left (x \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a \log \left (b x + a\right )}{b} - \frac {2 \, c \log \left (d x + c\right )}{d}\right )} B a^{2} g^{2} + {\left (x^{2} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac {2 \, a^{2} \log \left (b x + a\right )}{b^{2}} + \frac {2 \, c^{2} \log \left (d x + c\right )}{d^{2}} - \frac {2 \, {\left (b c - a d\right )} x}{b d}\right )} B a b g^{2} + \frac {1}{3} \, {\left (x^{3} \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + \frac {2 \, a^{3} \log \left (b x + a\right )}{b^{3}} - \frac {2 \, c^{3} \log \left (d x + c\right )}{d^{3}} - \frac {{\left (b^{2} c d - a b d^{2}\right )} x^{2} - 2 \, {\left (b^{2} c^{2} - a^{2} d^{2}\right )} x}{b^{2} d^{2}}\right )} B b^{2} g^{2} + A a^{2} g^{2} x \]

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)^2/(d*x+c)^2)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.59, size = 296, normalized size = 2.47 \[ x^2\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{6\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{6\,d}\right )-x\,\left (\frac {\left (3\,a\,d+3\,b\,c\right )\,\left (\frac {b\,g^2\,\left (9\,A\,a\,d+3\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{3\,d}-\frac {A\,b\,g^2\,\left (3\,a\,d+3\,b\,c\right )}{3\,d}\right )}{3\,b\,d}-\frac {a\,g^2\,\left (3\,A\,a\,d+3\,A\,b\,c+2\,B\,a\,d-2\,B\,b\,c\right )}{d}+\frac {A\,a\,b\,c\,g^2}{d}\right )+\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (B\,a^2\,g^2\,x+B\,a\,b\,g^2\,x^2+\frac {B\,b^2\,g^2\,x^3}{3}\right )-\frac {\ln \left (c+d\,x\right )\,\left (6\,B\,a^2\,c\,d^2\,g^2-6\,B\,a\,b\,c^2\,d\,g^2+2\,B\,b^2\,c^3\,g^2\right )}{3\,d^3}+\frac {A\,b^2\,g^2\,x^3}{3}+\frac {2\,B\,a^3\,g^2\,\ln \left (a+b\,x\right )}{3\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 3.51, size = 517, normalized size = 4.31 \[ \frac {A b^{2} g^{2} x^{3}}{3} + \frac {2 B a^{3} g^{2} \log {\left (x + \frac {\frac {2 B a^{4} d^{3} g^{2}}{b} + 6 B a^{3} c d^{2} g^{2} - 6 B a^{2} b c^{2} d g^{2} + 2 B a b^{2} c^{3} g^{2}}{2 B a^{3} d^{3} g^{2} + 6 B a^{2} b c d^{2} g^{2} - 6 B a b^{2} c^{2} d g^{2} + 2 B b^{3} c^{3} g^{2}} \right )}}{3 b} - \frac {2 B c g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) \log {\left (x + \frac {8 B a^{3} c d^{2} g^{2} - 6 B a^{2} b c^{2} d g^{2} + 2 B a b^{2} c^{3} g^{2} - 2 B a c g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right ) + \frac {2 B b c^{2} g^{2} \left (3 a^{2} d^{2} - 3 a b c d + b^{2} c^{2}\right )}{d}}{2 B a^{3} d^{3} g^{2} + 6 B a^{2} b c d^{2} g^{2} - 6 B a b^{2} c^{2} d g^{2} + 2 B b^{3} c^{3} g^{2}} \right )}}{3 d^{3}} + x^{2} \left (A a b g^{2} + \frac {B a b g^{2}}{3} - \frac {B b^{2} c g^{2}}{3 d}\right ) + x \left (A a^{2} g^{2} + \frac {4 B a^{2} g^{2}}{3} - \frac {2 B a b c g^{2}}{d} + \frac {2 B b^{2} c^{2} g^{2}}{3 d^{2}}\right ) + \left (B a^{2} g^{2} x + B a b g^{2} x^{2} + \frac {B b^{2} g^{2} x^{3}}{3}\right ) \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )} \]

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)**2/(d*x+c)**2)),x)

[Out]

A*b**2*g**2*x**3/3 + 2*B*a**3*g**2*log(x + (2*B*a**4*d**3*g**2/b + 6*B*a**3*c*d**2*g**2 - 6*B*a**2*b*c**2*d*g*
*2 + 2*B*a*b**2*c**3*g**2)/(2*B*a**3*d**3*g**2 + 6*B*a**2*b*c*d**2*g**2 - 6*B*a*b**2*c**2*d*g**2 + 2*B*b**3*c*
*3*g**2))/(3*b) - 2*B*c*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2)*log(x + (8*B*a**3*c*d**2*g**2 - 6*B*a**2*b*
c**2*d*g**2 + 2*B*a*b**2*c**3*g**2 - 2*B*a*c*g**2*(3*a**2*d**2 - 3*a*b*c*d + b**2*c**2) + 2*B*b*c**2*g**2*(3*a
**2*d**2 - 3*a*b*c*d + b**2*c**2)/d)/(2*B*a**3*d**3*g**2 + 6*B*a**2*b*c*d**2*g**2 - 6*B*a*b**2*c**2*d*g**2 + 2
*B*b**3*c**3*g**2))/(3*d**3) + x**2*(A*a*b*g**2 + B*a*b*g**2/3 - B*b**2*c*g**2/(3*d)) + x*(A*a**2*g**2 + 4*B*a
**2*g**2/3 - 2*B*a*b*c*g**2/d + 2*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)**2/(c + d*x)**2)

________________________________________________________________________________________