3.3.12 \(\int (a g+b g x)^2 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2 \, dx\) [212]

Optimal. Leaf size=343 \[ \frac {4 B^2 (b c-a d)^2 g^2 x}{3 d^2}-\frac {4 B^2 (b c-a d)^3 g^2 \log (a+b x)}{b d^3}-\frac {4 B^2 (b c-a d)^3 g^2 \log \left (\frac {c+d x}{a+b x}\right )}{3 b d^3}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}-\frac {4 B (b c-a d)^2 g^2 (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {4 B (b c-a d)^3 g^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3}+\frac {8 B^2 (b c-a d)^3 g^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3} \]

[Out]

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

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Rubi [A]
time = 0.25, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2552, 2356, 2389, 2379, 2438, 2351, 31, 46} \begin {gather*} \frac {8 B^2 g^2 (b c-a d)^3 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b d^3}-\frac {4 B g^2 (b c-a d)^3 \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b d^3}-\frac {4 B g^2 (c+d x) (b c-a d)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 d^3}+\frac {2 B g^2 (a+b x)^2 (b c-a d) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b d}+\frac {g^2 (a+b x)^3 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{3 b}-\frac {4 B^2 g^2 (b c-a d)^3 \log (a+b x)}{b d^3}-\frac {4 B^2 g^2 (b c-a d)^3 \log \left (\frac {c+d x}{a+b x}\right )}{3 b d^3}+\frac {4 B^2 g^2 x (b c-a d)^2}{3 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*B^2*(b*c - a*d)^2*g^2*x)/(3*d^2) - (4*B^2*(b*c - a*d)^3*g^2*Log[a + b*x])/(b*d^3) - (4*B^2*(b*c - a*d)^3*g^
2*Log[(c + d*x)/(a + b*x)])/(3*b*d^3) + (2*B*(b*c - a*d)*g^2*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^
2]))/(3*b*d) - (4*B*(b*c - a*d)^2*g^2*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*d^3) + (g^2*(a +
b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(3*b) - (4*B*(b*c - a*d)^3*g^2*(A + B*Log[(e*(c + d*x)^2)/(
a + b*x)^2])*Log[1 - (d*(a + b*x))/(b*(c + d*x))])/(3*b*d^3) + (8*B^2*(b*c - a*d)^3*g^2*PolyLog[2, (d*(a + b*x
))/(b*(c + d*x))])/(3*b*d^3)

Rule 31

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

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2351

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2552

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

Rubi steps

\begin {align*} \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx &=\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {(2 B) \int \frac {2 (b c-a d) g^3 (a+b x)^2 \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{c+d x} \, dx}{3 b g}\\ &=\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {\left (4 B (b c-a d) g^2\right ) \int \frac {(a+b x)^2 \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{c+d x} \, dx}{3 b}\\ &=\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {\left (4 B (b c-a d) g^2\right ) \int \left (-\frac {b (b c-a d) \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^2}+\frac {b (a+b x) \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{d}+\frac {(-b c+a d)^2 \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^2 (c+d x)}\right ) \, dx}{3 b}\\ &=\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {\left (4 B (b c-a d) g^2\right ) \int (a+b x) \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{3 d}+\frac {\left (4 B (b c-a d)^2 g^2\right ) \int \left (-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{3 d^2}-\frac {\left (4 B (b c-a d)^3 g^2\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{c+d x} \, dx}{3 b d^2}\\ &=-\frac {4 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}+\frac {4 B (b c-a d)^3 g^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {\left (2 B^2 (b c-a d) g^2\right ) \int \frac {2 (b c-a d) (-a-b x)}{c+d x} \, dx}{3 b d}-\frac {\left (4 B^2 (b c-a d)^2 g^2\right ) \int \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right ) \, dx}{3 d^2}-\frac {\left (4 B^2 (b c-a d)^3 g^2\right ) \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{e (c+d x)^2} \, dx}{3 b d^3}\\ &=-\frac {4 A B (b c-a d)^2 g^2 x}{3 d^2}-\frac {4 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{3 b d^2}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}+\frac {4 B (b c-a d)^3 g^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {\left (4 B^2 (b c-a d)^2 g^2\right ) \int \frac {-a-b x}{c+d x} \, dx}{3 b d}-\frac {\left (8 B^2 (b c-a d)^3 g^2\right ) \int \frac {1}{c+d x} \, dx}{3 b d^2}-\frac {\left (4 B^2 (b c-a d)^3 g^2\right ) \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{(c+d x)^2} \, dx}{3 b d^3 e}\\ &=-\frac {4 A B (b c-a d)^2 g^2 x}{3 d^2}-\frac {8 B^2 (b c-a d)^3 g^2 \log (c+d x)}{3 b d^3}-\frac {4 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{3 b d^2}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}+\frac {4 B (b c-a d)^3 g^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {\left (4 B^2 (b c-a d)^2 g^2\right ) \int \left (-\frac {b}{d}+\frac {b c-a d}{d (c+d x)}\right ) \, dx}{3 b d}-\frac {\left (4 B^2 (b c-a d)^3 g^2\right ) \int \left (-\frac {2 b e \log (c+d x)}{a+b x}+\frac {2 d e \log (c+d x)}{c+d x}\right ) \, dx}{3 b d^3 e}\\ &=-\frac {4 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac {4 B^2 (b c-a d)^2 g^2 x}{3 d^2}-\frac {4 B^2 (b c-a d)^3 g^2 \log (c+d x)}{b d^3}-\frac {4 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{3 b d^2}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}+\frac {4 B (b c-a d)^3 g^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}+\frac {\left (8 B^2 (b c-a d)^3 g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 d^3}-\frac {\left (8 B^2 (b c-a d)^3 g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b d^2}\\ &=-\frac {4 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac {4 B^2 (b c-a d)^2 g^2 x}{3 d^2}-\frac {4 B^2 (b c-a d)^3 g^2 \log (c+d x)}{b d^3}+\frac {8 B^2 (b c-a d)^3 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac {4 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{3 b d^2}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}+\frac {4 B (b c-a d)^3 g^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {\left (8 B^2 (b c-a d)^3 g^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b d^3}-\frac {\left (8 B^2 (b c-a d)^3 g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b d^2}\\ &=-\frac {4 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac {4 B^2 (b c-a d)^2 g^2 x}{3 d^2}-\frac {4 B^2 (b c-a d)^3 g^2 \log (c+d x)}{b d^3}+\frac {8 B^2 (b c-a d)^3 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac {4 B^2 (b c-a d)^3 g^2 \log ^2(c+d x)}{3 b d^3}-\frac {4 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{3 b d^2}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}+\frac {4 B (b c-a d)^3 g^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}-\frac {\left (8 B^2 (b c-a d)^3 g^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b d^3}\\ &=-\frac {4 A B (b c-a d)^2 g^2 x}{3 d^2}+\frac {4 B^2 (b c-a d)^2 g^2 x}{3 d^2}-\frac {4 B^2 (b c-a d)^3 g^2 \log (c+d x)}{b d^3}+\frac {8 B^2 (b c-a d)^3 g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b d^3}-\frac {4 B^2 (b c-a d)^3 g^2 \log ^2(c+d x)}{3 b d^3}-\frac {4 B^2 (b c-a d)^2 g^2 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{3 b d^2}+\frac {2 B (b c-a d) g^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d}+\frac {4 B (b c-a d)^3 g^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b d^3}+\frac {g^2 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b}+\frac {8 B^2 (b c-a d)^3 g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b d^3}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 298, normalized size = 0.87 \begin {gather*} \frac {g^2 \left ((a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2-\frac {2 B (b c-a d) \left (2 A b d (b c-a d) x+4 B (b c-a d)^2 \log (c+d x)-2 B (b c-a d) (b d x+(-b c+a d) \log (c+d x))+2 B d (b c-a d) (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )-d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-2 (b c-a d)^2 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-2 B (b c-a d)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{d^3}\right )}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(g^2*((a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2 - (2*B*(b*c - a*d)*(2*A*b*d*(b*c - a*d)*x + 4*B*(
b*c - a*d)^2*Log[c + d*x] - 2*B*(b*c - a*d)*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 2*B*d*(b*c - a*d)*(a + b*x
)*Log[(e*(c + d*x)^2)/(a + b*x)^2] - d^2*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) - 2*(b*c - a*d)^
2*Log[c + d*x]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) - 2*B*(b*c - a*d)^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*
d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/d^3))/(3*b)

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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{2} \left (A +B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1190 vs. \(2 (332) = 664\).
time = 0.43, size = 1190, normalized size = 3.47 \begin {gather*} \frac {1}{3} \, A^{2} b^{2} g^{2} x^{3} + A^{2} a b g^{2} x^{2} + 2 \, {\left (x \log \left (\frac {d^{2} x^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d x e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) - \frac {2 \, a \log \left (b x + a\right )}{b} + \frac {2 \, c \log \left (d x + c\right )}{d}\right )} A B a^{2} g^{2} + 2 \, {\left (x^{2} \log \left (\frac {d^{2} x^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d x e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{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 )} A B a b g^{2} + \frac {2}{3} \, {\left (x^{3} \log \left (\frac {d^{2} x^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {2 \, c d x e}{b^{2} x^{2} + 2 \, a b x + a^{2}} + \frac {c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{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 )} A B b^{2} g^{2} + A^{2} a^{2} g^{2} x - \frac {4 \, {\left (2 \, b^{2} c^{3} g^{2} - 4 \, a b c^{2} d g^{2} + a^{2} c d^{2} g^{2}\right )} B^{2} \log \left (d x + c\right )}{3 \, d^{3}} - \frac {8 \, {\left (b^{3} c^{3} g^{2} - 3 \, a b^{2} c^{2} d g^{2} + 3 \, a^{2} b c d^{2} g^{2} - a^{3} d^{3} g^{2}\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B^{2}}{3 \, b d^{3}} + \frac {B^{2} b^{3} d^{3} g^{2} x^{3} + {\left (2 \, b^{3} c d^{2} g^{2} + a b^{2} d^{3} g^{2}\right )} B^{2} x^{2} + {\left (4 \, a b^{2} c d^{2} g^{2} - a^{2} b d^{3} g^{2}\right )} B^{2} x + 4 \, {\left (B^{2} b^{3} d^{3} g^{2} x^{3} + 3 \, B^{2} a b^{2} d^{3} g^{2} x^{2} + 3 \, B^{2} a^{2} b d^{3} g^{2} x + B^{2} a^{3} d^{3} g^{2}\right )} \log \left (b x + a\right )^{2} + 4 \, {\left (B^{2} b^{3} d^{3} g^{2} x^{3} + 3 \, B^{2} a b^{2} d^{3} g^{2} x^{2} + 3 \, B^{2} a^{2} b d^{3} g^{2} x + {\left (b^{3} c^{3} g^{2} - 3 \, a b^{2} c^{2} d g^{2} + 3 \, a^{2} b c d^{2} g^{2}\right )} B^{2}\right )} \log \left (d x + c\right )^{2} - 4 \, {\left (B^{2} b^{3} d^{3} g^{2} x^{3} + {\left (b^{3} c d^{2} g^{2} + 2 \, a b^{2} d^{3} g^{2}\right )} B^{2} x^{2} - {\left (2 \, b^{3} c^{2} d g^{2} - 6 \, a b^{2} c d^{2} g^{2} + a^{2} b d^{3} g^{2}\right )} B^{2} x - {\left (2 \, a b^{2} c^{2} d g^{2} - 5 \, a^{2} b c d^{2} g^{2} + 2 \, a^{3} d^{3} g^{2}\right )} B^{2}\right )} \log \left (b x + a\right ) + 4 \, {\left (B^{2} b^{3} d^{3} g^{2} x^{3} + {\left (b^{3} c d^{2} g^{2} + 2 \, a b^{2} d^{3} g^{2}\right )} B^{2} x^{2} - {\left (2 \, b^{3} c^{2} d g^{2} - 6 \, a b^{2} c d^{2} g^{2} + a^{2} b d^{3} g^{2}\right )} B^{2} x - 2 \, {\left (B^{2} b^{3} d^{3} g^{2} x^{3} + 3 \, B^{2} a b^{2} d^{3} g^{2} x^{2} + 3 \, B^{2} a^{2} b d^{3} g^{2} x + B^{2} a^{3} d^{3} g^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{3 \, b d^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*A^2*b^2*g^2*x^3 + A^2*a*b*g^2*x^2 + 2*(x*log(d^2*x^2*e/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*x*e/(b^2*x^2 + 2*
a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) - 2*a*log(b*x + a)/b + 2*c*log(d*x + c)/d)*A*B*a^2*g^2 + 2*(x^
2*log(d^2*x^2*e/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*x*e/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a
^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))*A*B*a*b*g^2 + 2/3*(x^3*log(d^2
*x^2*e/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*x*e/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^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))
*A*B*b^2*g^2 + A^2*a^2*g^2*x - 4/3*(2*b^2*c^3*g^2 - 4*a*b*c^2*d*g^2 + a^2*c*d^2*g^2)*B^2*log(d*x + c)/d^3 - 8/
3*(b^3*c^3*g^2 - 3*a*b^2*c^2*d*g^2 + 3*a^2*b*c*d^2*g^2 - a^3*d^3*g^2)*(log(b*x + a)*log((b*d*x + a*d)/(b*c - a
*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B^2/(b*d^3) + 1/3*(B^2*b^3*d^3*g^2*x^3 + (2*b^3*c*d^2*g^2 + a*b^
2*d^3*g^2)*B^2*x^2 + (4*a*b^2*c*d^2*g^2 - a^2*b*d^3*g^2)*B^2*x + 4*(B^2*b^3*d^3*g^2*x^3 + 3*B^2*a*b^2*d^3*g^2*
x^2 + 3*B^2*a^2*b*d^3*g^2*x + B^2*a^3*d^3*g^2)*log(b*x + a)^2 + 4*(B^2*b^3*d^3*g^2*x^3 + 3*B^2*a*b^2*d^3*g^2*x
^2 + 3*B^2*a^2*b*d^3*g^2*x + (b^3*c^3*g^2 - 3*a*b^2*c^2*d*g^2 + 3*a^2*b*c*d^2*g^2)*B^2)*log(d*x + c)^2 - 4*(B^
2*b^3*d^3*g^2*x^3 + (b^3*c*d^2*g^2 + 2*a*b^2*d^3*g^2)*B^2*x^2 - (2*b^3*c^2*d*g^2 - 6*a*b^2*c*d^2*g^2 + a^2*b*d
^3*g^2)*B^2*x - (2*a*b^2*c^2*d*g^2 - 5*a^2*b*c*d^2*g^2 + 2*a^3*d^3*g^2)*B^2)*log(b*x + a) + 4*(B^2*b^3*d^3*g^2
*x^3 + (b^3*c*d^2*g^2 + 2*a*b^2*d^3*g^2)*B^2*x^2 - (2*b^3*c^2*d*g^2 - 6*a*b^2*c*d^2*g^2 + a^2*b*d^3*g^2)*B^2*x
- 2*(B^2*b^3*d^3*g^2*x^3 + 3*B^2*a*b^2*d^3*g^2*x^2 + 3*B^2*a^2*b*d^3*g^2*x + B^2*a^3*d^3*g^2)*log(b*x + a))*l
og(d*x + c))/(b*d^3)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a\,g+b\,g\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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