3.31 \(\int \frac {(a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))}{c i+d i x} \, dx\)

Optimal. Leaf size=252 \[ \frac {g^3 (b c-a d)^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+11 B\right )}{6 d^4 i}+\frac {g^3 (a+b x) (b c-a d)^2 \left (6 B \log \left (\frac {e (a+b x)}{c+d x}\right )+6 A+5 B\right )}{6 d^3 i}-\frac {g^3 (a+b x)^2 (b c-a d) \left (3 B \log \left (\frac {e (a+b x)}{c+d x}\right )+3 A+B\right )}{6 d^2 i}+\frac {g^3 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d i}+\frac {B g^3 (b c-a d)^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i} \]

[Out]

1/3*g^3*(b*x+a)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/d/i-1/6*(-a*d+b*c)*g^3*(b*x+a)^2*(3*A+B+3*B*ln(e*(b*x+a)/(d*x+c)
))/d^2/i+1/6*(-a*d+b*c)^2*g^3*(b*x+a)*(6*A+5*B+6*B*ln(e*(b*x+a)/(d*x+c)))/d^3/i+1/6*(-a*d+b*c)^3*g^3*ln((-a*d+
b*c)/b/(d*x+c))*(6*A+11*B+6*B*ln(e*(b*x+a)/(d*x+c)))/d^4/i+B*(-a*d+b*c)^3*g^3*polylog(2,d*(b*x+a)/b/(d*x+c))/d
^4/i

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Rubi [A]  time = 0.63, antiderivative size = 408, normalized size of antiderivative = 1.62, number of steps used = 23, number of rules used = 13, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.325, Rules used = {2528, 2486, 31, 2525, 12, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {B g^3 (b c-a d)^3 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^4 i}-\frac {g^3 (a+b x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^2 i}-\frac {g^3 (b c-a d)^3 \log (c i+d i x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4 i}+\frac {g^3 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 d i}+\frac {A b g^3 x (b c-a d)^2}{d^3 i}+\frac {B g^3 (a+b x) (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i}-\frac {B g^3 (a+b x)^2 (b c-a d)}{6 d^2 i}+\frac {5 b B g^3 x (b c-a d)^2}{6 d^3 i}-\frac {B g^3 (b c-a d)^3 \log ^2(i (c+d x))}{2 d^4 i}-\frac {11 B g^3 (b c-a d)^3 \log (c+d x)}{6 d^4 i}+\frac {B g^3 (b c-a d)^3 \log (c i+d i x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^4 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b*(b*c - a*d)^2*g^3*x)/(d^3*i) + (5*b*B*(b*c - a*d)^2*g^3*x)/(6*d^3*i) - (B*(b*c - a*d)*g^3*(a + b*x)^2)/(6
*d^2*i) + (B*(b*c - a*d)^2*g^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^3*i) - ((b*c - a*d)*g^3*(a + b*x)^2*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^2*i) + (g^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*d*
i) - (11*B*(b*c - a*d)^3*g^3*Log[c + d*x])/(6*d^4*i) - (B*(b*c - a*d)^3*g^3*Log[i*(c + d*x)]^2)/(2*d^4*i) + (B
*(b*c - a*d)^3*g^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c*i + d*i*x])/(d^4*i) - ((b*c - a*d)^3*g^3*(A + B*Log
[(e*(a + b*x))/(c + d*x)])*Log[c*i + d*i*x])/(d^4*i) + (B*(b*c - a*d)^3*g^3*PolyLog[2, (b*(c + d*x))/(b*c - a*
d)])/(d^4*i)

Rule 12

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, 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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

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

Rule 2391

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 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

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 2524

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

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]

Rubi steps

\begin {align*} \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{31 c+31 d x} \, dx &=\int \left (\frac {b (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{31 d^3}+\frac {(-b c+a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 (31 c+31 d x)}-\frac {b (b c-a d) g^2 (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{31 d^2}+\frac {b g (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{31 d}\right ) \, dx\\ &=\frac {(b g) \int (a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{31 d}-\frac {\left (b (b c-a d) g^2\right ) \int (a g+b g x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{31 d^2}+\frac {\left (b (b c-a d)^2 g^3\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{31 d^3}-\frac {\left ((b c-a d)^3 g^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{31 c+31 d x} \, dx}{d^3}\\ &=\frac {A b (b c-a d)^2 g^3 x}{31 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac {B \int \frac {(b c-a d) g^3 (a+b x)^2}{c+d x} \, dx}{93 d}+\frac {(B (b c-a d) g) \int \frac {(b c-a d) g^2 (a+b x)}{c+d x} \, dx}{62 d^2}+\frac {\left (b B (b c-a d)^2 g^3\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{31 d^3}+\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (31 c+31 d x)}{e (a+b x)} \, dx}{31 d^4}\\ &=\frac {A b (b c-a d)^2 g^3 x}{31 d^3}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{31 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac {\left (B (b c-a d) g^3\right ) \int \frac {(a+b x)^2}{c+d x} \, dx}{93 d}+\frac {\left (B (b c-a d)^2 g^3\right ) \int \frac {a+b x}{c+d x} \, dx}{62 d^2}-\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {1}{c+d x} \, dx}{31 d^3}+\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (31 c+31 d x)}{a+b x} \, dx}{31 d^4 e}\\ &=\frac {A b (b c-a d)^2 g^3 x}{31 d^3}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{31 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac {B (b c-a d)^3 g^3 \log (c+d x)}{31 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac {\left (B (b c-a d) g^3\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}{93 d}+\frac {\left (B (b c-a d)^2 g^3\right ) \int \left (\frac {b}{d}+\frac {-b c+a d}{d (c+d x)}\right ) \, dx}{62 d^2}+\frac {\left (B (b c-a d)^3 g^3\right ) \int \left (\frac {b e \log (31 c+31 d x)}{a+b x}-\frac {d e \log (31 c+31 d x)}{c+d x}\right ) \, dx}{31 d^4 e}\\ &=\frac {A b (b c-a d)^2 g^3 x}{31 d^3}+\frac {5 b B (b c-a d)^2 g^3 x}{186 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2}{186 d^2}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{31 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac {11 B (b c-a d)^3 g^3 \log (c+d x)}{186 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}+\frac {\left (b B (b c-a d)^3 g^3\right ) \int \frac {\log (31 c+31 d x)}{a+b x} \, dx}{31 d^4}-\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {\log (31 c+31 d x)}{c+d x} \, dx}{31 d^3}\\ &=\frac {A b (b c-a d)^2 g^3 x}{31 d^3}+\frac {5 b B (b c-a d)^2 g^3 x}{186 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2}{186 d^2}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{31 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac {11 B (b c-a d)^3 g^3 \log (c+d x)}{186 d^4}+\frac {B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (31 c+31 d x)}{31 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac {\left (B (b c-a d)^3 g^3\right ) \operatorname {Subst}\left (\int \frac {31 \log (x)}{x} \, dx,x,31 c+31 d x\right )}{961 d^4}-\frac {\left (B (b c-a d)^3 g^3\right ) \int \frac {\log \left (\frac {31 d (a+b x)}{-31 b c+31 a d}\right )}{31 c+31 d x} \, dx}{d^3}\\ &=\frac {A b (b c-a d)^2 g^3 x}{31 d^3}+\frac {5 b B (b c-a d)^2 g^3 x}{186 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2}{186 d^2}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{31 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac {11 B (b c-a d)^3 g^3 \log (c+d x)}{186 d^4}+\frac {B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (31 c+31 d x)}{31 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}-\frac {\left (B (b c-a d)^3 g^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,31 c+31 d x\right )}{31 d^4}-\frac {\left (B (b c-a d)^3 g^3\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-31 b c+31 a d}\right )}{x} \, dx,x,31 c+31 d x\right )}{31 d^4}\\ &=\frac {A b (b c-a d)^2 g^3 x}{31 d^3}+\frac {5 b B (b c-a d)^2 g^3 x}{186 d^3}-\frac {B (b c-a d) g^3 (a+b x)^2}{186 d^2}+\frac {B (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{31 d^3}-\frac {(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{62 d^2}+\frac {g^3 (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{93 d}-\frac {11 B (b c-a d)^3 g^3 \log (c+d x)}{186 d^4}-\frac {B (b c-a d)^3 g^3 \log ^2(31 (c+d x))}{62 d^4}+\frac {B (b c-a d)^3 g^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (31 c+31 d x)}{31 d^4}-\frac {(b c-a d)^3 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (31 c+31 d x)}{31 d^4}+\frac {B (b c-a d)^3 g^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{31 d^4}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 354, normalized size = 1.40 \[ \frac {g^3 \left (2 d^3 (a+b x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+3 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-6 (b c-a d)^3 \log (i (c+d x)) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+6 A b d x (b c-a d)^2+B (b c-a d) \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+3 B (b c-a d)^3 \left (\log (i (c+d x)) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (i (c+d x))\right )+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-6 B (b c-a d)^3 \log (c+d x)+3 B (b c-a d)^2 ((a d-b c) \log (c+d x)+b d x)\right )}{6 d^4 i} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)] + 3*d^2*(-(b*c) + a
*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*d^3*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])
- 6*B*(b*c - a*d)^3*Log[c + d*x] + B*(b*c - a*d)*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[
c + d*x]) + 3*B*(b*c - a*d)^2*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) - 6*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))
/(c + d*x)])*Log[i*(c + d*x)] + 3*B*(b*c - a*d)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[i*(c + d*x)])*Lo
g[i*(c + d*x)] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/(6*d^4*i)

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fricas [F]  time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b^{3} g^{3} x^{3} + 3 \, A a b^{2} g^{3} x^{2} + 3 \, A a^{2} b g^{3} x + A a^{3} g^{3} + {\left (B b^{3} g^{3} x^{3} + 3 \, B a b^{2} g^{3} x^{2} + 3 \, B a^{2} b g^{3} x + B a^{3} g^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{d i x + c i}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x, algorithm="fricas")

[Out]

integral((A*b^3*g^3*x^3 + 3*A*a*b^2*g^3*x^2 + 3*A*a^2*b*g^3*x + A*a^3*g^3 + (B*b^3*g^3*x^3 + 3*B*a*b^2*g^3*x^2
+ 3*B*a^2*b*g^3*x + B*a^3*g^3)*log((b*e*x + a*e)/(d*x + c)))/(d*i*x + c*i), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.16, size = 4297, normalized size = 17.05 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^3*(B*ln((b*x+a)/(d*x+c)*e)+A)/(d*i*x+c*i),x)

[Out]

-1/d*A*g^3/i*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^3-1/d*B*g^3/i*dilog(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*
e)*d)/b/e)*a^3-11/6/d*B*g^3/i*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^3-3/d^3*A*g^3/i*ln(-b*e+(b/d*e+(a*d-b
*c)/(d*x+c)/d*e)*d)*b^2*c^2*a-3/d^4*e*A*g^3/i*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*c^3+3/d^2*B*g^3/i*dilog(-(
-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*a^2*b*c+15/d^2*e^2*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+
c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^2*b^3*c^3/(d*x+c)^2+5/d^2*e^3*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)
*a*d*e-1/(d*x+c)*b*c*e)^3*b^4*c^4/(d*x+c)^3*a^2+18/d^2*e*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*
d*e-1/(d*x+c)*b*c*e)*b^2*c^2/(d*x+c)*a^2-15/d*e^2*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(
d*x+c)*b*c*e)^2*a^3*b^2*c^2/(d*x+c)^2-2/d^3*e^3*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*
x+c)*b*c*e)^3*b^5*c^5/(d*x+c)^3*a-2*d*e^3*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b
*c*e)^3*a^5*b*c/(d*x+c)^3-20/3/d*e^3*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)
^3*b^3*c^3/(d*x+c)^3*a^3-15/2/d^3*e^2*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e
)^2*a*b^4*c^4/(d*x+c)^2-12/d*e*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^3*b
*c/(d*x+c)-12/d^3*e*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*b^3*c^3/(d*x+c)*
a+11/6/d^4*B*g^3/i*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*b^3*c^3+1/d^4*A*g^3/i*ln(-b*e+(b/d*e+(a*d-b*c)/(d*
x+c)/d*e)*d)*b^3*c^3+1/d^4*B*g^3/i*dilog(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*b^3*c^3-1/d*B*g^3/i*ln(b
/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*a^3+7/2/d^3*e*B*g^3/i*b^3/(1/(d*x+
c)*a*d*e-1/(d*x+c)*b*c*e)*c^2*a+1/d^3*e^3*A*g^3/i*b^5/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^2*a-3/2/d^4*e^2*A*
g^3/i*b^5/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*c^3-7/6/d^4*e*B*g^3/i*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*c^3+
1/3/d*e^3*A*g^3/i*b^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^3+3/2/d*e^2*A*g^3/i*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)
*b*c*e)^2*a^3+1/6/d*e^2*B*g^3/i*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^3-1/2/d^2*e^2*B*g^3/i*b^3/(1/(d*x+c)
*a*d*e-1/(d*x+c)*b*c*e)^2*a^2*c-3/2/d^4*e^2*B*g^3/i*b^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*
x+c)*b*c*e)^2*c^3-3/2*d*e^2*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^5/(d
*x+c)^2+3/2/d*e^2*B*g^3/i*b^2*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^3+1/3/d*e^
3*B*g^3/i*b^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^3+3/d*e*B*g^3/i*ln(b/d*e+(
a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^3*b-3/d^4*e*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/
(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*b^4*c^3-11/2/d^3*B*g^3/i*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*b^2*c^2*a+
3*e*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^4/(d*x+c)+1/d^4*B*g^3/i*ln(b/d
*e+(a*d-b*c)/(d*x+c)/d*e)*ln(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*b^3*c^3+3/d*e*A*g^3/i*b/(1/(d*x+c)*a
*d*e-1/(d*x+c)*b*c*e)*a^3-9/d^2*e*A*g^3/i*b^2/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^2*c-7/2/d^2*e*B*g^3/i*b^2/(1
/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^2*c+3/d^2*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(-(-b*e+(b/d*e+(a*d-b*c)
/(d*x+c)/d*e)*d)/b/e)*a^2*b*c-3/d^3*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*ln(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)
/d*e)*d)/b/e)*a*b^2*c^2+1/3*d^2*e^3*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^
3*a^6/(d*x+c)^3+1/2/d^3*e^2*B*g^3/i*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*c^2*a-9/2/d^2*e^2*A*g^3/i*b^3/(1/(
d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^2*c+9/2/d^3*e^2*A*g^3/i*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a*c^2-1/d^2*
e^3*A*g^3/i*b^4/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^2*c+7/6/d*e*B*g^3/i*b/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*
a^3-1/3/d^4*e^3*A*g^3/i*b^6/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^3+3/2/d^4*e^2*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*
x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*b^5*c^5/(d*x+c)^2+1/d^3*e^3*B*g^3/i*b^5*ln(b/d*e+(a*d-b*c)/(d*x+
c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^2*a-9/2/d^2*e^2*B*g^3/i*b^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(
d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^2*c+1/3/d^4*e^3*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(
d*x+c)*b*c*e)^3*b^6*c^6/(d*x+c)^3+9/2/d^3*e^2*B*g^3/i*b^4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(
d*x+c)*b*c*e)^2*c^2*a+5*e^3*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^4*b^
2*c^2/(d*x+c)^3+15/2*e^2*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^2*a^4*b*c/(
d*x+c)^2-1/d^2*e^3*B*g^3/i*b^4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*a^2*c-9/d^2
*e*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a^2*b^2*c+9/d^3*e*B*g^3/i*ln(b/d*
e+(a*d-b*c)/(d*x+c)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*a*b^3*c^2+3/d^4*e*B*g^3/i*ln(b/d*e+(a*d-b*c)/(d*x+c
)/d*e)/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*b^4*c^4/(d*x+c)-1/6/d^4*e^2*B*g^3/i*b^5/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*
c*e)^2*c^3+11/2/d^2*B*g^3/i*ln(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)*a^2*b*c+3/d^2*A*g^3/i*ln(-b*e+(b/d*e+(a*d
-b*c)/(d*x+c)/d*e)*d)*a^2*b*c-3/d^3*B*g^3/i*dilog(-(-b*e+(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*d)/b/e)*b^2*c^2*a+9/d^3
*e*A*g^3/i*b^3/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)*c^2*a-1/3/d^4*e^3*B*g^3/i*b^6*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)
/(1/(d*x+c)*a*d*e-1/(d*x+c)*b*c*e)^3*c^3

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maxima [B]  time = 1.79, size = 790, normalized size = 3.13 \[ 3 \, A a^{2} b g^{3} {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} - \frac {1}{6} \, A b^{3} g^{3} {\left (\frac {6 \, c^{3} \log \left (d x + c\right )}{d^{4} i} - \frac {2 \, d^{2} x^{3} - 3 \, c d x^{2} + 6 \, c^{2} x}{d^{3} i}\right )} + \frac {3}{2} \, A a b^{2} g^{3} {\left (\frac {2 \, c^{2} \log \left (d x + c\right )}{d^{3} i} + \frac {d x^{2} - 2 \, c x}{d^{2} i}\right )} + \frac {A a^{3} g^{3} \log \left (d i x + c i\right )}{d i} - \frac {{\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\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}{d^{4} i} + \frac {{\left (6 \, a^{3} d^{3} g^{3} \log \relax (e) - {\left (6 \, g^{3} \log \relax (e) + 11 \, g^{3}\right )} b^{3} c^{3} + 9 \, {\left (2 \, g^{3} \log \relax (e) + 3 \, g^{3}\right )} a b^{2} c^{2} d - 18 \, {\left (g^{3} \log \relax (e) + g^{3}\right )} a^{2} b c d^{2}\right )} B \log \left (d x + c\right )}{6 \, d^{4} i} + \frac {2 \, B b^{3} d^{3} g^{3} x^{3} \log \relax (e) - {\left ({\left (3 \, g^{3} \log \relax (e) + g^{3}\right )} b^{3} c d^{2} - {\left (9 \, g^{3} \log \relax (e) + g^{3}\right )} a b^{2} d^{3}\right )} B x^{2} + 3 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} B \log \left (d x + c\right )^{2} + {\left ({\left (6 \, g^{3} \log \relax (e) + 5 \, g^{3}\right )} b^{3} c^{2} d - 6 \, {\left (3 \, g^{3} \log \relax (e) + 2 \, g^{3}\right )} a b^{2} c d^{2} + {\left (18 \, g^{3} \log \relax (e) + 7 \, g^{3}\right )} a^{2} b d^{3}\right )} B x + {\left (2 \, B b^{3} d^{3} g^{3} x^{3} - 3 \, {\left (b^{3} c d^{2} g^{3} - 3 \, a b^{2} d^{3} g^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{2} d g^{3} - 3 \, a b^{2} c d^{2} g^{3} + 3 \, a^{2} b d^{3} g^{3}\right )} B x + {\left (6 \, a b^{2} c^{2} d g^{3} - 15 \, a^{2} b c d^{2} g^{3} + 11 \, a^{3} d^{3} g^{3}\right )} B\right )} \log \left (b x + a\right ) - {\left (2 \, B b^{3} d^{3} g^{3} x^{3} - 3 \, {\left (b^{3} c d^{2} g^{3} - 3 \, a b^{2} d^{3} g^{3}\right )} B x^{2} + 6 \, {\left (b^{3} c^{2} d g^{3} - 3 \, a b^{2} c d^{2} g^{3} + 3 \, a^{2} b d^{3} g^{3}\right )} B x\right )} \log \left (d x + c\right )}{6 \, d^{4} i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^3*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x, algorithm="maxima")

[Out]

3*A*a^2*b*g^3*(x/(d*i) - c*log(d*x + c)/(d^2*i)) - 1/6*A*b^3*g^3*(6*c^3*log(d*x + c)/(d^4*i) - (2*d^2*x^3 - 3*
c*d*x^2 + 6*c^2*x)/(d^3*i)) + 3/2*A*a*b^2*g^3*(2*c^2*log(d*x + c)/(d^3*i) + (d*x^2 - 2*c*x)/(d^2*i)) + A*a^3*g
^3*log(d*i*x + c*i)/(d*i) - (b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*(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/(d^4*i) + 1/6*(6*a^3*d^3*g^3*log(e)
- (6*g^3*log(e) + 11*g^3)*b^3*c^3 + 9*(2*g^3*log(e) + 3*g^3)*a*b^2*c^2*d - 18*(g^3*log(e) + g^3)*a^2*b*c*d^2)*
B*log(d*x + c)/(d^4*i) + 1/6*(2*B*b^3*d^3*g^3*x^3*log(e) - ((3*g^3*log(e) + g^3)*b^3*c*d^2 - (9*g^3*log(e) + g
^3)*a*b^2*d^3)*B*x^2 + 3*(b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*B*log(d*x + c)^2
+ ((6*g^3*log(e) + 5*g^3)*b^3*c^2*d - 6*(3*g^3*log(e) + 2*g^3)*a*b^2*c*d^2 + (18*g^3*log(e) + 7*g^3)*a^2*b*d^3
)*B*x + (2*B*b^3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^
3 + 3*a^2*b*d^3*g^3)*B*x + (6*a*b^2*c^2*d*g^3 - 15*a^2*b*c*d^2*g^3 + 11*a^3*d^3*g^3)*B)*log(b*x + a) - (2*B*b^
3*d^3*g^3*x^3 - 3*(b^3*c*d^2*g^3 - 3*a*b^2*d^3*g^3)*B*x^2 + 6*(b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^3 + 3*a^2*b*d^3
*g^3)*B*x)*log(d*x + c))/(d^4*i)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^3\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{c\,i+d\,i\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a*g + b*g*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(c*i + d*i*x),x)

[Out]

int(((a*g + b*g*x)^3*(A + B*log((e*(a + b*x))/(c + d*x))))/(c*i + d*i*x), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i),x)

[Out]

Timed out

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