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

Optimal. Leaf size=247 \[ \frac {d^2 i^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2}-\frac {2 d i^2 (b c-a d) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2}-\frac {i^2 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2 (a+b x)}+\frac {2 B d i^2 (b c-a d) \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}-\frac {B d i^2 (b c-a d) \log (c+d x)}{b^3 g^2}-\frac {B i^2 (c+d x) (b c-a d)}{b^2 g^2 (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.52, antiderivative size = 313, normalized size of antiderivative = 1.27, number of steps used = 18, 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, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {2 B d i^2 (b c-a d) \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}+\frac {2 d i^2 (b c-a d) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2}-\frac {i^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2 (a+b x)}+\frac {B d^2 i^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {B i^2 (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {B d i^2 (b c-a d) \log ^2(a+b x)}{b^3 g^2}-\frac {B d i^2 (b c-a d) \log (a+b x)}{b^3 g^2}+\frac {2 B d i^2 (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {A d^2 i^2 x}{b^2 g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 44

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] && L
tQ[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 {(15 c+15 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx &=\int \left (\frac {225 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2}+\frac {225 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)^2}+\frac {450 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}\right ) \, dx\\ &=\frac {\left (225 d^2\right ) \int \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \, dx}{b^2 g^2}+\frac {(450 d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^2}+\frac {\left (225 (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^2}\\ &=\frac {225 A d^2 x}{b^2 g^2}-\frac {225 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {450 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {\left (225 B d^2\right ) \int \log \left (\frac {e (a+b x)}{c+d x}\right ) \, dx}{b^2 g^2}-\frac {(450 B d (b c-a d)) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^3 g^2}+\frac {\left (225 B (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}\\ &=\frac {225 A d^2 x}{b^2 g^2}+\frac {225 B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {225 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {450 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac {\left (225 B d^2 (b c-a d)\right ) \int \frac {1}{c+d x} \, dx}{b^3 g^2}+\frac {\left (225 B (b c-a d)^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^2}-\frac {(450 B d (b c-a d)) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 e g^2}\\ &=\frac {225 A d^2 x}{b^2 g^2}+\frac {225 B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {225 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {450 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac {225 B d (b c-a d) \log (c+d x)}{b^3 g^2}+\frac {\left (225 B (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^2}-\frac {(450 B d (b c-a d)) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 e g^2}\\ &=\frac {225 A d^2 x}{b^2 g^2}-\frac {225 B (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {225 B d (b c-a d) \log (a+b x)}{b^3 g^2}+\frac {225 B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {225 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {450 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac {(450 B d (b c-a d)) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^2}+\frac {\left (450 B d^2 (b c-a d)\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^2}\\ &=\frac {225 A d^2 x}{b^2 g^2}-\frac {225 B (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {225 B d (b c-a d) \log (a+b x)}{b^3 g^2}+\frac {225 B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {225 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {450 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {450 B d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {(450 B d (b c-a d)) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^2}-\frac {(450 B d (b c-a d)) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^2}\\ &=\frac {225 A d^2 x}{b^2 g^2}-\frac {225 B (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {225 B d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac {225 B d (b c-a d) \log ^2(a+b x)}{b^3 g^2}+\frac {225 B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {225 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {450 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {450 B d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}-\frac {(450 B d (b c-a d)) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 g^2}\\ &=\frac {225 A d^2 x}{b^2 g^2}-\frac {225 B (b c-a d)^2}{b^3 g^2 (a+b x)}-\frac {225 B d (b c-a d) \log (a+b x)}{b^3 g^2}-\frac {225 B d (b c-a d) \log ^2(a+b x)}{b^3 g^2}+\frac {225 B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{b^3 g^2}-\frac {225 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2 (a+b x)}+\frac {450 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}+\frac {450 B d (b c-a d) \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^2}+\frac {450 B d (b c-a d) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^2}\\ \end {align*}

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Mathematica [A]  time = 0.23, size = 221, normalized size = 0.89 \[ \frac {i^2 \left (2 d (b c-a d) \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )-\frac {(b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+B d (a d-b c) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-\frac {B (b c-a d)^2}{a+b x}+B d (a d-b c) \log (a+b x)+A b d^2 x\right )}{b^3 g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d^{2} i^{2} x^{2} + 2 \, A c d i^{2} x + A c^{2} i^{2} + {\left (B d^{2} i^{2} x^{2} + 2 \, B c d i^{2} x + B c^{2} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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maple [B]  time = 0.14, size = 1465, normalized size = 5.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.91, size = 992, normalized size = 4.02 \[ -A {\left (\frac {a^{2}}{b^{4} g^{2} x + a b^{3} g^{2}} - \frac {x}{b^{2} g^{2}} + \frac {2 \, a \log \left (b x + a\right )}{b^{3} g^{2}}\right )} d^{2} i^{2} + 2 \, A c d i^{2} {\left (\frac {a}{b^{3} g^{2} x + a b^{2} g^{2}} + \frac {\log \left (b x + a\right )}{b^{2} g^{2}}\right )} - B c^{2} i^{2} {\left (\frac {\log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {A c^{2} i^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac {{\left (b^{2} c^{2} d i^{2} + a b c d^{2} i^{2} - a^{2} d^{3} i^{2}\right )} B \log \left (d x + c\right )}{b^{4} c g^{2} - a b^{3} d g^{2}} + \frac {{\left (b^{3} c d^{2} i^{2} \log \relax (e) - a b^{2} d^{3} i^{2} \log \relax (e)\right )} B x^{2} + {\left (a b^{2} c d^{2} i^{2} \log \relax (e) - a^{2} b d^{3} i^{2} \log \relax (e)\right )} B x + {\left ({\left (b^{3} c^{2} d i^{2} - 2 \, a b^{2} c d^{2} i^{2} + a^{2} b d^{3} i^{2}\right )} B x + {\left (a b^{2} c^{2} d i^{2} - 2 \, a^{2} b c d^{2} i^{2} + a^{3} d^{3} i^{2}\right )} B\right )} \log \left (b x + a\right )^{2} + {\left (2 \, {\left (i^{2} \log \relax (e) + i^{2}\right )} a b^{2} c^{2} d - 3 \, {\left (i^{2} \log \relax (e) + i^{2}\right )} a^{2} b c d^{2} + {\left (i^{2} \log \relax (e) + i^{2}\right )} a^{3} d^{3}\right )} B + {\left ({\left (b^{3} c d^{2} i^{2} - a b^{2} d^{3} i^{2}\right )} B x^{2} + {\left (2 \, b^{3} c^{2} d i^{2} \log \relax (e) - 4 \, {\left (i^{2} \log \relax (e) - i^{2}\right )} a b^{2} c d^{2} + {\left (2 \, i^{2} \log \relax (e) - 3 \, i^{2}\right )} a^{2} b d^{3}\right )} B x - {\left (4 \, a^{2} b c d^{2} i^{2} \log \relax (e) - 2 \, {\left (i^{2} \log \relax (e) + i^{2}\right )} a b^{2} c^{2} d - {\left (2 \, i^{2} \log \relax (e) - i^{2}\right )} a^{3} d^{3}\right )} B\right )} \log \left (b x + a\right ) - {\left ({\left (b^{3} c d^{2} i^{2} - a b^{2} d^{3} i^{2}\right )} B x^{2} + {\left (a b^{2} c d^{2} i^{2} - a^{2} b d^{3} i^{2}\right )} B x + {\left (2 \, a b^{2} c^{2} d i^{2} - 3 \, a^{2} b c d^{2} i^{2} + a^{3} d^{3} i^{2}\right )} B + 2 \, {\left ({\left (b^{3} c^{2} d i^{2} - 2 \, a b^{2} c d^{2} i^{2} + a^{2} b d^{3} i^{2}\right )} B x + {\left (a b^{2} c^{2} d i^{2} - 2 \, a^{2} b c d^{2} i^{2} + a^{3} d^{3} i^{2}\right )} B\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{4} c g^{2} - a^{2} b^{3} d g^{2} + {\left (b^{5} c g^{2} - a b^{4} d g^{2}\right )} x} + \frac {2 \, {\left (b c d i^{2} - a d^{2} i^{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}{b^{3} g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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