3.16 \(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^3} \, dx\)

Optimal. Leaf size=230 \[ -\frac {d^2 i^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3}-\frac {d i^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b g^3 (a+b x)^2}+\frac {B d^2 i^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}-\frac {B d i^2 (c+d x)}{b^2 g^3 (a+b x)}-\frac {B i^2 (c+d x)^2}{4 b g^3 (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.59, antiderivative size = 338, normalized size of antiderivative = 1.47, number of steps used = 19, number of rules used = 11, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B d^2 i^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^3}+\frac {d^2 i^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3}-\frac {2 d i^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3 (a+b x)}-\frac {i^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^3 g^3 (a+b x)^2}+\frac {B d^2 i^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {3 B d i^2 (b c-a d)}{2 b^3 g^3 (a+b x)}-\frac {B i^2 (b c-a d)^2}{4 b^3 g^3 (a+b x)^2}-\frac {B d^2 i^2 \log ^2(a+b x)}{2 b^3 g^3}-\frac {3 B d^2 i^2 \log (a+b x)}{2 b^3 g^3}+\frac {3 B d^2 i^2 \log (c+d x)}{2 b^3 g^3} \]

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)^3,x]

[Out]

-(B*(b*c - a*d)^2*i^2)/(4*b^3*g^3*(a + b*x)^2) - (3*B*d*(b*c - a*d)*i^2)/(2*b^3*g^3*(a + b*x)) - (3*B*d^2*i^2*
Log[a + b*x])/(2*b^3*g^3) - (B*d^2*i^2*Log[a + b*x]^2)/(2*b^3*g^3) - ((b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x
))/(c + d*x)]))/(2*b^3*g^3*(a + b*x)^2) - (2*d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^3*
(a + b*x)) + (d^2*i^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(b^3*g^3) + (3*B*d^2*i^2*Log[c + d*x]
)/(2*b^3*g^3) + (B*d^2*i^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(b^3*g^3) + (B*d^2*i^2*PolyLog[2, -((d
*(a + b*x))/(b*c - a*d))])/(b^3*g^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 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 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 {(16 c+16 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx &=\int \left (\frac {256 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)^3}+\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)^2}+\frac {256 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}\right ) \, dx\\ &=\frac {\left (256 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 g^3}+\frac {(512 d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 g^3}+\frac {\left (256 (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^3}\\ &=-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}-\frac {\left (256 B d^2\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 (a+b x)}{e (a+b x)} \, dx}{b^3 g^3}+\frac {(512 B d (b c-a d)) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (128 B (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}\\ &=-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {\left (512 B d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^3}+\frac {\left (128 B (b c-a d)^3\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^3}-\frac {\left (256 B d^2\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 (a+b x)}{a+b x} \, dx}{b^3 e g^3}\\ &=-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {\left (512 B d (b c-a d)^2\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^3}+\frac {\left (128 B (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^3}-\frac {\left (256 B d^2\right ) \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^3}\\ &=-\frac {64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {384 B d^2 \log (a+b x)}{b^3 g^3}-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {384 B d^2 \log (c+d x)}{b^3 g^3}-\frac {\left (256 B d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 g^3}+\frac {\left (256 B d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 g^3}\\ &=-\frac {64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {384 B d^2 \log (a+b x)}{b^3 g^3}-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {384 B d^2 \log (c+d x)}{b^3 g^3}+\frac {256 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {\left (256 B d^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 g^3}-\frac {\left (256 B d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 g^3}\\ &=-\frac {64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {384 B d^2 \log (a+b x)}{b^3 g^3}-\frac {128 B d^2 \log ^2(a+b x)}{b^3 g^3}-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {384 B d^2 \log (c+d x)}{b^3 g^3}+\frac {256 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}-\frac {\left (256 B d^2\right ) \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^3}\\ &=-\frac {64 B (b c-a d)^2}{b^3 g^3 (a+b x)^2}-\frac {384 B d (b c-a d)}{b^3 g^3 (a+b x)}-\frac {384 B d^2 \log (a+b x)}{b^3 g^3}-\frac {128 B d^2 \log ^2(a+b x)}{b^3 g^3}-\frac {128 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)^2}-\frac {512 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3 (a+b x)}+\frac {256 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^3}+\frac {384 B d^2 \log (c+d x)}{b^3 g^3}+\frac {256 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^3 g^3}+\frac {256 B d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^3 g^3}\\ \end {align*}

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Mathematica [A]  time = 0.31, size = 244, normalized size = 1.06 \[ \frac {i^2 \left (4 d^2 \log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {8 d (a d-b c) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}-2 B d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+\frac {6 B d (a d-b c)}{a+b x}-\frac {B (b c-a d)^2}{(a+b x)^2}-6 B d^2 \log (a+b x)+6 B d^2 \log (c+d x)\right )}{4 b^3 g^3} \]

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)^3,x]

[Out]

(i^2*(-((B*(b*c - a*d)^2)/(a + b*x)^2) + (6*B*d*(-(b*c) + a*d))/(a + b*x) - 6*B*d^2*Log[a + b*x] - (2*(b*c - a
*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 + (8*d*(-(b*c) + a*d)*(A + B*Log[(e*(a + b*x))/(c + d*
x)]))/(a + b*x) + 4*d^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 6*B*d^2*Log[c + d*x] - 2*B*d^2*(Lo
g[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)])))/(
4*b^3*g^3)

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fricas [F]  time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d^{2} i^{2} x^{2} + 2 \, A c d i^{2} x + A c^{2} i^{2} + {\left (B d^{2} i^{2} x^{2} + 2 \, B c d i^{2} x + B c^{2} i^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{3} g^{3} x^{3} + 3 \, a b^{2} g^{3} x^{2} + 3 \, a^{2} b g^{3} x + a^{3} g^{3}}, x\right ) \]

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)^3,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^3*g^3*x^3 + 3*a*b^2*g^3*x^2 + 3*a^2*b*g^3*x + a^3*g^3), 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)^3,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.06, size = 1495, normalized size = 6.50 \[ \text {result too large to display} \]

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)^3,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, B d^{2} i^{2} {\left (\frac {{\left (4 \, a b x + 3 \, a^{2} + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b^{5} g^{3} x^{2} + 2 \, a b^{4} g^{3} x + a^{2} b^{3} g^{3}} - 2 \, \int \frac {2 \, b^{3} d x^{3} \log \relax (e) + 7 \, a^{2} b d x + 3 \, a^{3} d + 2 \, {\left (b^{3} c \log \relax (e) + 2 \, a b^{2} d\right )} x^{2} + 2 \, {\left (2 \, b^{3} d x^{3} + 3 \, a^{2} b d x + a^{3} d + {\left (b^{3} c + 3 \, a b^{2} d\right )} x^{2}\right )} \log \left (b x + a\right )}{2 \, {\left (b^{6} d g^{3} x^{4} + a^{3} b^{3} c g^{3} + {\left (b^{6} c g^{3} + 3 \, a b^{5} d g^{3}\right )} x^{3} + 3 \, {\left (a b^{5} c g^{3} + a^{2} b^{4} d g^{3}\right )} x^{2} + {\left (3 \, a^{2} b^{4} c g^{3} + a^{3} b^{3} d g^{3}\right )} x\right )}}\,{d x}\right )} - \frac {1}{2} \, B c d i^{2} {\left (\frac {2 \, {\left (2 \, b x + a\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}} + \frac {3 \, a b c - a^{2} d + 2 \, {\left (2 \, b^{2} c - a b d\right )} x}{{\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}} + \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (b x + a\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}} - \frac {2 \, {\left (2 \, b c d - a d^{2}\right )} \log \left (d x + c\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3}}\right )} + \frac {1}{2} \, A d^{2} i^{2} {\left (\frac {4 \, a b x + 3 \, a^{2}}{b^{5} g^{3} x^{2} + 2 \, a b^{4} g^{3} x + a^{2} b^{3} g^{3}} + \frac {2 \, \log \left (b x + a\right )}{b^{3} g^{3}}\right )} + \frac {1}{4} \, B c^{2} i^{2} {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c - a b^{3} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{3} c - a^{2} b^{2} d\right )} g^{3} x + {\left (a^{2} b^{2} c - a^{3} b d\right )} g^{3}} - \frac {2 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{3}}\right )} - \frac {{\left (2 \, b x + a\right )} A c d i^{2}}{b^{4} g^{3} x^{2} + 2 \, a b^{3} g^{3} x + a^{2} b^{2} g^{3}} - \frac {A c^{2} i^{2}}{2 \, {\left (b^{3} g^{3} x^{2} + 2 \, a b^{2} g^{3} x + a^{2} b g^{3}\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)^3,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (a\,g+b\,g\,x\right )}^3} \,d x \]

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)^3,x)

[Out]

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

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

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)**3,x)

[Out]

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

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