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

Optimal. Leaf size=251 \[ -\frac {b^2 g^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 i^3}-\frac {g^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d i^3 (c+d x)^2}-\frac {A b g^2 (a+b x)}{d^2 i^3 (c+d x)}-\frac {b^2 B g^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^3 i^3}-\frac {b B g^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^2 i^3 (c+d x)}+\frac {b B g^2 (a+b x)}{d^2 i^3 (c+d x)}+\frac {B g^2 (a+b x)^2}{4 d i^3 (c+d x)^2} \]

[Out]

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

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Rubi [A]  time = 0.61, antiderivative size = 340, normalized size of antiderivative = 1.35, 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, 2394, 2393, 2391, 2390, 2301} \[ -\frac {b^2 B g^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^3 i^3}+\frac {b^2 g^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 i^3}+\frac {2 b g^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 i^3 (c+d x)}-\frac {g^2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 d^3 i^3 (c+d x)^2}-\frac {b^2 B g^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^3 i^3}-\frac {3 b^2 B g^2 \log (a+b x)}{2 d^3 i^3}-\frac {3 b B g^2 (b c-a d)}{2 d^3 i^3 (c+d x)}+\frac {B g^2 (b c-a d)^2}{4 d^3 i^3 (c+d x)^2}+\frac {b^2 B g^2 \log ^2(c+d x)}{2 d^3 i^3}+\frac {3 b^2 B g^2 \log (c+d x)}{2 d^3 i^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(B*(b*c - a*d)^2*g^2)/(4*d^3*i^3*(c + d*x)^2) - (3*b*B*(b*c - a*d)*g^2)/(2*d^3*i^3*(c + d*x)) - (3*b^2*B*g^2*L
og[a + b*x])/(2*d^3*i^3) - ((b*c - a*d)^2*g^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d^3*i^3*(c + d*x)^2) +
(2*b*(b*c - a*d)*g^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*i^3*(c + d*x)) + (3*b^2*B*g^2*Log[c + d*x])/(2
*d^3*i^3) - (b^2*B*g^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d^3*i^3) + (b^2*g^2*(A + B*Log[(e*(a +
b*x))/(c + d*x)])*Log[c + d*x])/(d^3*i^3) + (b^2*B*g^2*Log[c + d*x]^2)/(2*d^3*i^3) - (b^2*B*g^2*PolyLog[2, (b
*(c + d*x))/(b*c - a*d)])/(d^3*i^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 {(a g+b g x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(48 c+48 d x)^3} \, dx &=\int \left (\frac {(-b c+a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{110592 d^2 (c+d x)^3}-\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^2 (c+d x)^2}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{110592 d^2 (c+d x)}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{110592 d^2}-\frac {\left (b (b c-a d) g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{55296 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{110592 d^2}\\ &=-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{221184 d^3 (c+d x)^2}+\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}-\frac {\left (b^2 B g^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 (c+d x)}{e (a+b x)} \, dx}{110592 d^3}-\frac {\left (b B (b c-a d) g^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{55296 d^3}+\frac {\left (B (b c-a d)^2 g^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{221184 d^3}\\ &=-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{221184 d^3 (c+d x)^2}+\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}-\frac {\left (b B (b c-a d)^2 g^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{55296 d^3}+\frac {\left (B (b c-a d)^3 g^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{221184 d^3}-\frac {\left (b^2 B g^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 (c+d x)}{a+b x} \, dx}{110592 d^3 e}\\ &=-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{221184 d^3 (c+d x)^2}+\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^3 (c+d x)}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}-\frac {\left (b B (b c-a d)^2 g^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{55296 d^3}+\frac {\left (B (b c-a d)^3 g^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{221184 d^3}-\frac {\left (b^2 B g^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{110592 d^3 e}\\ &=\frac {B (b c-a d)^2 g^2}{442368 d^3 (c+d x)^2}-\frac {b B (b c-a d) g^2}{73728 d^3 (c+d x)}-\frac {b^2 B g^2 \log (a+b x)}{73728 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{221184 d^3 (c+d x)^2}+\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^3 (c+d x)}+\frac {b^2 B g^2 \log (c+d x)}{73728 d^3}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}-\frac {\left (b^3 B g^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{110592 d^3}+\frac {\left (b^2 B g^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{110592 d^2}\\ &=\frac {B (b c-a d)^2 g^2}{442368 d^3 (c+d x)^2}-\frac {b B (b c-a d) g^2}{73728 d^3 (c+d x)}-\frac {b^2 B g^2 \log (a+b x)}{73728 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{221184 d^3 (c+d x)^2}+\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^3 (c+d x)}+\frac {b^2 B g^2 \log (c+d x)}{73728 d^3}-\frac {b^2 B g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}+\frac {\left (b^2 B g^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{110592 d^3}+\frac {\left (b^2 B g^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{110592 d^2}\\ &=\frac {B (b c-a d)^2 g^2}{442368 d^3 (c+d x)^2}-\frac {b B (b c-a d) g^2}{73728 d^3 (c+d x)}-\frac {b^2 B g^2 \log (a+b x)}{73728 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{221184 d^3 (c+d x)^2}+\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^3 (c+d x)}+\frac {b^2 B g^2 \log (c+d x)}{73728 d^3}-\frac {b^2 B g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 B g^2 \log ^2(c+d x)}{221184 d^3}+\frac {\left (b^2 B g^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{110592 d^3}\\ &=\frac {B (b c-a d)^2 g^2}{442368 d^3 (c+d x)^2}-\frac {b B (b c-a d) g^2}{73728 d^3 (c+d x)}-\frac {b^2 B g^2 \log (a+b x)}{73728 d^3}-\frac {(b c-a d)^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{221184 d^3 (c+d x)^2}+\frac {b (b c-a d) g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{55296 d^3 (c+d x)}+\frac {b^2 B g^2 \log (c+d x)}{73728 d^3}-\frac {b^2 B g^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 g^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{110592 d^3}+\frac {b^2 B g^2 \log ^2(c+d x)}{221184 d^3}-\frac {b^2 B g^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{110592 d^3}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 245, normalized size = 0.98 \[ \frac {g^2 \left (4 b^2 \log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )+\frac {8 b (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-\frac {2 (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(c+d x)^2}-2 b^2 B \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-6 b^2 B \log (a+b x)-\frac {6 b B (b c-a d)}{c+d x}+\frac {B (b c-a d)^2}{(c+d x)^2}+6 b^2 B \log (c+d x)\right )}{4 d^3 i^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b^{2} g^{2} x^{2} + 2 \, A a b g^{2} x + A a^{2} g^{2} + {\left (B b^{2} g^{2} x^{2} + 2 \, B a b g^{2} x + B a^{2} g^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{d^{3} i^{3} x^{3} + 3 \, c d^{2} i^{3} x^{2} + 3 \, c^{2} d i^{3} x + c^{3} i^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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maple [B]  time = 0.06, size = 1569, normalized size = 6.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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