\(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^2} \, dx\) [15]

Optimal result

Integrand size = 40, antiderivative size = 247 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=-\frac {B (b c-a d) i^2 (c+d x)}{b^2 g^2 (a+b x)}+\frac {d^2 i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac {(b c-a d) i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac {B d (b c-a d) i^2 \log (c+d x)}{b^3 g^2}-\frac {2 d (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {2 B d (b c-a d) i^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2} \]

[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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2562, 46, 2393, 2341, 2351, 31, 2379, 2438} \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\frac {d^2 i^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2}-\frac {2 d i^2 (b c-a d) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^2}-\frac {i^2 (c+d x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^2 (a+b x)}+\frac {2 B d i^2 (b c-a d) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}-\frac {B d i^2 (b c-a d) \log (c+d x)}{b^3 g^2}-\frac {B i^2 (c+d x) (b c-a d)}{b^2 g^2 (a+b x)} \]

[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]

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

Rule 31

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2351

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

Rule 2379

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

Rule 2393

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2438

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

Rule 2562

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

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.89 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\frac {i^2 \left (A b d^2 x-\frac {B (b c-a d)^2}{a+b x}+B d (-b c+a d) \log (a+b x)+B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-\frac {(b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+2 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+B d (-b c+a d) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3 g^2} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(567\) vs. \(2(247)=494\).

Time = 1.56 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.30

method	result	size
parts	\(\frac {i^{2} A \left (\frac {x \,d^{2}}{b^{2}}-\frac {2 d \left (a d -c b \right ) \ln \left (b x +a \right )}{b^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{b^{3} \left (b x +a \right )}\right )}{g^{2}}-\frac {i^{2} B \left (a d -c b \right )^{3} e^{3} \left (\frac {d^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{2} b^{2} e^{2}}+\frac {d^{5} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{\left (a d -c b \right )^{2} b^{3} e^{3}}-\frac {2 d^{6} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{\left (a d -c b \right )^{2} b^{3} e^{3}}+\frac {d^{6} \left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right )}{\left (a d -c b \right )^{2} b^{2} e^{2}}\right )}{g^{2} d^{4}}\)	\(568\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {i^{2} d^{2} e^{2} A \left (\frac {d}{b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {2 d \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{3} e^{3}}-\frac {1}{b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {2 d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b^{3} e^{3}}\right )}{g^{2}}+\frac {i^{2} d^{2} e^{2} B \left (\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}}{b^{2} e^{2}}+\frac {d^{2} \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{b^{2} e^{2}}-\frac {2 d^{2} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{b^{3} e^{3}}+\frac {d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{b^{3} e^{3}}\right )}{g^{2}}\right )}{d^{2}}\)	\(632\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {i^{2} d^{2} e^{2} A \left (\frac {d}{b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}-\frac {2 d \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{3} e^{3}}-\frac {1}{b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {2 d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b^{3} e^{3}}\right )}{g^{2}}+\frac {i^{2} d^{2} e^{2} B \left (\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}}{b^{2} e^{2}}+\frac {d^{2} \left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right )}{b^{2} e^{2}}-\frac {2 d^{2} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{b^{3} e^{3}}+\frac {d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{b^{3} e^{3}}\right )}{g^{2}}\right )}{d^{2}}\)	\(632\)
risch	\(\text {Expression too large to display}\)	\(2648\)

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\text {Timed out} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 992 vs. \(2 (246) = 492\).

Time = 0.27 (sec) , antiderivative size = 992, normalized size of antiderivative = 4.02 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=-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 \left (e\right ) - a b^{2} d^{3} i^{2} \log \left (e\right )\right )} B x^{2} + {\left (a b^{2} c d^{2} i^{2} \log \left (e\right ) - a^{2} b d^{3} i^{2} \log \left (e\right )\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 \left (e\right ) + i^{2}\right )} a b^{2} c^{2} d - 3 \, {\left (i^{2} \log \left (e\right ) + i^{2}\right )} a^{2} b c d^{2} + {\left (i^{2} \log \left (e\right ) + 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 \left (e\right ) - 4 \, {\left (i^{2} \log \left (e\right ) - i^{2}\right )} a b^{2} c d^{2} + {\left (2 \, i^{2} \log \left (e\right ) - 3 \, i^{2}\right )} a^{2} b d^{3}\right )} B x - {\left (4 \, a^{2} b c d^{2} i^{2} \log \left (e\right ) - 2 \, {\left (i^{2} \log \left (e\right ) + i^{2}\right )} a b^{2} c^{2} d - {\left (2 \, i^{2} \log \left (e\right ) - 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}} \]

[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)

Giac [F]

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

[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]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \]

[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)