\(\int \frac {(c i+d i x)^2 (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^3} \, dx\) [16]

Optimal result

Integrand size = 40, antiderivative size = 230 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {B d i^2 (c+d x)}{b^2 g^3 (a+b x)}-\frac {B i^2 (c+d x)^2}{4 b g^3 (a+b x)^2}-\frac {d i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}-\frac {d^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}+\frac {B d^2 i^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3} \]

[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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2562, 2380, 2341, 2379, 2438} \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=-\frac {d^2 i^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 g^3}-\frac {d i^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b g^3 (a+b x)^2}+\frac {B d^2 i^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3}-\frac {B d i^2 (c+d x)}{b^2 g^3 (a+b x)}-\frac {B i^2 (c+d x)^2}{4 b g^3 (a+b x)^2} \]

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

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 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 2380

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

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 {i^2 \text {Subst}\left (\int \frac {A+B \log (e x)}{x^3 (b-d x)} \, dx,x,\frac {a+b x}{c+d x}\right )}{g^3} \\ & = \frac {i^2 \text {Subst}\left (\int \frac {A+B \log (e x)}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{b g^3}+\frac {\left (d i^2\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^2 (b-d x)} \, dx,x,\frac {a+b x}{c+d x}\right )}{b g^3} \\ & = -\frac {B i^2 (c+d x)^2}{4 b g^3 (a+b x)^2}-\frac {i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}+\frac {\left (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^3}+\frac {\left (d^2 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^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}-\frac {d i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}-\frac {d^2 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^3}+\frac {\left (B d^2 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^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}-\frac {d i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^3 (a+b x)}-\frac {i^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 b g^3 (a+b x)^2}-\frac {d^2 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^3}+\frac {B d^2 i^2 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.06 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=\frac {i^2 \left (-\frac {B (b c-a d)^2}{(a+b x)^2}+\frac {6 B d (-b c+a d)}{a+b x}-6 B d^2 \log (a+b x)-\frac {2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}+\frac {8 d (-b c+a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+4 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+6 B d^2 \log (c+d x)-2 B d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{4 b^3 g^3} \]

[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)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(521\) vs. \(2(226)=452\).

Time = 1.49 (sec) , antiderivative size = 522, normalized size of antiderivative = 2.27

method	result	size
parts	\(\frac {i^{2} A \left (\frac {d^{2} \ln \left (b x +a \right )}{b^{3}}-\frac {a^{2} d^{2}-2 a b c d +b^{2} c^{2}}{2 b^{3} \left (b x +a \right )^{2}}+\frac {2 d \left (a d -c b \right )}{b^{3} \left (b x +a \right )}\right )}{g^{3}}-\frac {i^{2} B \left (a d -c b \right )^{3} e^{3} \left (-\frac {d^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{3} b^{2} e^{2}}-\frac {d^{6} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{3} b^{3} e^{3}}+\frac {d^{7} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{\left (a d -c b \right )^{3} b^{3} e^{3}}-\frac {d^{4} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} b e}\right )}{g^{3} d^{4}}\)	\(522\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{2} d^{2} e^{2} A \left (-\frac {1}{2 b e \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b^{3} e^{3}}-\frac {d}{b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}-\frac {d^{2} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{3} e^{3}}\right )}{\left (a d -c b \right ) g^{3}}-\frac {i^{2} d^{2} e^{2} B \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 b^{3} e^{3}}-\frac {d^{3} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{b^{3} e^{3}}+\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}}{b e}+\frac {d \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{b^{2} e^{2}}\right )}{\left (a d -c b \right ) g^{3}}\right )}{d^{2}}\)	\(599\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {i^{2} d^{2} e^{2} A \left (-\frac {1}{2 b e \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{b^{3} e^{3}}-\frac {d}{b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}-\frac {d^{2} \ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{3} e^{3}}\right )}{\left (a d -c b \right ) g^{3}}-\frac {i^{2} d^{2} e^{2} B \left (\frac {d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 b^{3} e^{3}}-\frac {d^{3} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{b^{3} e^{3}}+\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}}{b e}+\frac {d \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{b^{2} e^{2}}\right )}{\left (a d -c b \right ) g^{3}}\right )}{d^{2}}\)	\(599\)
risch	\(\text {Expression too large to display}\)	\(2261\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

Sympy [F]

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

[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

Maxima [F]

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

[In]

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

Giac [F]

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

[In]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^3} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (a\,g+b\,g\,x\right )}^3} \,d x \]

[In]

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