Integrand size = 40, antiderivative size = 247 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=-\frac {B (b c-a d) i^2 (c+d x)}{b^2 g^2 (a+b x)}+\frac {d^2 i^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^3 g^2}-\frac {(b c-a d) i^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^2 (a+b x)}-\frac {B d (b c-a d) i^2 \log (c+d x)}{b^3 g^2}-\frac {2 d (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2}+\frac {2 B d (b c-a d) i^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3 g^2} \] Output:
-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
Time = 0.26 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.89 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\frac {i^2 \left (A b d^2 x-\frac {B (b c-a d)^2}{a+b x}+B d (-b c+a d) \log (a+b x)+B d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-\frac {(b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}+2 d (b c-a d) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+B d (-b c+a d) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3 g^2} \] Input:
Integrate[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b* g*x)^2,x]
Output:
(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)
Time = 0.52 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.90, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2962, 2793, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(c i+d i x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{(a g+b g x)^2} \, dx\) |
| \(\Big \downarrow \) 2962 |
| \(\displaystyle \frac {i^2 (b c-a d) \int \frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{g^2}\) |
| \(\Big \downarrow \) 2793 |
| \(\displaystyle \frac {i^2 (b c-a d) \int \left (\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d^2}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d}{b^2 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {(c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 (a+b x)^2}\right )d\frac {a+b x}{c+d x}}{g^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i^2 (b c-a d) \left (\frac {d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3 (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 d \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^3}-\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{b^2 (a+b x)}+\frac {2 B d \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^3}+\frac {B d \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}-\frac {B (c+d x)}{b^2 (a+b x)}\right )}{g^2}\) |
Input:
Int[((c*i + d*i*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a*g + b*g*x)^2 ,x]
Output:
((b*c - a*d)*i^2*(-((B*(c + d*x))/(b^2*(a + b*x))) - ((c + d*x)*(A + B*Log [(e*(a + b*x))/(c + d*x)]))/(b^2*(a + b*x)) + (d^2*(a + b*x)*(A + B*Log[(e *(a + b*x))/(c + d*x)]))/(b^3*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + ( B*d*Log[b - (d*(a + b*x))/(c + d*x)])/b^3 - (2*d*(A + B*Log[(e*(a + b*x))/ (c + d*x)])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/b^3 + (2*B*d*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b^3))/g^2
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_.), x_Symbol] :> With[{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] && Integer Q[r]))
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Sy mbol] :> Simp[(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] && IGt Q[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && I ntegersQ[m, q]
Leaf count of result is larger than twice the leaf count of optimal. \(567\) vs. \(2(247)=494\).
Time = 3.06 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.30
| method | result | size |
| parts | \(\frac {i^{2} A \left (\frac {x \,d^{2}}{b^{2}}-\frac {2 d \left (d a -b c \right ) \ln \left (b x +a \right )}{b^{3}}-\frac {a^{2} d^{2}-2 a c d b +c^{2} b^{2}}{b^{3} \left (b x +a \right )}\right )}{g^{2}}-\frac {i^{2} B \left (d a -b c \right )^{3} e^{3} \left (\frac {\left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}\right ) d^{4}}{\left (d a -b c \right )^{2} b^{2} e^{2}}+\frac {\left (\frac {\ln \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}{b e d}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e \right )}\right ) d^{6}}{\left (d a -b c \right )^{2} b^{2} e^{2}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d^{5}}{\left (d a -b c \right )^{2} b^{3} e^{3}}-\frac {2 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{6}}{\left (d a -b c \right )^{2} b^{3} e^{3}}\right )}{g^{2} d^{4}}\) | \(568\) |
| derivativedivides | \(-\frac {e \left (d a -b c \right ) \left (\frac {i^{2} d^{2} e^{2} A \left (-\frac {1}{b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {2 d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b^{3} e^{3}}+\frac {d}{b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \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 (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{3} e^{3}}\right )}{g^{2}}+\frac {i^{2} d^{2} e^{2} B \left (-\frac {2 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \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 (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{2}}{b^{3} e^{3}}+\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}}{b^{2} e^{2}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d}{b^{3} e^{3}}+\frac {\left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right ) d^{2}}{b^{2} e^{2}}\right )}{g^{2}}\right )}{d^{2}}\) | \(632\) |
| default | \(-\frac {e \left (d a -b c \right ) \left (\frac {i^{2} d^{2} e^{2} A \left (-\frac {1}{b^{2} e^{2} \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}+\frac {2 d \ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b^{3} e^{3}}+\frac {d}{b^{2} e^{2} \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \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 (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b^{3} e^{3}}\right )}{g^{2}}+\frac {i^{2} d^{2} e^{2} B \left (-\frac {2 \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \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 (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right ) d^{2}}{b^{3} e^{3}}+\frac {-\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}}}{b^{2} e^{2}}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )^{2} d}{b^{3} e^{3}}+\frac {\left (\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}{b e d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) \left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right )}{b e \left (b e -\left (\frac {b e}{d}+\frac {\left (d a -b c \right ) e}{d \left (d x +c \right )}\right ) d \right )}\right ) d^{2}}{b^{2} e^{2}}\right )}{g^{2}}\right )}{d^{2}}\) | \(632\) |
| risch | \(\text {Expression too large to display}\) | \(2648\) |
Input:
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=_RETU RNVERBOSE)
Output:
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/(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)))/ (a*d-b*c)^2*d^4/b^2/e^2+(1/b/e*ln((b*e/d+(a*d-b*c)*e/d/(d*x+c))*d-b*e)/d-l n(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))/(a*d-b*c)^2*d^6/b^2/e^2+ln(b*e/d+(a*d-b*c)*e /d/(d*x+c))^2/(a*d-b*c)^2*d^5/b^3/e^3-2*(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)/(a*d-b*c)^2*d^6/b^3/e^3)
\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\int { \frac {{\left (d i x + c i\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (b g x + a g\right )}^{2}} \,d x } \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algo rithm="fricas")
Output:
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)
Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\text {Timed out} \] Input:
integrate((d*i*x+c*i)**2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 992 vs. \(2 (246) = 492\).
Time = 0.11 (sec) , antiderivative size = 992, normalized size of antiderivative = 4.02 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx =\text {Too large to display} \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algo rithm="maxima")
Output:
-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*log(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*log(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)*lo g(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))*log(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(...
\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\int { \frac {{\left (d i x + c i\right )}^{2} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}}{{\left (b g x + a g\right )}^{2}} \,d x } \] Input:
integrate((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x, algo rithm="giac")
Output:
integrate((d*i*x + c*i)^2*(B*log((b*x + a)*e/(d*x + c)) + A)/(b*g*x + a*g) ^2, x)
Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^2\,\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \] Input:
int(((c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^2 ,x)
Output:
int(((c*i + d*i*x)^2*(A + B*log((e*(a + b*x))/(c + d*x))))/(a*g + b*g*x)^2 , x)
\[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^2} \, dx =\text {Too large to display} \] Input:
int((d*i*x+c*i)^2*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^2,x)
Output:
( - int((log((a*e + b*e*x)/(c + d*x))*x**2)/(a**2 + 2*a*b*x + b**2*x**2),x )*a**3*b**4*d**3 + int((log((a*e + b*e*x)/(c + d*x))*x**2)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**5*c*d**2 - int((log((a*e + b*e*x)/(c + d*x))*x**2 )/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**5*d**3*x + int((log((a*e + b*e*x )/(c + d*x))*x**2)/(a**2 + 2*a*b*x + b**2*x**2),x)*a*b**6*c*d**2*x - 2*int ((log((a*e + b*e*x)/(c + d*x))*x)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**3*b** 4*c*d**2 + 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a**2 + 2*a*b*x + b**2*x **2),x)*a**2*b**5*c**2*d - 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a**2 + 2*a*b*x + b**2*x**2),x)*a**2*b**5*c*d**2*x + 2*int((log((a*e + b*e*x)/(c + d*x))*x)/(a**2 + 2*a*b*x + b**2*x**2),x)*a*b**6*c**2*d*x + 2*log(a + b*x) *a**5*d**3 - 4*log(a + b*x)*a**4*b*c*d**2 + 2*log(a + b*x)*a**4*b*d**3*x + 2*log(a + b*x)*a**3*b**2*c**2*d - 4*log(a + b*x)*a**3*b**2*c*d**2*x + 2*l og(a + b*x)*a**2*b**3*c**2*d*x - log(a + b*x)*a*b**4*c**3 - log(a + b*x)*b **5*c**3*x + log(c + d*x)*a*b**4*c**3 + log(c + d*x)*b**5*c**3*x - log((a* e + b*e*x)/(c + d*x))*a*b**4*c**2*d*x + log((a*e + b*e*x)/(c + d*x))*b**5* c**3*x - 2*a**4*b*d**3*x + 4*a**3*b**2*c*d**2*x - a**3*b**2*d**3*x**2 - 3* a**2*b**3*c**2*d*x + a**2*b**3*c*d**2*x**2 + a*b**4*c**3*x - a*b**4*c**2*d *x + b**5*c**3*x)/(a*b**3*g**2*(a**2*d - a*b*c + a*b*d*x - b**2*c*x))