Integrand size = 34, antiderivative size = 34 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\text {Int}\left (\frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2},x\right ) \] Output:
Defer(Int)((b*g*x+a*g)^2/(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
Not integrable
Time = 0.51 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx \] Input:
Integrate[(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2,x]
Output:
Integrate[(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2, x]
Not integrable
Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {2956}
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.
| \(\displaystyle \int \frac {(a g+b g x)^2}{\left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2} \, dx\) |
| \(\Big \downarrow \) 2956 |
| \(\displaystyle \int \frac {(a g+b g x)^2}{\left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}dx\) |
Input:
Int[(a*g + b*g*x)^2/(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2,x]
Output:
$Aborted
Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Unintegrable[(f + g*x)^m*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p, x] /; FreeQ[{a, b, c, d, e, f, g, A, B, m, n, p}, x] && EqQ[n + mn, 0] && IntegerQ[n]
Not integrable
Time = 1.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00
\[\int \frac {\left (b g x +a g \right )^{2}}{{\left (A +B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )\right )}^{2}}d x\]
Input:
int((b*g*x+a*g)^2/(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
Output:
int((b*g*x+a*g)^2/(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2,x)
Not integrable
Time = 0.07 (sec) , antiderivative size = 125, normalized size of antiderivative = 3.68 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2}}{{\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \] Input:
integrate((b*g*x+a*g)^2/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="f ricas")
Output:
integral((b^2*g^2*x^2 + 2*a*b*g^2*x + a^2*g^2)/(B^2*log((d^2*e*x^2 + 2*c*d *e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2))^2 + 2*A*B*log((d^2*e*x^2 + 2*c*d* e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2)) + A^2), x)
Not integrable
Time = 15.33 (sec) , antiderivative size = 792, normalized size of antiderivative = 23.29 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\frac {- a^{3} c g^{2} - a^{3} d g^{2} x - 3 a^{2} b c g^{2} x - 3 a^{2} b d g^{2} x^{2} - 3 a b^{2} c g^{2} x^{2} - 3 a b^{2} d g^{2} x^{3} - b^{3} c g^{2} x^{3} - b^{3} d g^{2} x^{4}}{2 A B a d - 2 A B b c + \left (2 B^{2} a d - 2 B^{2} b c\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}} + \frac {g^{2} \left (\int \frac {a^{3} d}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {3 a^{2} b c}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {3 b^{3} c x^{2}}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {4 b^{3} d x^{3}}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {6 a b^{2} c x}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {9 a b^{2} d x^{2}}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {6 a^{2} b d x}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx\right )}{2 B \left (a d - b c\right )} \] Input:
integrate((b*g*x+a*g)**2/(A+B*ln(e*(d*x+c)**2/(b*x+a)**2))**2,x)
Output:
(-a**3*c*g**2 - a**3*d*g**2*x - 3*a**2*b*c*g**2*x - 3*a**2*b*d*g**2*x**2 - 3*a*b**2*c*g**2*x**2 - 3*a*b**2*d*g**2*x**3 - b**3*c*g**2*x**3 - b**3*d*g **2*x**4)/(2*A*B*a*d - 2*A*B*b*c + (2*B**2*a*d - 2*B**2*b*c)*log(e*(c + d* x)**2/(a + b*x)**2)) + g**2*(Integral(a**3*d/(A + B*log(c**2*e/(a**2 + 2*a *b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e*x**2/( a**2 + 2*a*b*x + b**2*x**2))), x) + Integral(3*a**2*b*c/(A + B*log(c**2*e/ (a**2 + 2*a*b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d* *2*e*x**2/(a**2 + 2*a*b*x + b**2*x**2))), x) + Integral(3*b**3*c*x**2/(A + B*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b **2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b**2*x**2))), x) + Integral(4*b* *3*d*x**3/(A + B*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b**2*x**2))), x) + Integral(6*a*b**2*c*x/(A + B*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) + 2* c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b**2* x**2))), x) + Integral(9*a*b**2*d*x**2/(A + B*log(c**2*e/(a**2 + 2*a*b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e*x**2/(a**2 + 2*a*b*x + b**2*x**2))), x) + Integral(6*a**2*b*d*x/(A + B*log(c**2*e/(a** 2 + 2*a*b*x + b**2*x**2) + 2*c*d*e*x/(a**2 + 2*a*b*x + b**2*x**2) + d**2*e *x**2/(a**2 + 2*a*b*x + b**2*x**2))), x))/(2*B*(a*d - b*c))
Not integrable
Time = 0.07 (sec) , antiderivative size = 312, normalized size of antiderivative = 9.18 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2}}{{\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \] Input:
integrate((b*g*x+a*g)^2/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="m axima")
Output:
-1/2*(b^3*d*g^2*x^4 + a^3*c*g^2 + (b^3*c*g^2 + 3*a*b^2*d*g^2)*x^3 + 3*(a*b ^2*c*g^2 + a^2*b*d*g^2)*x^2 + (3*a^2*b*c*g^2 + a^3*d*g^2)*x)/(2*(b*c - a*d )*B^2*log(b*x + a) - 2*(b*c - a*d)*B^2*log(d*x + c) - (b*c - a*d)*A*B - (b *c*log(e) - a*d*log(e))*B^2) + integrate(1/2*(4*b^3*d*g^2*x^3 + 3*a^2*b*c* g^2 + a^3*d*g^2 + 3*(b^3*c*g^2 + 3*a*b^2*d*g^2)*x^2 + 6*(a*b^2*c*g^2 + a^2 *b*d*g^2)*x)/(2*(b*c - a*d)*B^2*log(b*x + a) - 2*(b*c - a*d)*B^2*log(d*x + c) - (b*c - a*d)*A*B - (b*c*log(e) - a*d*log(e))*B^2), x)
Not integrable
Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{2}}{{\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2}} \,d x } \] Input:
integrate((b*g*x+a*g)^2/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="g iac")
Output:
integrate((b*g*x + a*g)^2/(B*log((d*x + c)^2*e/(b*x + a)^2) + A)^2, x)
Not integrable
Time = 34.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^2}{{\left (A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\right )}^2} \,d x \] Input:
int((a*g + b*g*x)^2/(A + B*log((e*(c + d*x)^2)/(a + b*x)^2))^2,x)
Output:
int((a*g + b*g*x)^2/(A + B*log((e*(c + d*x)^2)/(a + b*x)^2))^2, x)
Not integrable
Time = 0.26 (sec) , antiderivative size = 9212, normalized size of antiderivative = 270.94 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx =\text {Too large to display} \] Input:
int((b*g*x+a*g)^2/(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x)
Output:
(g**2*(2*int(x**4/(log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*a*b**2*c + log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*a*b**2*d*x + log((c**2*e + 2*c*d*e*x + d**2*e*x* *2)/(a**2 + 2*a*b*x + b**2*x**2))**2*b**3*c*x + log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*b**3*d*x**2 + 2*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**2*b*c + 2*log( (c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**2*b*d* x + 2*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2)) *a*b**2*c*x + 2*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b **2*x**2))*a*b**2*d*x**2 + a**3*c + a**3*d*x + a**2*b*c*x + a**2*b*d*x**2) ,x)*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a *b**4*d**2 - 2*int(x**4/(log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2* a*b*x + b**2*x**2))**2*a*b**2*c + log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/( a**2 + 2*a*b*x + b**2*x**2))**2*a*b**2*d*x + log((c**2*e + 2*c*d*e*x + d** 2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*b**3*c*x + log((c**2*e + 2*c*d* e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))**2*b**3*d*x**2 + 2*log((c **2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a**2*b*c + 2*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2*x**2))*a** 2*b*d*x + 2*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*a*b*x + b**2* x**2))*a*b**2*c*x + 2*log((c**2*e + 2*c*d*e*x + d**2*e*x**2)/(a**2 + 2*...