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

Optimal. Leaf size=37 \[ \text {Int}\left (\frac {(a g+b g x)^2}{\left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.21, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \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 \left (\frac {a^2 g^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}+\frac {2 a b g^2 x}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}+\frac {b^2 g^2 x^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}\right ) \, dx\\ &=\left (a^2 g^2\right ) \int \frac {1}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx+\left (2 a b g^2\right ) \int \frac {x}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx+\left (b^2 g^2\right ) \int \frac {x^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {(a g+b g x)^2}{\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}{B^{2} \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 2 \, A B \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + A^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.76, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right )^{2}}{\left (B \ln \left (\frac {\left (d x +c \right )^{2} e}{\left (b x +a \right )^{2}}\right )+A \right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {b^{3} d g^{2} x^{4} + a^{3} c g^{2} + {\left (b^{3} c g^{2} + 3 \, a b^{2} d g^{2}\right )} x^{3} + 3 \, {\left (a b^{2} c g^{2} + a^{2} b d g^{2}\right )} x^{2} + {\left (3 \, a^{2} b c g^{2} + a^{3} d g^{2}\right )} x}{2 \, {\left (2 \, {\left (b c - a d\right )} B^{2} \log \left (b x + a\right ) - 2 \, {\left (b c - a d\right )} B^{2} \log \left (d x + c\right ) - {\left (b c - a d\right )} A B - {\left (b c \log \relax (e) - a d \log \relax (e)\right )} B^{2}\right )}} + \int \frac {4 \, b^{3} d g^{2} x^{3} + 3 \, a^{2} b c g^{2} + a^{3} d g^{2} + 3 \, {\left (b^{3} c g^{2} + 3 \, a b^{2} d g^{2}\right )} x^{2} + 6 \, {\left (a b^{2} c g^{2} + a^{2} b d g^{2}\right )} x}{2 \, {\left (2 \, {\left (b c - a d\right )} B^{2} \log \left (b x + a\right ) - 2 \, {\left (b c - a d\right )} B^{2} \log \left (d x + c\right ) - {\left (b c - a d\right )} A B - {\left (b c \log \relax (e) - a d \log \relax (e)\right )} B^{2}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (a\,g+b\,g\,x\right )}^2}{{\left (A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {- a^{3} c g^{2} - a^{3} d g^{2} x - 3 a^{2} b c g^{2} x - 3 a^{2} b d g^{2} x^{2} - 3 a b^{2} c g^{2} x^{2} - 3 a b^{2} d g^{2} x^{3} - b^{3} c g^{2} x^{3} - b^{3} d g^{2} x^{4}}{2 A B a d - 2 A B b c + \left (2 B^{2} a d - 2 B^{2} b c\right ) \log {\left (\frac {e \left (c + d x\right )^{2}}{\left (a + b x\right )^{2}} \right )}} + \frac {g^{2} \left (\int \frac {a^{3} d}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {3 a^{2} b c}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {3 b^{3} c x^{2}}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {4 b^{3} d x^{3}}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {6 a b^{2} c x}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {9 a b^{2} d x^{2}}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx + \int \frac {6 a^{2} b d x}{A + B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}\, dx\right )}{2 B \left (a d - b c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-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))

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