3.2.24 \(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{(a g+b g x)^2} \, dx\) [124]

Optimal. Leaf size=65 \[ -\frac {2 B}{b g^2 (a+b x)}-\frac {(c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) g^2 (a+b x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2550, 2341} \begin {gather*} -\frac {(c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac {2 B (c+d x)}{g^2 (a+b x) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2550

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.04, size = 111, normalized size = 1.71 \begin {gather*} -\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{b g^2 (a+b x)}+\frac {2 B (b c-a d) \left (-\frac {1}{(b c-a d) (a+b x)}-\frac {d \log (a+b x)}{(b c-a d)^2}+\frac {d \log (c+d x)}{(b c-a d)^2}\right )}{b g^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(144\) vs. \(2(65)=130\).
time = 0.30, size = 145, normalized size = 2.23

method	result	size
norman	\(\frac {\frac {\left (A +2 B \right ) x}{g a}+\frac {c B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (a d -c b \right ) g}+\frac {B d x \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{\left (a d -c b \right ) g}}{g \left (b x +a \right )}\)	\(93\)
risch	\(-\frac {B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )}{b \,g^{2} \left (b x +a \right )}-\frac {-2 B \ln \left (-b x -a \right ) b d x +2 B \ln \left (d x +c \right ) b d x -2 B \ln \left (-b x -a \right ) a d +2 B \ln \left (d x +c \right ) a d +A a d -A b c +2 B a d -2 B b c}{g^{2} \left (b x +a \right ) b \left (a d -c b \right )}\)	\(132\)
derivativedivides	\(-\frac {-\frac {d^{2} A}{g^{2} \left (\frac {a d -c b}{d x +c}+b \right ) \left (a d -c b \right )}+\frac {\frac {2 d^{2} B}{b g \left (d x +c \right )}-\frac {d^{2} B \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}}{d}\)	\(145\)
default	\(-\frac {-\frac {d^{2} A}{g^{2} \left (\frac {a d -c b}{d x +c}+b \right ) \left (a d -c b \right )}+\frac {\frac {2 d^{2} B}{b g \left (d x +c \right )}-\frac {d^{2} B \ln \left (\frac {e \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2}}{d^{2}}\right )}{g \left (a d -c b \right )}}{g \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )}}{d}\)	\(145\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d*(-d^2/g^2*A/((a*d-b*c)/(d*x+c)+b)/(a*d-b*c)+(2*d^2*B/b/g/(d*x+c)-d^2*B/g/(a*d-b*c)*ln(e*(a/(d*x+c)*d-b*c/
(d*x+c)+b)^2/d^2))/g/(a/(d*x+c)*d-b*c/(d*x+c)+b))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (66) = 132\).
time = 0.31, size = 190, normalized size = 2.92 \begin {gather*} -B {\left (\frac {\log \left (\frac {b^{2} x^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b x e}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac {2}{b^{2} g^{2} x + a b g^{2}} + \frac {2 \, d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {2 \, d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac {A}{b^{2} g^{2} x + a b g^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.34, size = 108, normalized size = 1.66 \begin {gather*} -\frac {{\left (A + 2 \, B\right )} b c - {\left (A + 2 \, B\right )} a d + {\left (B b d x + B b c\right )} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x + {\left (a b^{2} c - a^{2} b d\right )} g^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (54) = 108\).
time = 1.04, size = 255, normalized size = 3.92 \begin {gather*} - \frac {B \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{a b g^{2} + b^{2} g^{2} x} - \frac {2 B d \log {\left (x + \frac {- \frac {2 B a^{2} d^{3}}{a d - b c} + \frac {4 B a b c d^{2}}{a d - b c} + 2 B a d^{2} - \frac {2 B b^{2} c^{2} d}{a d - b c} + 2 B b c d}{4 B b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {2 B d \log {\left (x + \frac {\frac {2 B a^{2} d^{3}}{a d - b c} - \frac {4 B a b c d^{2}}{a d - b c} + 2 B a d^{2} + \frac {2 B b^{2} c^{2} d}{a d - b c} + 2 B b c d}{4 B b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac {- A - 2 B}{a b g^{2} + b^{2} g^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*log(e*(a + b*x)**2/(c + d*x)**2)/(a*b*g**2 + b**2*g**2*x) - 2*B*d*log(x + (-2*B*a**2*d**3/(a*d - b*c) + 4*B
*a*b*c*d**2/(a*d - b*c) + 2*B*a*d**2 - 2*B*b**2*c**2*d/(a*d - b*c) + 2*B*b*c*d)/(4*B*b*d**2))/(b*g**2*(a*d - b
*c)) + 2*B*d*log(x + (2*B*a**2*d**3/(a*d - b*c) - 4*B*a*b*c*d**2/(a*d - b*c) + 2*B*a*d**2 + 2*B*b**2*c**2*d/(a
*d - b*c) + 2*B*b*c*d)/(4*B*b*d**2))/(b*g**2*(a*d - b*c)) + (-A - 2*B)/(a*b*g**2 + b**2*g**2*x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (66) = 132\).
time = 2.40, size = 188, normalized size = 2.89 \begin {gather*} {\left (2 \, {\left (b^{2} c g^{2} - a b d g^{2}\right )} {\left (\frac {d \log \left ({\left | \frac {b c g}{b g x + a g} - \frac {a d g}{b g x + a g} + d \right |}\right )}{b^{4} c^{2} g^{4} - 2 \, a b^{3} c d g^{4} + a^{2} b^{2} d^{2} g^{4}} - \frac {1}{{\left (b^{2} c g^{2} - a b d g^{2}\right )} {\left (b g x + a g\right )} b g}\right )} - \frac {\log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right )}{{\left (b g x + a g\right )} b g}\right )} B - \frac {A}{{\left (b g x + a g\right )} b g} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(b^2*c*g^2 - a*b*d*g^2)*(d*log(abs(b*c*g/(b*g*x + a*g) - a*d*g/(b*g*x + a*g) + d))/(b^4*c^2*g^4 - 2*a*b^3*c
*d*g^4 + a^2*b^2*d^2*g^4) - 1/((b^2*c*g^2 - a*b*d*g^2)*(b*g*x + a*g)*b*g)) - log((b*x + a)^2*e/(d*x + c)^2)/((
b*g*x + a*g)*b*g))*B - A/((b*g*x + a*g)*b*g)

________________________________________________________________________________________

Mupad [B]
time = 5.25, size = 108, normalized size = 1.66 \begin {gather*} -\frac {A+2\,B}{x\,b^2\,g^2+a\,b\,g^2}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{b^2\,g^2\,\left (x+\frac {a}{b}\right )}-\frac {B\,d\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{b\,g^2\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (A + 2*B)/(b^2*g^2*x + a*b*g^2) - (B*log((e*(a + b*x)^2)/(c + d*x)^2))/(b^2*g^2*(x + a/b)) - (B*d*atan((b*c*
2i + b*d*x*2i)/(a*d - b*c) + 1i)*4i)/(b*g^2*(a*d - b*c))

________________________________________________________________________________________