3.126 \(\int \frac {A+B \log (\frac {e (a+b x)^2}{(c+d x)^2})}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=177 \[ -\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{3 b g^4 (a+b x)^3}-\frac {2 B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}+\frac {2 B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}-\frac {2 B d^2}{3 b g^4 (a+b x) (b c-a d)^2}+\frac {B d}{3 b g^4 (a+b x)^2 (b c-a d)}-\frac {2 B}{9 b g^4 (a+b x)^3} \]

[Out]

-2/9*B/b/g^4/(b*x+a)^3+1/3*B*d/b/(-a*d+b*c)/g^4/(b*x+a)^2-2/3*B*d^2/b/(-a*d+b*c)^2/g^4/(b*x+a)-2/3*B*d^3*ln(b*
x+a)/b/(-a*d+b*c)^3/g^4+1/3*(-A-B*ln(e*(b*x+a)^2/(d*x+c)^2))/b/g^4/(b*x+a)^3+2/3*B*d^3*ln(d*x+c)/b/(-a*d+b*c)^
3/g^4

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Rubi [A]  time = 0.11, antiderivative size = 177, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2525, 12, 44} \[ -\frac {B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A}{3 b g^4 (a+b x)^3}-\frac {2 B d^2}{3 b g^4 (a+b x) (b c-a d)^2}-\frac {2 B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}+\frac {2 B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}+\frac {B d}{3 b g^4 (a+b x)^2 (b c-a d)}-\frac {2 B}{9 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B)/(9*b*g^4*(a + b*x)^3) + (B*d)/(3*b*(b*c - a*d)*g^4*(a + b*x)^2) - (2*B*d^2)/(3*b*(b*c - a*d)^2*g^4*(a +
b*x)) - (2*B*d^3*Log[a + b*x])/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])/(3*b*g^4*(a
+ b*x)^3) + (2*B*d^3*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[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)^4} \, dx &=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{3 b g^4 (a+b x)^3}+\frac {B \int \frac {2 (b c-a d)}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{3 b g^4 (a+b x)^3}+\frac {(2 B (b c-a d)) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{3 b g^4 (a+b x)^3}+\frac {(2 B (b c-a d)) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^4}\\ &=-\frac {2 B}{9 b g^4 (a+b x)^3}+\frac {B d}{3 b (b c-a d) g^4 (a+b x)^2}-\frac {2 B d^2}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac {2 B d^3 \log (a+b x)}{3 b (b c-a d)^3 g^4}-\frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{3 b g^4 (a+b x)^3}+\frac {2 B d^3 \log (c+d x)}{3 b (b c-a d)^3 g^4}\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 140, normalized size = 0.79 \[ -\frac {3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )+\frac {B \left (-6 d^3 (a+b x)^3 \log (c+d x)+6 d^2 (a+b x)^2 (b c-a d)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )}{(b c-a d)^3}}{9 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*(3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + (B*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)^2*(a + b*x) + 6*d^2*(
b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]))/(b*c - a*d)^3)/(b*g
^4*(a + b*x)^3)

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fricas [B]  time = 0.59, size = 430, normalized size = 2.43 \[ -\frac {{\left (3 \, A + 2 \, B\right )} b^{3} c^{3} - 9 \, {\left (A + B\right )} a b^{2} c^{2} d + 9 \, {\left (A + 2 \, B\right )} a^{2} b c d^{2} - {\left (3 \, A + 11 \, B\right )} a^{3} d^{3} + 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} x + 3 \, {\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{9 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \]

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)^4,x, algorithm="fricas")

[Out]

-1/9*((3*A + 2*B)*b^3*c^3 - 9*(A + B)*a*b^2*c^2*d + 9*(A + 2*B)*a^2*b*c*d^2 - (3*A + 11*B)*a^3*d^3 + 6*(B*b^3*
c*d^2 - B*a*b^2*d^3)*x^2 - 3*(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b*d^3)*x + 3*(B*b^3*d^3*x^3 + 3*B*a*b^2*
d^3*x^2 + 3*B*a^2*b*d^3*x + B*b^3*c^3 - 3*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)
/(d^2*x^2 + 2*c*d*x + c^2)))/((b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3*(a*b^6*c^3
- 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d
^2 - a^5*b^2*d^3)*g^4*x + (a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)

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giac [B]  time = 0.33, size = 473, normalized size = 2.67 \[ -\frac {2 \, B d^{3} \log \left (b x + a\right )}{3 \, {\left (b^{4} c^{3} g^{4} - 3 \, a b^{3} c^{2} d g^{4} + 3 \, a^{2} b^{2} c d^{2} g^{4} - a^{3} b d^{3} g^{4}\right )}} + \frac {2 \, B d^{3} \log \left (d x + c\right )}{3 \, {\left (b^{4} c^{3} g^{4} - 3 \, a b^{3} c^{2} d g^{4} + 3 \, a^{2} b^{2} c d^{2} g^{4} - a^{3} b d^{3} g^{4}\right )}} - \frac {B \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} - \frac {6 \, B b^{2} d^{2} x^{2} - 3 \, B b^{2} c d x + 15 \, B a b d^{2} x + 3 \, A b^{2} c^{2} + 5 \, B b^{2} c^{2} - 6 \, A a b c d - 13 \, B a b c d + 3 \, A a^{2} d^{2} + 14 \, B a^{2} d^{2}}{9 \, {\left (b^{6} c^{2} g^{4} x^{3} - 2 \, a b^{5} c d g^{4} x^{3} + a^{2} b^{4} d^{2} g^{4} x^{3} + 3 \, a b^{5} c^{2} g^{4} x^{2} - 6 \, a^{2} b^{4} c d g^{4} x^{2} + 3 \, a^{3} b^{3} d^{2} g^{4} x^{2} + 3 \, a^{2} b^{4} c^{2} g^{4} x - 6 \, a^{3} b^{3} c d g^{4} x + 3 \, a^{4} b^{2} d^{2} g^{4} x + a^{3} b^{3} c^{2} g^{4} - 2 \, a^{4} b^{2} c d g^{4} + a^{5} b d^{2} g^{4}\right )}} \]

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)^4,x, algorithm="giac")

[Out]

-2/3*B*d^3*log(b*x + a)/(b^4*c^3*g^4 - 3*a*b^3*c^2*d*g^4 + 3*a^2*b^2*c*d^2*g^4 - a^3*b*d^3*g^4) + 2/3*B*d^3*lo
g(d*x + c)/(b^4*c^3*g^4 - 3*a*b^3*c^2*d*g^4 + 3*a^2*b^2*c*d^2*g^4 - a^3*b*d^3*g^4) - 1/3*B*log((b^2*x^2 + 2*a*
b*x + a^2)/(d^2*x^2 + 2*c*d*x + c^2))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/9*(6*B
*b^2*d^2*x^2 - 3*B*b^2*c*d*x + 15*B*a*b*d^2*x + 3*A*b^2*c^2 + 5*B*b^2*c^2 - 6*A*a*b*c*d - 13*B*a*b*c*d + 3*A*a
^2*d^2 + 14*B*a^2*d^2)/(b^6*c^2*g^4*x^3 - 2*a*b^5*c*d*g^4*x^3 + a^2*b^4*d^2*g^4*x^3 + 3*a*b^5*c^2*g^4*x^2 - 6*
a^2*b^4*c*d*g^4*x^2 + 3*a^3*b^3*d^2*g^4*x^2 + 3*a^2*b^4*c^2*g^4*x - 6*a^3*b^3*c*d*g^4*x + 3*a^4*b^2*d^2*g^4*x
+ a^3*b^3*c^2*g^4 - 2*a^4*b^2*c*d*g^4 + a^5*b*d^2*g^4)

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maple [B]  time = 0.16, size = 579, normalized size = 3.27 \[ \frac {B \,b^{2} d^{3} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{3 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{3} \left (a^{3} d^{3}-3 a^{2} c \,d^{2} b +3 a \,c^{2} d \,b^{2}-b^{3} c^{3}\right ) g^{4}}+\frac {A \,b^{2} d^{3}}{3 \left (a d -b c \right )^{3} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{3} g^{4}}+\frac {B b \,d^{3} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d x +c \right ) g^{4}}-\frac {A b \,d^{3}}{\left (a d -b c \right )^{3} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} g^{4}}-\frac {2 B b \,d^{3}}{3 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (d x +c \right ) g^{4}}+\frac {B \,d^{3} \ln \left (\frac {\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{2} e}{d^{2}}\right )}{\left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{3} \left (a d -b c \right ) \left (d x +c \right )^{2} g^{4}}+\frac {A \,d^{3}}{\left (a d -b c \right )^{3} \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right ) g^{4}}-\frac {5 B \,d^{3}}{3 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{3} \left (a d -b c \right ) \left (d x +c \right )^{2} g^{4}}-\frac {11 B \,d^{3}}{9 \left (\frac {a d}{d x +c}-\frac {b c}{d x +c}+b \right )^{3} \left (d x +c \right )^{3} b \,g^{4}} \]

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)^4,x)

[Out]

1/3*d^3/g^4*A*b^2/(a*d-b*c)^3/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3+d^3/g^4*A/(a*d-b*c)^3/(1/(d*x+c)*a*d-1/(d*x+c)
*b*c+b)-d^3/g^4*A*b/(a*d-b*c)^3/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2-11/9*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)
^3*B/b/(d*x+c)^3+1/3*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*b^2*B/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*
c^3)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)-2/3*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B*b/(a^2*d^2-2*
a*b*c*d+b^2*c^2)/(d*x+c)-5/3*d^3/g^4/(1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^3*B/(a*d-b*c)/(d*x+c)^2+d^3/g^4/(1/(d*x+c
)*a*d-1/(d*x+c)*b*c+b)^3*B/(a*d-b*c)/(d*x+c)^2*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)+d^3/g^4/(1/(d*x+c)*
a*d-1/(d*x+c)*b*c+b)^3*B*b/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)*ln((1/(d*x+c)*a*d-1/(d*x+c)*b*c+b)^2/d^2*e)

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maxima [B]  time = 1.39, size = 480, normalized size = 2.71 \[ -\frac {1}{9} \, B {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {3 \, \log \left (\frac {b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac {a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {A}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \]

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)^4,x, algorithm="maxima")

[Out]

-1/9*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c
*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c
*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 3*log(b^2*e*x^2/(d^2*x^2 + 2*c*d*x
+ c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2
+ 3*a^2*b^2*g^4*x + a^3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*
g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/3*A/(b^4*g^4*x^3
+ 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

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mupad [B]  time = 5.80, size = 341, normalized size = 1.93 \[ \frac {2\,A\,a\,c\,d}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,b\,c^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {2\,B\,b\,c^2}{9\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,a^2\,d^2}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {11\,B\,a^2\,d^2}{9\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {5\,B\,a\,d^2\,x}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {2\,B\,b\,d^2\,x^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}{3\,b\,g^4\,{\left (a+b\,x\right )}^3}+\frac {7\,B\,a\,c\,d}{9\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,b\,c\,d\,x}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,d^3\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,4{}\mathrm {i}}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^3} \]

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)^4,x)

[Out]

(2*A*a*c*d)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) - (B*d^3*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*4i)/(3*b
*g^4*(a*d - b*c)^3) - (A*b*c^2)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) - (2*B*b*c^2)/(9*g^4*(a*d - b*c)^2*(a + b*x)
^3) - (A*a^2*d^2)/(3*b*g^4*(a*d - b*c)^2*(a + b*x)^3) - (11*B*a^2*d^2)/(9*b*g^4*(a*d - b*c)^2*(a + b*x)^3) - (
5*B*a*d^2*x)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) - (2*B*b*d^2*x^2)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) - (B*log((e
*(a + b*x)^2)/(c + d*x)^2))/(3*b*g^4*(a + b*x)^3) + (7*B*a*c*d)/(9*g^4*(a*d - b*c)^2*(a + b*x)^3) + (B*b*c*d*x
)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3)

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sympy [B]  time = 4.24, size = 677, normalized size = 3.82 \[ - \frac {B \log {\left (\frac {e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{3 a^{3} b g^{4} + 9 a^{2} b^{2} g^{4} x + 9 a b^{3} g^{4} x^{2} + 3 b^{4} g^{4} x^{3}} - \frac {2 B d^{3} \log {\left (x + \frac {- \frac {2 B a^{4} d^{7}}{\left (a d - b c\right )^{3}} + \frac {8 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} - \frac {12 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} + \frac {8 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + 2 B a d^{4} - \frac {2 B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + 2 B b c d^{3}}{4 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac {2 B d^{3} \log {\left (x + \frac {\frac {2 B a^{4} d^{7}}{\left (a d - b c\right )^{3}} - \frac {8 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} + \frac {12 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac {8 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + 2 B a d^{4} + \frac {2 B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + 2 B b c d^{3}}{4 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac {- 3 A a^{2} d^{2} + 6 A a b c d - 3 A b^{2} c^{2} - 11 B a^{2} d^{2} + 7 B a b c d - 2 B b^{2} c^{2} - 6 B b^{2} d^{2} x^{2} + x \left (- 15 B a b d^{2} + 3 B b^{2} c d\right )}{9 a^{5} b d^{2} g^{4} - 18 a^{4} b^{2} c d g^{4} + 9 a^{3} b^{3} c^{2} g^{4} + x^{3} \left (9 a^{2} b^{4} d^{2} g^{4} - 18 a b^{5} c d g^{4} + 9 b^{6} c^{2} g^{4}\right ) + x^{2} \left (27 a^{3} b^{3} d^{2} g^{4} - 54 a^{2} b^{4} c d g^{4} + 27 a b^{5} c^{2} g^{4}\right ) + x \left (27 a^{4} b^{2} d^{2} g^{4} - 54 a^{3} b^{3} c d g^{4} + 27 a^{2} b^{4} c^{2} g^{4}\right )} \]

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)**4,x)

[Out]

-B*log(e*(a + b*x)**2/(c + d*x)**2)/(3*a**3*b*g**4 + 9*a**2*b**2*g**4*x + 9*a*b**3*g**4*x**2 + 3*b**4*g**4*x**
3) - 2*B*d**3*log(x + (-2*B*a**4*d**7/(a*d - b*c)**3 + 8*B*a**3*b*c*d**6/(a*d - b*c)**3 - 12*B*a**2*b**2*c**2*
d**5/(a*d - b*c)**3 + 8*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + 2*B*a*d**4 - 2*B*b**4*c**4*d**3/(a*d - b*c)**3 + 2
*B*b*c*d**3)/(4*B*b*d**4))/(3*b*g**4*(a*d - b*c)**3) + 2*B*d**3*log(x + (2*B*a**4*d**7/(a*d - b*c)**3 - 8*B*a*
*3*b*c*d**6/(a*d - b*c)**3 + 12*B*a**2*b**2*c**2*d**5/(a*d - b*c)**3 - 8*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + 2
*B*a*d**4 + 2*B*b**4*c**4*d**3/(a*d - b*c)**3 + 2*B*b*c*d**3)/(4*B*b*d**4))/(3*b*g**4*(a*d - b*c)**3) + (-3*A*
a**2*d**2 + 6*A*a*b*c*d - 3*A*b**2*c**2 - 11*B*a**2*d**2 + 7*B*a*b*c*d - 2*B*b**2*c**2 - 6*B*b**2*d**2*x**2 +
x*(-15*B*a*b*d**2 + 3*B*b**2*c*d))/(9*a**5*b*d**2*g**4 - 18*a**4*b**2*c*d*g**4 + 9*a**3*b**3*c**2*g**4 + x**3*
(9*a**2*b**4*d**2*g**4 - 18*a*b**5*c*d*g**4 + 9*b**6*c**2*g**4) + x**2*(27*a**3*b**3*d**2*g**4 - 54*a**2*b**4*
c*d*g**4 + 27*a*b**5*c**2*g**4) + x*(27*a**4*b**2*d**2*g**4 - 54*a**3*b**3*c*d*g**4 + 27*a**2*b**4*c**2*g**4))

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