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

Optimal. Leaf size=175 \[ -\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b g^4 (a+b x)^3}+\frac {B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}-\frac {B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}+\frac {B d^2}{3 b g^4 (a+b x) (b c-a d)^2}-\frac {B d}{6 b g^4 (a+b x)^2 (b c-a d)}+\frac {B}{9 b g^4 (a+b x)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2525, 12, 44} \[ -\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{3 b g^4 (a+b x)^3}+\frac {B d^2}{3 b g^4 (a+b x) (b c-a d)^2}+\frac {B d^3 \log (a+b x)}{3 b g^4 (b c-a d)^3}-\frac {B d^3 \log (c+d x)}{3 b g^4 (b c-a d)^3}-\frac {B d}{6 b g^4 (a+b x)^2 (b c-a d)}+\frac {B}{9 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 (c+d x)}{a+b x}\right )}{(a g+b g x)^4} \, dx &=-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{3 b g^4 (a+b x)^3}+\frac {B \int \frac {-b c+a d}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{3 b g^4 (a+b x)^3}-\frac {(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 (c+d x)}{a+b x}\right )}{3 b g^4 (a+b x)^3}-\frac {(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 {B}{9 b g^4 (a+b x)^3}-\frac {B d}{6 b (b c-a d) g^4 (a+b x)^2}+\frac {B d^2}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac {B d^3 \log (a+b x)}{3 b (b c-a d)^3 g^4}-\frac {B d^3 \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{3 b g^4 (a+b x)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 141, normalized size = 0.81 \[ \frac {\frac {B \left ((b c-a d) \left (11 a^2 d^2+a b d (15 d x-7 c)+b^2 \left (2 c^2-3 c d x+6 d^2 x^2\right )\right )-6 d^3 (a+b x)^3 \log (c+d x)+6 d^3 (a+b x)^3 \log (a+b x)\right )}{(b c-a d)^3}-6 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{18 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((B*((b*c - a*d)*(11*a^2*d^2 + a*b*d*(-7*c + 15*d*x) + b^2*(2*c^2 - 3*c*d*x + 6*d^2*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 - 6*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(18*b*
g^4*(a + b*x)^3)

________________________________________________________________________________________

fricas [B]  time = 1.12, size = 412, normalized size = 2.35 \[ -\frac {2 \, {\left (3 \, A - B\right )} b^{3} c^{3} - 9 \, {\left (2 \, A - B\right )} a b^{2} c^{2} d + 18 \, {\left (A - B\right )} a^{2} b c d^{2} - {\left (6 \, A - 11 \, B\right )} a^{3} d^{3} - 6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} + 5 \, B a^{2} b d^{3}\right )} x + 6 \, {\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d + 3 \, B a^{2} b c d^{2}\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{18 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

-1/18*(2*(3*A - B)*b^3*c^3 - 9*(2*A - B)*a*b^2*c^2*d + 18*(A - B)*a^2*b*c*d^2 - (6*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 + 6*(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((d*e*x + c*e)/(b*x + a)))/((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)

________________________________________________________________________________________

giac [B]  time = 1.06, size = 382, normalized size = 2.18 \[ -\frac {{\left (\frac {18 \, {\left (d x e + c e\right )} B d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{b x + a} - \frac {18 \, {\left (d x e + c e\right )}^{2} B b d e \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{2}} + \frac {18 \, {\left (d x e + c e\right )} A d^{2} e^{2}}{b x + a} - \frac {18 \, {\left (d x e + c e\right )} B d^{2} e^{2}}{b x + a} - \frac {18 \, {\left (d x e + c e\right )}^{2} A b d e}{{\left (b x + a\right )}^{2}} + \frac {9 \, {\left (d x e + c e\right )}^{2} B b d e}{{\left (b x + a\right )}^{2}} + \frac {6 \, {\left (d x e + c e\right )}^{3} B b^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{3}} + \frac {6 \, {\left (d x e + c e\right )}^{3} A b^{2}}{{\left (b x + a\right )}^{3}} - \frac {2 \, {\left (d x e + c e\right )}^{3} B b^{2}}{{\left (b x + a\right )}^{3}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{18 \, {\left (b^{2} c^{2} g^{4} e^{2} - 2 \, a b c d g^{4} e^{2} + a^{2} d^{2} g^{4} e^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

-1/18*(18*(d*x*e + c*e)*B*d^2*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a) - 18*(d*x*e + c*e)^2*B*b*d*e*log((d*x
*e + c*e)/(b*x + a))/(b*x + a)^2 + 18*(d*x*e + c*e)*A*d^2*e^2/(b*x + a) - 18*(d*x*e + c*e)*B*d^2*e^2/(b*x + a)
- 18*(d*x*e + c*e)^2*A*b*d*e/(b*x + a)^2 + 9*(d*x*e + c*e)^2*B*b*d*e/(b*x + a)^2 + 6*(d*x*e + c*e)^3*B*b^2*lo
g((d*x*e + c*e)/(b*x + a))/(b*x + a)^3 + 6*(d*x*e + c*e)^3*A*b^2/(b*x + a)^3 - 2*(d*x*e + c*e)^3*B*b^2/(b*x +
a)^3)*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/(b^2*c^2*g^4*e^2 - 2*a*b*c*d*g^4
*e^2 + a^2*d^2*g^4*e^2)

________________________________________________________________________________________

maple [B]  time = 0.05, size = 1012, normalized size = 5.78 \[ -\frac {B \,a^{4} d^{4} \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} b \,g^{4}}+\frac {4 B \,a^{3} c \,d^{3} \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}-\frac {2 B \,a^{2} b \,c^{2} d^{2} \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{\left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}+\frac {4 B a \,b^{2} c^{3} d \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}-\frac {B \,b^{3} c^{4} \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}-\frac {A \,a^{4} d^{4}}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} b \,g^{4}}+\frac {4 A \,a^{3} c \,d^{3}}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}-\frac {2 A \,a^{2} b \,c^{2} d^{2}}{\left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}+\frac {4 A a \,b^{2} c^{3} d}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}-\frac {A \,b^{3} c^{4}}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}+\frac {B \,a^{4} d^{4}}{9 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} b \,g^{4}}-\frac {4 B \,a^{3} c \,d^{3}}{9 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}+\frac {2 B \,a^{2} b \,c^{2} d^{2}}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}-\frac {4 B a \,b^{2} c^{3} d}{9 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}+\frac {B \,b^{3} c^{4}}{9 \left (a d -b c \right )^{4} \left (b x +a \right )^{3} g^{4}}+\frac {B \,a^{3} d^{4}}{6 \left (a d -b c \right )^{4} \left (b x +a \right )^{2} b \,g^{4}}-\frac {B \,a^{2} c \,d^{3}}{2 \left (a d -b c \right )^{4} \left (b x +a \right )^{2} g^{4}}+\frac {B a b \,c^{2} d^{2}}{2 \left (a d -b c \right )^{4} \left (b x +a \right )^{2} g^{4}}-\frac {B \,b^{2} c^{3} d}{6 \left (a d -b c \right )^{4} \left (b x +a \right )^{2} g^{4}}+\frac {B \,a^{2} d^{4}}{3 \left (a d -b c \right )^{4} \left (b x +a \right ) b \,g^{4}}+\frac {B a \,d^{4} \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{3 \left (a d -b c \right )^{4} b \,g^{4}}-\frac {2 B a c \,d^{3}}{3 \left (a d -b c \right )^{4} \left (b x +a \right ) g^{4}}+\frac {B b \,c^{2} d^{2}}{3 \left (a d -b c \right )^{4} \left (b x +a \right ) g^{4}}-\frac {B c \,d^{3} \ln \left (\frac {d e}{b}-\frac {\left (a d -b c \right ) e}{\left (b x +a \right ) b}\right )}{3 \left (a d -b c \right )^{4} g^{4}}+\frac {A a \,d^{4}}{3 \left (a d -b c \right )^{4} b \,g^{4}}-\frac {A c \,d^{3}}{3 \left (a d -b c \right )^{4} g^{4}}-\frac {11 B a \,d^{4}}{18 \left (a d -b c \right )^{4} b \,g^{4}}+\frac {11 B c \,d^{3}}{18 \left (a d -b c \right )^{4} g^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^4,x)

[Out]

1/3/b/(a*d-b*c)^4/g^4*A*d^4*a+4/3/(a*d-b*c)^4/g^4*B*ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)/(b*x+a)^3*a^3*d^3*c-1/3/
b/(a*d-b*c)^4/g^4*B*ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)/(b*x+a)^3*a^4*d^4+1/2*b/(a*d-b*c)^4/g^4*B*d^2/(b*x+a)^2*
c^2*a-4/9*b^2/(a*d-b*c)^4/g^4*B/(b*x+a)^3*c^3*a*d-1/3/(a*d-b*c)^4/g^4*B*ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)*d^3*
c-1/3*b^3/(a*d-b*c)^4/g^4*A/(b*x+a)^3*c^4+1/9*b^3/(a*d-b*c)^4/g^4*B/(b*x+a)^3*c^4+2/3*b/(a*d-b*c)^4/g^4*B/(b*x
+a)^3*a^2*d^2*c^2-2*b/(a*d-b*c)^4/g^4*A/(b*x+a)^3*a^2*d^2*c^2+4/3*b^2/(a*d-b*c)^4/g^4*A/(b*x+a)^3*a*d*c^3+11/1
8/(a*d-b*c)^4/g^4*B*d^3*c+1/9/b/(a*d-b*c)^4/g^4*B/(b*x+a)^3*a^4*d^4-1/3/b/(a*d-b*c)^4/g^4*A/(b*x+a)^3*a^4*d^4+
1/3*b/(a*d-b*c)^4/g^4*B*d^2/(b*x+a)*c^2+1/6/b/(a*d-b*c)^4/g^4*B*d^4/(b*x+a)^2*a^3-1/6*b^2/(a*d-b*c)^4/g^4*B*d/
(b*x+a)^2*c^3+1/3/b/(a*d-b*c)^4/g^4*B*d^4/(b*x+a)*a^2+4/3/(a*d-b*c)^4/g^4*A/(b*x+a)^3*a^3*d^3*c+1/3/b/(a*d-b*c
)^4/g^4*B*ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)*d^4*a-1/3*b^3/(a*d-b*c)^4/g^4*B*ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)/
(b*x+a)^3*c^4-4/9/(a*d-b*c)^4/g^4*B/(b*x+a)^3*a^3*d^3*c-2/3/(a*d-b*c)^4/g^4*B*d^3/(b*x+a)*c*a-1/2/(a*d-b*c)^4/
g^4*B*d^3/(b*x+a)^2*a^2*c-1/3/(a*d-b*c)^4/g^4*A*d^3*c-11/18/b/(a*d-b*c)^4/g^4*B*d^4*a-2*b/(a*d-b*c)^4/g^4*B*ln
(1/b*d*e-(a*d-b*c)/(b*x+a)/b*e)/(b*x+a)^3*a^2*d^2*c^2+4/3*b^2/(a*d-b*c)^4/g^4*B*ln(1/b*d*e-(a*d-b*c)/(b*x+a)/b
*e)/(b*x+a)^3*c^3*a*d

________________________________________________________________________________________

maxima [B]  time = 1.37, size = 428, normalized size = 2.45 \[ \frac {1}{18} \, B {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} - \frac {6 \, \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {A}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

1/18*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) - 6*log(d*e*x/(b*x + a) + c*e/(b*x +
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) + 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)

________________________________________________________________________________________

mupad [B]  time = 5.88, size = 339, normalized size = 1.94 \[ \frac {B\,b\,c^2}{9\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {A\,b\,c^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )}{3\,b\,g^4\,{\left (a+b\,x\right )}^3}-\frac {A\,a^2\,d^2}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {11\,B\,a^2\,d^2}{18\,b\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {5\,B\,a\,d^2\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,b\,d^2\,x^2}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {2\,A\,a\,c\,d}{3\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {7\,B\,a\,c\,d}{18\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}-\frac {B\,b\,c\,d\,x}{6\,g^4\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^3}+\frac {B\,d^3\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{3\,b\,g^4\,{\left (a\,d-b\,c\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log((e*(c + d*x))/(a + b*x)))/(a*g + b*g*x)^4,x)

[Out]

(B*d^3*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(3*b*g^4*(a*d - b*c)^3) - (B*log((e*(c + d*x))/(a +
b*x)))/(3*b*g^4*(a + b*x)^3) - (A*b*c^2)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) + (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)/(18*b*g^4*(a*d - b*c)^2*(a + b*x
)^3) + (5*B*a*d^2*x)/(6*g^4*(a*d - b*c)^2*(a + b*x)^3) + (B*b*d^2*x^2)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) + (2*
A*a*c*d)/(3*g^4*(a*d - b*c)^2*(a + b*x)^3) - (7*B*a*c*d)/(18*g^4*(a*d - b*c)^2*(a + b*x)^3) - (B*b*c*d*x)/(6*g
^4*(a*d - b*c)^2*(a + b*x)^3)

________________________________________________________________________________________

sympy [B]  time = 4.06, size = 656, normalized size = 3.75 \[ - \frac {B \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}}{3 a^{3} b g^{4} + 9 a^{2} b^{2} g^{4} x + 9 a b^{3} g^{4} x^{2} + 3 b^{4} g^{4} x^{3}} + \frac {B d^{3} \log {\left (x + \frac {- \frac {B a^{4} d^{7}}{\left (a d - b c\right )^{3}} + \frac {4 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} - \frac {6 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} + \frac {4 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + B a d^{4} - \frac {B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + B b c d^{3}}{2 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} - \frac {B d^{3} \log {\left (x + \frac {\frac {B a^{4} d^{7}}{\left (a d - b c\right )^{3}} - \frac {4 B a^{3} b c d^{6}}{\left (a d - b c\right )^{3}} + \frac {6 B a^{2} b^{2} c^{2} d^{5}}{\left (a d - b c\right )^{3}} - \frac {4 B a b^{3} c^{3} d^{4}}{\left (a d - b c\right )^{3}} + B a d^{4} + \frac {B b^{4} c^{4} d^{3}}{\left (a d - b c\right )^{3}} + B b c d^{3}}{2 B b d^{4}} \right )}}{3 b g^{4} \left (a d - b c\right )^{3}} + \frac {- 6 A a^{2} d^{2} + 12 A a b c d - 6 A b^{2} c^{2} + 11 B a^{2} d^{2} - 7 B a b c d + 2 B b^{2} c^{2} + 6 B b^{2} d^{2} x^{2} + x \left (15 B a b d^{2} - 3 B b^{2} c d\right )}{18 a^{5} b d^{2} g^{4} - 36 a^{4} b^{2} c d g^{4} + 18 a^{3} b^{3} c^{2} g^{4} + x^{3} \left (18 a^{2} b^{4} d^{2} g^{4} - 36 a b^{5} c d g^{4} + 18 b^{6} c^{2} g^{4}\right ) + x^{2} \left (54 a^{3} b^{3} d^{2} g^{4} - 108 a^{2} b^{4} c d g^{4} + 54 a b^{5} c^{2} g^{4}\right ) + x \left (54 a^{4} b^{2} d^{2} g^{4} - 108 a^{3} b^{3} c d g^{4} + 54 a^{2} b^{4} c^{2} g^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)**4,x)

[Out]

-B*log(e*(c + d*x)/(a + b*x))/(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) + B
*d**3*log(x + (-B*a**4*d**7/(a*d - b*c)**3 + 4*B*a**3*b*c*d**6/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**5/(a*d -
b*c)**3 + 4*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + B*a*d**4 - B*b**4*c**4*d**3/(a*d - b*c)**3 + B*b*c*d**3)/(2*B
*b*d**4))/(3*b*g**4*(a*d - b*c)**3) - B*d**3*log(x + (B*a**4*d**7/(a*d - b*c)**3 - 4*B*a**3*b*c*d**6/(a*d - b*
c)**3 + 6*B*a**2*b**2*c**2*d**5/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**4/(a*d - b*c)**3 + B*a*d**4 + B*b**4*c**4*
d**3/(a*d - b*c)**3 + B*b*c*d**3)/(2*B*b*d**4))/(3*b*g**4*(a*d - b*c)**3) + (-6*A*a**2*d**2 + 12*A*a*b*c*d - 6
*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))/(18*a**5*b*d**2*g**4 - 36*a**4*b**2*c*d*g**4 + 18*a**3*b**3*c**2*g**4 + x**3*(18*a**2*b**4*d**2*g**4 -
36*a*b**5*c*d*g**4 + 18*b**6*c**2*g**4) + x**2*(54*a**3*b**3*d**2*g**4 - 108*a**2*b**4*c*d*g**4 + 54*a*b**5*c*
*2*g**4) + x*(54*a**4*b**2*d**2*g**4 - 108*a**3*b**3*c*d*g**4 + 54*a**2*b**4*c**2*g**4))

________________________________________________________________________________________