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

Optimal. Leaf size=173 \[ -\frac {b i (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 (a+b x)^3 (b c-a d)^2}+\frac {d i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 (a+b x)^2 (b c-a d)^2}-\frac {b B i (c+d x)^3}{9 g^4 (a+b x)^3 (b c-a d)^2}+\frac {B d i (c+d x)^2}{4 g^4 (a+b x)^2 (b c-a d)^2} \]

[Out]

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

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Rubi [A]  time = 0.34, antiderivative size = 225, normalized size of antiderivative = 1.30, number of steps used = 10, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2528, 2525, 12, 44} \[ -\frac {d i \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 b^2 g^4 (a+b x)^2}-\frac {i (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 b^2 g^4 (a+b x)^3}+\frac {B d^2 i}{6 b^2 g^4 (a+b x) (b c-a d)}+\frac {B d^3 i \log (a+b x)}{6 b^2 g^4 (b c-a d)^2}-\frac {B d^3 i \log (c+d x)}{6 b^2 g^4 (b c-a d)^2}-\frac {B i (b c-a d)}{9 b^2 g^4 (a+b x)^3}-\frac {B d i}{12 b^2 g^4 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)*i)/(9*b^2*g^4*(a + b*x)^3) - (B*d*i)/(12*b^2*g^4*(a + b*x)^2) + (B*d^2*i)/(6*b^2*(b*c - a*d)*g
^4*(a + b*x)) + (B*d^3*i*Log[a + b*x])/(6*b^2*(b*c - a*d)^2*g^4) - ((b*c - a*d)*i*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(3*b^2*g^4*(a + b*x)^3) - (d*i*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*b^2*g^4*(a + b*x)^2) - (B*d^
3*i*Log[c + d*x])/(6*b^2*(b*c - a*d)^2*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]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(8 c+8 d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a g+b g x)^4} \, dx &=\int \left (\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^4 (a+b x)^4}+\frac {8 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b g^4 (a+b x)^3}\right ) \, dx\\ &=\frac {(8 d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b g^4}+\frac {(8 (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b g^4}\\ &=-\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {4 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac {(4 B d) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac {(8 B (b c-a d)) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {4 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac {(4 B d (b c-a d)) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^2 g^4}+\frac {\left (8 B (b c-a d)^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^2 g^4}\\ &=-\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {4 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}+\frac {(4 B d (b c-a d)) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^2 g^4}+\frac {\left (8 B (b c-a d)^2\right ) \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^2 g^4}\\ &=-\frac {8 B (b c-a d)}{9 b^2 g^4 (a+b x)^3}-\frac {2 B d}{3 b^2 g^4 (a+b x)^2}+\frac {4 B d^2}{3 b^2 (b c-a d) g^4 (a+b x)}+\frac {4 B d^3 \log (a+b x)}{3 b^2 (b c-a d)^2 g^4}-\frac {8 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 b^2 g^4 (a+b x)^3}-\frac {4 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{b^2 g^4 (a+b x)^2}-\frac {4 B d^3 \log (c+d x)}{3 b^2 (b c-a d)^2 g^4}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 187, normalized size = 1.08 \[ -\frac {i \left (\frac {12 A b c}{(a+b x)^3}+\frac {18 A d}{(a+b x)^2}-\frac {12 a A d}{(a+b x)^3}-\frac {6 B d^3 \log (a+b x)}{(b c-a d)^2}+\frac {6 B d^3 \log (c+d x)}{(b c-a d)^2}-\frac {6 B d^2}{(a+b x) (b c-a d)}+\frac {6 B (a d+2 b c+3 b d x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac {4 b B c}{(a+b x)^3}+\frac {3 B d}{(a+b x)^2}-\frac {4 a B d}{(a+b x)^3}\right )}{36 b^2 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/36*(i*((12*A*b*c)/(a + b*x)^3 + (4*b*B*c)/(a + b*x)^3 - (12*a*A*d)/(a + b*x)^3 - (4*a*B*d)/(a + b*x)^3 + (1
8*A*d)/(a + b*x)^2 + (3*B*d)/(a + b*x)^2 - (6*B*d^2)/((b*c - a*d)*(a + b*x)) - (6*B*d^3*Log[a + b*x])/(b*c - a
*d)^2 + (6*B*(2*b*c + a*d + 3*b*d*x)*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x)^3 + (6*B*d^3*Log[c + d*x])/(b*c -
a*d)^2))/(b^2*g^4)

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fricas [B]  time = 1.08, size = 363, normalized size = 2.10 \[ \frac {6 \, {\left (B b^{3} c d^{2} - B a b^{2} d^{3}\right )} i x^{2} - 3 \, {\left ({\left (6 \, A + B\right )} b^{3} c^{2} d - 6 \, {\left (2 \, A + B\right )} a b^{2} c d^{2} + {\left (6 \, A + 5 \, B\right )} a^{2} b d^{3}\right )} i x - {\left (4 \, {\left (3 \, A + B\right )} b^{3} c^{3} - 9 \, {\left (2 \, A + B\right )} a b^{2} c^{2} d + {\left (6 \, A + 5 \, B\right )} a^{3} d^{3}\right )} i + 6 \, {\left (B b^{3} d^{3} i x^{3} + 3 \, B a b^{2} d^{3} i x^{2} - 3 \, {\left (B b^{3} c^{2} d - 2 \, B a b^{2} c d^{2}\right )} i x - {\left (2 \, B b^{3} c^{3} - 3 \, B a b^{2} c^{2} d\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(6*(B*b^3*c*d^2 - B*a*b^2*d^3)*i*x^2 - 3*((6*A + B)*b^3*c^2*d - 6*(2*A + B)*a*b^2*c*d^2 + (6*A + 5*B)*a^2
*b*d^3)*i*x - (4*(3*A + B)*b^3*c^3 - 9*(2*A + B)*a*b^2*c^2*d + (6*A + 5*B)*a^3*d^3)*i + 6*(B*b^3*d^3*i*x^3 + 3
*B*a*b^2*d^3*i*x^2 - 3*(B*b^3*c^2*d - 2*B*a*b^2*c*d^2)*i*x - (2*B*b^3*c^3 - 3*B*a*b^2*c^2*d)*i)*log((b*e*x + a
*e)/(d*x + c)))/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g
^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4
)

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giac [A]  time = 1.46, size = 244, normalized size = 1.41 \[ -\frac {{\left (12 \, B b i e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {18 \, {\left (b x e + a e\right )} B d i e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + 12 \, A b i e^{4} + 4 \, B b i e^{4} - \frac {18 \, {\left (b x e + a e\right )} A d i e^{3}}{d x + c} - \frac {9 \, {\left (b x e + a e\right )} B d i e^{3}}{d x + c}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{36 \, {\left (\frac {{\left (b x e + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x e + a e\right )}^{3} a d g^{4}}{{\left (d x + c\right )}^{3}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36*(12*B*b*i*e^4*log((b*x*e + a*e)/(d*x + c)) - 18*(b*x*e + a*e)*B*d*i*e^3*log((b*x*e + a*e)/(d*x + c))/(d*
x + c) + 12*A*b*i*e^4 + 4*B*b*i*e^4 - 18*(b*x*e + a*e)*A*d*i*e^3/(d*x + c) - 9*(b*x*e + a*e)*B*d*i*e^3/(d*x +
c))*(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*x*e + a*e)^3*b*c*g^4/(d*x + c)
^3 - (b*x*e + a*e)^3*a*d*g^4/(d*x + c)^3)

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maple [B]  time = 0.05, size = 804, normalized size = 4.65 \[ -\frac {B a b d \,e^{3} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{3 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}+\frac {B \,b^{2} c \,e^{3} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{3 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {A a b d \,e^{3} i}{3 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}+\frac {A \,b^{2} c \,e^{3} i}{3 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {B a b d \,e^{3} i}{9 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}+\frac {B a \,d^{2} e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}+\frac {B \,b^{2} c \,e^{3} i}{9 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{3} g^{4}}-\frac {B b c d \,e^{2} i \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{2 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}+\frac {A a \,d^{2} e^{2} i}{2 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}-\frac {A b c d \,e^{2} i}{2 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}+\frac {B a \,d^{2} e^{2} i}{4 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}}-\frac {B b c d \,e^{2} i}{4 \left (a d -b c \right )^{3} \left (\frac {a e}{d x +c}-\frac {b c e}{\left (d x +c \right ) d}+\frac {b e}{d}\right )^{2} g^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.38, size = 933, normalized size = 5.39 \[ -\frac {1}{36} \, B d i {\left (\frac {6 \, {\left (3 \, b x + a\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{5} g^{4} x^{3} + 3 \, a b^{4} g^{4} x^{2} + 3 \, a^{2} b^{3} g^{4} x + a^{3} b^{2} g^{4}} + \frac {5 \, a b^{2} c^{2} - 22 \, a^{2} b c d + 5 \, a^{3} d^{2} - 6 \, {\left (3 \, b^{3} c d - a b^{2} d^{2}\right )} x^{2} + 3 \, {\left (3 \, b^{3} c^{2} - 16 \, a b^{2} c d + 5 \, a^{2} b d^{2}\right )} x}{{\left (b^{7} c^{2} - 2 \, a b^{6} c d + a^{2} b^{5} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{2} - 2 \, a^{2} b^{5} c d + a^{3} b^{4} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{2} - 2 \, a^{3} b^{4} c d + a^{4} b^{3} d^{2}\right )} g^{4} x + {\left (a^{3} b^{4} c^{2} - 2 \, a^{4} b^{3} c d + a^{5} b^{2} d^{2}\right )} g^{4}} - \frac {6 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (b x + a\right )}{{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{4}} + \frac {6 \, {\left (3 \, b c d^{2} - a d^{3}\right )} \log \left (d x + c\right )}{{\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{4}}\right )} - \frac {1}{18} \, B c i {\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 {b e x}{d x + c} + \frac {a e}{d x + c}\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 {{\left (3 \, b x + a\right )} A d i}{6 \, {\left (b^{5} g^{4} x^{3} + 3 \, a b^{4} g^{4} x^{2} + 3 \, a^{2} b^{3} g^{4} x + a^{3} b^{2} g^{4}\right )}} - \frac {A c i}{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((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-1/36*B*d*i*(6*(3*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4
*x + a^3*b^2*g^4) + (5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 1
6*a*b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d +
a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^
5*b^2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3
)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) -
1/18*B*c*i*((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(b*e*x/(d*x + c) + a*e/(d
*x + c))/(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)) - 1/6*(3*b*x + a)*A*d*i/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/
3*A*c*i/(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 = 5.87, size = 361, normalized size = 2.09 \[ -\frac {\frac {6\,A\,a^2\,d^2\,i-12\,A\,b^2\,c^2\,i+5\,B\,a^2\,d^2\,i-4\,B\,b^2\,c^2\,i+6\,A\,a\,b\,c\,d\,i+5\,B\,a\,b\,c\,d\,i}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (6\,A\,a\,b\,d^2\,i+5\,B\,a\,b\,d^2\,i-6\,A\,b^2\,c\,d\,i-B\,b^2\,c\,d\,i\right )}{2\,\left (a\,d-b\,c\right )}+\frac {B\,b^2\,d^2\,i\,x^2}{a\,d-b\,c}}{6\,a^3\,b^2\,g^4+18\,a^2\,b^3\,g^4\,x+18\,a\,b^4\,g^4\,x^2+6\,b^5\,g^4\,x^3}-\frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {B\,c\,i}{3\,b^2\,g^4}+\frac {B\,a\,d\,i}{6\,b^3\,g^4}+\frac {B\,d\,i\,x}{2\,b^2\,g^4}\right )}{3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2}-\frac {B\,d^3\,i\,\mathrm {atanh}\left (\frac {6\,b^4\,c^2\,g^4-6\,a^2\,b^2\,d^2\,g^4}{6\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2}-\frac {2\,b\,d\,x}{a\,d-b\,c}\right )}{3\,b^2\,g^4\,{\left (a\,d-b\,c\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- ((6*A*a^2*d^2*i - 12*A*b^2*c^2*i + 5*B*a^2*d^2*i - 4*B*b^2*c^2*i + 6*A*a*b*c*d*i + 5*B*a*b*c*d*i)/(6*(a*d -
b*c)) + (x*(6*A*a*b*d^2*i + 5*B*a*b*d^2*i - 6*A*b^2*c*d*i - B*b^2*c*d*i))/(2*(a*d - b*c)) + (B*b^2*d^2*i*x^2)/
(a*d - b*c))/(6*a^3*b^2*g^4 + 6*b^5*g^4*x^3 + 18*a^2*b^3*g^4*x + 18*a*b^4*g^4*x^2) - (log((e*(a + b*x))/(c + d
*x))*((B*c*i)/(3*b^2*g^4) + (B*a*d*i)/(6*b^3*g^4) + (B*d*i*x)/(2*b^2*g^4)))/(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b
*x^2) - (B*d^3*i*atanh((6*b^4*c^2*g^4 - 6*a^2*b^2*d^2*g^4)/(6*b^2*g^4*(a*d - b*c)^2) - (2*b*d*x)/(a*d - b*c)))
/(3*b^2*g^4*(a*d - b*c)^2)

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sympy [B]  time = 11.30, size = 629, normalized size = 3.64 \[ - \frac {B d^{3} i \log {\left (x + \frac {- \frac {B a^{3} d^{6} i}{\left (a d - b c\right )^{2}} + \frac {3 B a^{2} b c d^{5} i}{\left (a d - b c\right )^{2}} - \frac {3 B a b^{2} c^{2} d^{4} i}{\left (a d - b c\right )^{2}} + B a d^{4} i + \frac {B b^{3} c^{3} d^{3} i}{\left (a d - b c\right )^{2}} + B b c d^{3} i}{2 B b d^{4} i} \right )}}{6 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {B d^{3} i \log {\left (x + \frac {\frac {B a^{3} d^{6} i}{\left (a d - b c\right )^{2}} - \frac {3 B a^{2} b c d^{5} i}{\left (a d - b c\right )^{2}} + \frac {3 B a b^{2} c^{2} d^{4} i}{\left (a d - b c\right )^{2}} + B a d^{4} i - \frac {B b^{3} c^{3} d^{3} i}{\left (a d - b c\right )^{2}} + B b c d^{3} i}{2 B b d^{4} i} \right )}}{6 b^{2} g^{4} \left (a d - b c\right )^{2}} + \frac {\left (- B a d i - 2 B b c i - 3 B b d i x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{6 a^{3} b^{2} g^{4} + 18 a^{2} b^{3} g^{4} x + 18 a b^{4} g^{4} x^{2} + 6 b^{5} g^{4} x^{3}} + \frac {- 6 A a^{2} d^{2} i - 6 A a b c d i + 12 A b^{2} c^{2} i - 5 B a^{2} d^{2} i - 5 B a b c d i + 4 B b^{2} c^{2} i - 6 B b^{2} d^{2} i x^{2} + x \left (- 18 A a b d^{2} i + 18 A b^{2} c d i - 15 B a b d^{2} i + 3 B b^{2} c d i\right )}{36 a^{4} b^{2} d g^{4} - 36 a^{3} b^{3} c g^{4} + x^{3} \left (36 a b^{5} d g^{4} - 36 b^{6} c g^{4}\right ) + x^{2} \left (108 a^{2} b^{4} d g^{4} - 108 a b^{5} c g^{4}\right ) + x \left (108 a^{3} b^{3} d g^{4} - 108 a^{2} b^{4} c g^{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-B*d**3*i*log(x + (-B*a**3*d**6*i/(a*d - b*c)**2 + 3*B*a**2*b*c*d**5*i/(a*d - b*c)**2 - 3*B*a*b**2*c**2*d**4*i
/(a*d - b*c)**2 + B*a*d**4*i + B*b**3*c**3*d**3*i/(a*d - b*c)**2 + B*b*c*d**3*i)/(2*B*b*d**4*i))/(6*b**2*g**4*
(a*d - b*c)**2) + B*d**3*i*log(x + (B*a**3*d**6*i/(a*d - b*c)**2 - 3*B*a**2*b*c*d**5*i/(a*d - b*c)**2 + 3*B*a*
b**2*c**2*d**4*i/(a*d - b*c)**2 + B*a*d**4*i - B*b**3*c**3*d**3*i/(a*d - b*c)**2 + B*b*c*d**3*i)/(2*B*b*d**4*i
))/(6*b**2*g**4*(a*d - b*c)**2) + (-B*a*d*i - 2*B*b*c*i - 3*B*b*d*i*x)*log(e*(a + b*x)/(c + d*x))/(6*a**3*b**2
*g**4 + 18*a**2*b**3*g**4*x + 18*a*b**4*g**4*x**2 + 6*b**5*g**4*x**3) + (-6*A*a**2*d**2*i - 6*A*a*b*c*d*i + 12
*A*b**2*c**2*i - 5*B*a**2*d**2*i - 5*B*a*b*c*d*i + 4*B*b**2*c**2*i - 6*B*b**2*d**2*i*x**2 + x*(-18*A*a*b*d**2*
i + 18*A*b**2*c*d*i - 15*B*a*b*d**2*i + 3*B*b**2*c*d*i))/(36*a**4*b**2*d*g**4 - 36*a**3*b**3*c*g**4 + x**3*(36
*a*b**5*d*g**4 - 36*b**6*c*g**4) + x**2*(108*a**2*b**4*d*g**4 - 108*a*b**5*c*g**4) + x*(108*a**3*b**3*d*g**4 -
108*a**2*b**4*c*g**4))

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