3.1.8 \(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))}{(a g+b g x)^4} \, dx\) [8]

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

[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 = 0.09, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2562, 45, 2372, 12} \begin {gather*} -\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} \end {gather*}

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*d*i*(c + d*x)^2)/(4*(b*c - a*d)^2*g^4*(a + b*x)^2) - (b*B*i*(c + d*x)^3)/(9*(b*c - a*d)^2*g^4*(a + b*x)^3)
+ (d*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^2*g^4*(a + b*x)^2) - (b*i*(c + d*x)^3*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^2*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 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2562

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

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 = 0.27, size = 187, normalized size = 1.08 \begin {gather*} -\frac {i \left (\frac {12 A b c}{(a+b x)^3}+\frac {4 b B c}{(a+b x)^3}-\frac {12 a A d}{(a+b x)^3}-\frac {4 a B d}{(a+b x)^3}+\frac {18 A d}{(a+b x)^2}+\frac {3 B d}{(a+b x)^2}-\frac {6 B d^2}{(b c-a d) (a+b x)}-\frac {6 B d^3 \log (a+b x)}{(b c-a d)^2}+\frac {6 B (2 b c+a d+3 b d x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac {6 B d^3 \log (c+d x)}{(b c-a d)^2}\right )}{36 b^2 g^4} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(344\) vs. \(2(165)=330\).
time = 0.54, size = 345, normalized size = 1.99

method	result	size
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {i \,d^{2} e^{2} A b}{3 \left (a d -c b \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{3} e A}{2 \left (a d -c b \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i \,d^{2} e^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{3} g^{4}}+\frac {i \,d^{3} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{4}}\right )}{d^{2}}\)	\(345\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {i \,d^{2} e^{2} A b}{3 \left (a d -c b \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {i \,d^{3} e A}{2 \left (a d -c b \right )^{3} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {i \,d^{2} e^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{3} g^{4}}+\frac {i \,d^{3} e B \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3} g^{4}}\right )}{d^{2}}\)	\(345\)
norman	\(\frac {-\frac {6 A \,a^{2} d^{2} i +6 A a b c d i -12 A \,b^{2} c^{2} i +3 B \,a^{2} d^{2} i +5 B a b c d i -4 B \,b^{2} c^{2} i}{36 b^{2} g \left (a d -c b \right )}-\frac {\left (6 A a \,d^{2} i -6 A b c d i +3 B a \,d^{2} i -B b c d i \right ) x}{12 g \left (a d -c b \right ) b}+\frac {B \,d^{2} i b \,x^{3}}{18 a \left (a d -c b \right ) g}+\frac {B i \,c^{2} \left (3 a d -2 c b \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}+\frac {B a i \,d^{3} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B i \,d^{3} b \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {B i c d \left (2 a d -c b \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) g}}{g^{3} \left (b x +a \right )^{3}}\)	\(383\)
risch	\(-\frac {B i \left (3 b d x +a d +2 c b \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{6 \left (b x +a \right )^{3} b^{2} g^{4}}-\frac {\left (6 B \ln \left (d x +c \right ) b^{3} d^{3} x^{3}-6 B \ln \left (-b x -a \right ) b^{3} d^{3} x^{3}+18 B \ln \left (d x +c \right ) a \,b^{2} d^{3} x^{2}-18 B \ln \left (-b x -a \right ) a \,b^{2} d^{3} x^{2}+18 B \ln \left (d x +c \right ) a^{2} b \,d^{3} x -18 B \ln \left (-b x -a \right ) a^{2} b \,d^{3} x +6 B a \,b^{2} d^{3} x^{2}-6 B \,b^{3} c \,d^{2} x^{2}+18 A \,a^{2} b \,d^{3} x -36 A a \,b^{2} c \,d^{2} x +18 A \,b^{3} c^{2} d x +6 B \ln \left (d x +c \right ) a^{3} d^{3}-6 B \ln \left (-b x -a \right ) a^{3} d^{3}+15 B \,a^{2} b \,d^{3} x -18 B a \,b^{2} c \,d^{2} x +3 B \,b^{3} c^{2} d x +6 A \,a^{3} d^{3}-18 A a \,b^{2} c^{2} d +12 A \,b^{3} c^{3}+5 B \,a^{3} d^{3}-9 B a \,b^{2} c^{2} d +4 B \,c^{3} b^{3}\right ) i}{36 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (b x +a \right )^{3} b^{2} g^{4}}\)	\(389\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (163) = 326\).
time = 0.32, size = 933, normalized size = 5.39 \begin {gather*} -\frac {1}{36} i \, B d {\left (\frac {6 \, {\left (3 \, b x + a\right )} \log \left (\frac {b x e}{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} i \, B c {\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 x e}{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 {i \, {\left (3 \, b x + a\right )} A d}{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 {i \, A c}{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 )}} \end {gather*}

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*I*B*d*(6*(3*b*x + a)*log(b*x*e/(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*I*B*c*((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*x*e/(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*I*(3*b*x + a)*A*d/(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*I*A*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)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (163) = 326\).
time = 0.39, size = 361, normalized size = 2.09 \begin {gather*} -\frac {4 \, {\left (3 i \, A + i \, B\right )} b^{3} c^{3} + 9 \, {\left (-2 i \, A - i \, B\right )} a b^{2} c^{2} d - {\left (-6 i \, A - 5 i \, B\right )} a^{3} d^{3} + 6 \, {\left (-i \, B b^{3} c d^{2} + i \, B a b^{2} d^{3}\right )} x^{2} + 3 \, {\left ({\left (6 i \, A + i \, B\right )} b^{3} c^{2} d + 6 \, {\left (-2 i \, A - i \, B\right )} a b^{2} c d^{2} + {\left (6 i \, A + 5 i \, B\right )} a^{2} b d^{3}\right )} x + 6 \, {\left (-i \, B b^{3} d^{3} x^{3} - 3 i \, B a b^{2} d^{3} x^{2} + 2 i \, B b^{3} c^{3} - 3 i \, B a b^{2} c^{2} d + 3 \, {\left (i \, B b^{3} c^{2} d - 2 i \, B a b^{2} c d^{2}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} 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 )}} \end {gather*}

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*(4*(3*I*A + I*B)*b^3*c^3 + 9*(-2*I*A - I*B)*a*b^2*c^2*d - (-6*I*A - 5*I*B)*a^3*d^3 + 6*(-I*B*b^3*c*d^2 +
I*B*a*b^2*d^3)*x^2 + 3*((6*I*A + I*B)*b^3*c^2*d + 6*(-2*I*A - I*B)*a*b^2*c*d^2 + (6*I*A + 5*I*B)*a^2*b*d^3)*x
+ 6*(-I*B*b^3*d^3*x^3 - 3*I*B*a*b^2*d^3*x^2 + 2*I*B*b^3*c^3 - 3*I*B*a*b^2*c^2*d + 3*(I*B*b^3*c^2*d - 2*I*B*a*
b^2*c*d^2)*x)*log((b*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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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (158) = 316\).
time = 5.08, size = 629, normalized size = 3.64 \begin {gather*} - \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} \cdot \left (36 a b^{5} d g^{4} - 36 b^{6} c g^{4}\right ) + x^{2} \cdot \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 )} \end {gather*}

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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Giac [A]
time = 4.47, size = 238, normalized size = 1.38 \begin {gather*} -\frac {{\left (12 i \, B b e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {18 i \, {\left (b x e + a e\right )} B d e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + 12 i \, A b e^{4} + 4 i \, B b e^{4} - \frac {18 i \, {\left (b x e + a e\right )} A d e^{3}}{d x + c} - \frac {9 i \, {\left (b x e + a e\right )} B d 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 )}} \end {gather*}

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*I*B*b*e^4*log((b*x*e + a*e)/(d*x + c)) - 18*I*(b*x*e + a*e)*B*d*e^3*log((b*x*e + a*e)/(d*x + c))/(d*
x + c) + 12*I*A*b*e^4 + 4*I*B*b*e^4 - 18*I*(b*x*e + a*e)*A*d*e^3/(d*x + c) - 9*I*(b*x*e + a*e)*B*d*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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Mupad [B]
time = 5.87, size = 361, normalized size = 2.09 \begin {gather*} -\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} \end {gather*}

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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