3.1.36 \(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^2 (c i+d i x)} \, dx\) [36]

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

[Out]

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

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Rubi [A]
time = 0.12, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2562, 45, 2372, 14, 2338} \begin {gather*} -\frac {d \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (b c-a d)^2}-\frac {b (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i (a+b x) (b c-a d)^2}-\frac {b B (c+d x)}{g^2 i (a+b x) (b c-a d)^2}+\frac {B d \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^2 i (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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 {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(36 c+36 d x) (a g+b g x)^2} \, dx &=\int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d) g^2 (a+b x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2 (c+d x)}\right ) \, dx\\ &=-\frac {(b d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{36 (b c-a d)^2 g^2}+\frac {d^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{36 (b c-a d)^2 g^2}+\frac {b \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{36 (b c-a d) g^2}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {(B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{36 (b c-a d)^2 g^2}-\frac {(B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{36 (b c-a d)^2 g^2}+\frac {B \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{36 (b c-a d) g^2}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {B \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{36 g^2}+\frac {(B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{36 (b c-a d)^2 e g^2}-\frac {(B d) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{36 (b c-a d)^2 e g^2}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {B \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}{36 g^2}+\frac {(B d) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{36 (b c-a d)^2 e g^2}-\frac {(B d) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{36 (b c-a d)^2 e g^2}\\ &=-\frac {B}{36 (b c-a d) g^2 (a+b x)}-\frac {B d \log (a+b x)}{36 (b c-a d)^2 g^2}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac {B d \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {(b B d) \int \frac {\log (a+b x)}{a+b x} \, dx}{36 (b c-a d)^2 g^2}-\frac {(b B d) \int \frac {\log (c+d x)}{a+b x} \, dx}{36 (b c-a d)^2 g^2}-\frac {\left (B d^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{36 (b c-a d)^2 g^2}+\frac {\left (B d^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{36 (b c-a d)^2 g^2}\\ &=-\frac {B}{36 (b c-a d) g^2 (a+b x)}-\frac {B d \log (a+b x)}{36 (b c-a d)^2 g^2}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac {B d \log (c+d x)}{36 (b c-a d)^2 g^2}-\frac {B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}-\frac {B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}+\frac {(B d) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{36 (b c-a d)^2 g^2}+\frac {(B d) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{36 (b c-a d)^2 g^2}+\frac {(b B d) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{36 (b c-a d)^2 g^2}+\frac {\left (B d^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{36 (b c-a d)^2 g^2}\\ &=-\frac {B}{36 (b c-a d) g^2 (a+b x)}-\frac {B d \log (a+b x)}{36 (b c-a d)^2 g^2}+\frac {B d \log ^2(a+b x)}{72 (b c-a d)^2 g^2}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac {B d \log (c+d x)}{36 (b c-a d)^2 g^2}-\frac {B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {B d \log ^2(c+d x)}{72 (b c-a d)^2 g^2}-\frac {B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}+\frac {(B d) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{36 (b c-a d)^2 g^2}+\frac {(B d) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{36 (b c-a d)^2 g^2}\\ &=-\frac {B}{36 (b c-a d) g^2 (a+b x)}-\frac {B d \log (a+b x)}{36 (b c-a d)^2 g^2}+\frac {B d \log ^2(a+b x)}{72 (b c-a d)^2 g^2}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{36 (b c-a d) g^2 (a+b x)}-\frac {d \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{36 (b c-a d)^2 g^2}+\frac {B d \log (c+d x)}{36 (b c-a d)^2 g^2}-\frac {B d \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{36 (b c-a d)^2 g^2}+\frac {B d \log ^2(c+d x)}{72 (b c-a d)^2 g^2}-\frac {B d \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}-\frac {B d \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}-\frac {B d \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^2 g^2}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.21, size = 292, normalized size = 1.69 \begin {gather*} -\frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )}{2 (b c-a d)^2 g^2 i (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.62, size = 288, normalized size = 1.66

method	result	size
norman	\(\frac {-\frac {\left (A a d +B b c \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {B a d \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {b \left (A d +B d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (A +B \right ) b x}{g i a \left (a d -c b \right )}-\frac {b B d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{g \left (b x +a \right )}\)	\(252\)
risch	\(\frac {A d \ln \left (d x +c \right )}{g^{2} i \left (a d -c b \right )^{2}}+\frac {A}{g^{2} i \left (a d -c b \right ) \left (b x +a \right )}-\frac {A d \ln \left (b x +a \right )}{g^{2} i \left (a d -c b \right )^{2}}-\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{2} i \left (a d -c b \right )^{2}}-\frac {B b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {B b e}{g^{2} i \left (a d -c b \right )^{2} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}\)	\(266\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A b}{i \left (a d -c b \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {d^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (a d -c b \right )^{3} g^{2}}\right )}{d^{2}}\)	\(288\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A b}{i \left (a d -c b \right )^{3} g^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{3} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{3} g^{2}}-\frac {d^{2} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{3} g^{2}}+\frac {d^{3} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (a d -c b \right )^{3} g^{2}}\right )}{d^{2}}\)	\(288\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d^2*e*(a*d-b*c)*(d^2/i/(a*d-b*c)^3/g^2*A*b/(b*e/d+(a*d-b*c)*e/d/(d*x+c))+d^3/e/i/(a*d-b*c)^3/g^2*A*ln(b*e/d
+(a*d-b*c)*e/d/(d*x+c))-d^2/i/(a*d-b*c)^3/g^2*B*b*(-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*
x+c))-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c)))+1/2*d^3/e/i/(a*d-b*c)^3/g^2*B*ln(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. 423 vs. \(2 (161) = 322\).
time = 0.33, size = 423, normalized size = 2.45 \begin {gather*} B {\left (\frac {1}{{\left (-i \, b^{2} c + i \, a b d\right )} g^{2} x + {\left (-i \, a b c + i \, a^{2} d\right )} g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}}\right )} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) + A {\left (\frac {1}{{\left (-i \, b^{2} c + i \, a b d\right )} g^{2} x + {\left (-i \, a b c + i \, a^{2} d\right )} g^{2}} - \frac {d \log \left (b x + a\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}} + \frac {d \log \left (d x + c\right )}{{\left (i \, b^{2} c^{2} - 2 i \, a b c d + i \, a^{2} d^{2}\right )} g^{2}}\right )} - \frac {{\left ({\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )^{2} + {\left (i \, b d x + i \, a d\right )} \log \left (d x + c\right )^{2} - 2 i \, b c + 2 i \, a d - 2 \, {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right ) - 2 \, {\left (-i \, b d x - i \, a d + {\left (i \, b d x + i \, a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{2 \, {\left (a b^{2} c^{2} g^{2} - 2 \, a^{2} b c d g^{2} + a^{3} d^{2} g^{2} + {\left (b^{3} c^{2} g^{2} - 2 \, a b^{2} c d g^{2} + a^{2} b d^{2} g^{2}\right )} x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.41, size = 157, normalized size = 0.91 \begin {gather*} -\frac {2 \, {\left (-i \, A - i \, B\right )} b c + 2 \, {\left (i \, A + i \, B\right )} a d - {\left (i \, B b d x + i \, B a d\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} + 2 \, {\left ({\left (-i \, A - i \, B\right )} b d x - i \, B b c - i \, A a d\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{2 \, {\left ({\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} g^{2} x + {\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} g^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (144) = 288\).
time = 0.73, size = 386, normalized size = 2.23 \begin {gather*} - \frac {B d \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{2} d^{2} g^{2} i - 4 a b c d g^{2} i + 2 b^{2} c^{2} g^{2} i} + \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a^{2} d g^{2} i - a b c g^{2} i + a b d g^{2} i x - b^{2} c g^{2} i x} + \left (A + B\right ) \left (\frac {d \log {\left (x + \frac {- \frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} + \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} - \frac {d \log {\left (x + \frac {\frac {a^{3} d^{4}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b c d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{2} c^{2} d^{2}}{\left (a d - b c\right )^{2}} + a d^{2} - \frac {b^{3} c^{3} d}{\left (a d - b c\right )^{2}} + b c d}{2 b d^{2}} \right )}}{g^{2} i \left (a d - b c\right )^{2}} + \frac {1}{a^{2} d g^{2} i - a b c g^{2} i + x \left (a b d g^{2} i - b^{2} c g^{2} i\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.84, size = 241, normalized size = 1.39 \begin {gather*} \frac {A+B}{\left (a\,d-b\,c\right )\,\left (a\,g^2\,i+b\,g^2\,i\,x\right )}-\frac {B\,d\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^2\,i\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )}{b\,d\,g^2\,i\,\left (\frac {x}{d}+\frac {a}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a^2\,d^2\,g^2\,i-b^2\,c^2\,g^2\,i}{g^2\,i\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A+B\right )\,2{}\mathrm {i}}{g^2\,i\,{\left (a\,d-b\,c\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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