3.1.95 \(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(c i+d i x)^2} \, dx\) [95]

Optimal. Leaf size=152 \[ -\frac {2 A B (a+b x)}{(b c-a d) i^2 (c+d x)}+\frac {2 B^2 (a+b x)}{(b c-a d) i^2 (c+d x)}-\frac {2 B^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d) i^2 (c+d x)}+\frac {(a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d) i^2 (c+d x)} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 152, 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 = {2552, 2333, 2332} \begin {gather*} \frac {(a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{i^2 (c+d x) (b c-a d)}-\frac {2 A B (a+b x)}{i^2 (c+d x) (b c-a d)}-\frac {2 B^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{i^2 (c+d x) (b c-a d)}+\frac {2 B^2 (a+b x)}{i^2 (c+d x) (b c-a d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2552

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

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(95 c+95 d x)^2} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {(2 B) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{95 (a+b x) (c+d x)^2} \, dx}{95 d}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {(2 B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x) (c+d x)^2} \, dx}{9025 d}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {(2 B (b c-a d)) \int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^2}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{9025 d}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac {(2 B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{9025}-\frac {(2 b B) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{9025 (b c-a d)}+\frac {\left (2 b^2 B\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {\left (2 B^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{9025 d}-\frac {\left (2 b B^2\right ) \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}{9025 d (b c-a d)}+\frac {\left (2 b B^2\right ) \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}{9025 d (b c-a d)}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {\left (2 B^2 (b c-a d)\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{9025 d}-\frac {\left (2 b B^2\right ) \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}{9025 d (b c-a d) e}+\frac {\left (2 b B^2\right ) \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}{9025 d (b c-a d) e}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {\left (2 B^2 (b c-a d)\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{9025 d}-\frac {\left (2 b B^2\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{9025 d (b c-a d) e}+\frac {\left (2 b B^2\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{9025 d (b c-a d) e}\\ &=-\frac {2 B^2}{9025 d (c+d x)}-\frac {2 b B^2 \log (a+b x)}{9025 d (b c-a d)}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {2 b B^2 \log (c+d x)}{9025 d (b c-a d)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}+\frac {\left (2 b B^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{9025 (b c-a d)}-\frac {\left (2 b B^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{9025 (b c-a d)}-\frac {\left (2 b^2 B^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{9025 d (b c-a d)}+\frac {\left (2 b^2 B^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=-\frac {2 B^2}{9025 d (c+d x)}-\frac {2 b B^2 \log (a+b x)}{9025 d (b c-a d)}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}-\frac {\left (2 b B^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{9025 (b c-a d)}-\frac {\left (2 b B^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{9025 d (b c-a d)}-\frac {\left (2 b B^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{9025 d (b c-a d)}-\frac {\left (2 b^2 B^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=-\frac {2 B^2}{9025 d (c+d x)}-\frac {2 b B^2 \log (a+b x)}{9025 d (b c-a d)}-\frac {b B^2 \log ^2(a+b x)}{9025 d (b c-a d)}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {b B^2 \log ^2(c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}-\frac {\left (2 b B^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{9025 d (b c-a d)}-\frac {\left (2 b B^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{9025 d (b c-a d)}\\ &=-\frac {2 B^2}{9025 d (c+d x)}-\frac {2 b B^2 \log (a+b x)}{9025 d (b c-a d)}-\frac {b B^2 \log ^2(a+b x)}{9025 d (b c-a d)}+\frac {2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac {2 b B \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac {2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {2 b B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac {b B^2 \log ^2(c+d x)}{9025 d (b c-a d)}+\frac {2 b B^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}+\frac {2 b B^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9025 d (b c-a d)}+\frac {2 b B^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.30, size = 315, normalized size = 2.07 \begin {gather*} \frac {-\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b (c+d x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 b (c+d 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+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-b B (c+d 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 B (c+d 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 )\right )}{b c-a d}}{d i^2 (c+d x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(340\) vs. \(2(152)=304\).
time = 0.57, size = 341, normalized size = 2.24

method	result	size
norman	\(\frac {\frac {\left (A^{2}-2 A B +2 B^{2}\right ) x}{i c}-\frac {B^{2} a \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i \left (a d -c b \right )}-\frac {B^{2} b x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{\left (a d -c b \right ) i}-\frac {2 a B \left (A -B \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i \left (a d -c b \right )}-\frac {2 B b \left (A -B \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i \left (a d -c b \right )}}{i \left (d x +c \right )}\)	\(182\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right )^{2} e^{2} i^{2}}+\frac {2 d^{2} A B \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (a d -c b \right )^{2} e^{2} i^{2}}+\frac {d^{2} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{\left (a d -c b \right )^{2} e^{2} i^{2}}\right )}{d^{2}}\)	\(341\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A^{2} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\left (a d -c b \right )^{2} e^{2} i^{2}}+\frac {2 d^{2} A B \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{\left (a d -c b \right )^{2} e^{2} i^{2}}+\frac {d^{2} B^{2} \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}-2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+\frac {2 \left (a d -c b \right ) e}{d \left (d x +c \right )}+\frac {2 b e}{d}\right )}{\left (a d -c b \right )^{2} e^{2} i^{2}}\right )}{d^{2}}\)	\(341\)
risch	\(-\frac {A^{2}}{i^{2} \left (d x +c \right ) d}-\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} b}{i^{2} \left (a d -c b \right ) d}-\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} a}{i^{2} \left (a d -c b \right ) \left (d x +c \right )}+\frac {B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} c b}{i^{2} \left (a d -c b \right ) d \left (d x +c \right )}+\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b}{i^{2} \left (a d -c b \right ) d}+\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a}{i^{2} \left (a d -c b \right ) \left (d x +c \right )}-\frac {2 B^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c b}{i^{2} \left (a d -c b \right ) d \left (d x +c \right )}-\frac {2 B^{2} a}{i^{2} \left (a d -c b \right ) \left (d x +c \right )}+\frac {2 B^{2} c b}{i^{2} \left (a d -c b \right ) d \left (d x +c \right )}-\frac {2 B^{2} b}{i^{2} d \left (a d -c b \right )}-\frac {2 A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b}{i^{2} \left (a d -c b \right ) d}-\frac {2 A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a}{i^{2} \left (a d -c b \right ) \left (d x +c \right )}+\frac {2 A B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c b}{i^{2} \left (a d -c b \right ) d \left (d x +c \right )}+\frac {2 A B a}{i^{2} \left (a d -c b \right ) \left (d x +c \right )}-\frac {2 A B c b}{i^{2} \left (a d -c b \right ) d \left (d x +c \right )}+\frac {2 A B b}{i^{2} \left (a d -c b \right ) d}\)	\(661\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 366 vs. \(2 (143) = 286\).
time = 0.28, size = 366, normalized size = 2.41 \begin {gather*} -{\left (2 \, {\left (\frac {b \log \left (b x + a\right )}{b c d - a d^{2}} - \frac {b \log \left (d x + c\right )}{b c d - a d^{2}} + \frac {1}{d^{2} x + c d}\right )} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {{\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{b c^{2} d - a c d^{2} + {\left (b c d^{2} - a d^{3}\right )} x}\right )} B^{2} - 2 \, A B {\left (\frac {b \log \left (b x + a\right )}{b c d - a d^{2}} - \frac {b \log \left (d x + c\right )}{b c d - a d^{2}} - \frac {\log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )}{d^{2} x + c d} + \frac {1}{d^{2} x + c d}\right )} + \frac {B^{2} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right )^{2}}{d^{2} x + c d} + \frac {A^{2}}{d^{2} x + c d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.43, size = 144, normalized size = 0.95 \begin {gather*} \frac {{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c - {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d - {\left (B^{2} b d x + B^{2} a d\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} - 2 \, {\left ({\left (A B - B^{2}\right )} b d x + {\left (A B - B^{2}\right )} a d\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{b c^{2} d - a c d^{2} + {\left (b c d^{2} - a d^{3}\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (128) = 256\).
time = 1.47, size = 432, normalized size = 2.84 \begin {gather*} \frac {2 B b \left (A - B\right ) \log {\left (x + \frac {2 A B a b d + 2 A B b^{2} c - 2 B^{2} a b d - 2 B^{2} b^{2} c - \frac {2 B a^{2} b d^{2} \left (A - B\right )}{a d - b c} + \frac {4 B a b^{2} c d \left (A - B\right )}{a d - b c} - \frac {2 B b^{3} c^{2} \left (A - B\right )}{a d - b c}}{4 A B b^{2} d - 4 B^{2} b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac {2 B b \left (A - B\right ) \log {\left (x + \frac {2 A B a b d + 2 A B b^{2} c - 2 B^{2} a b d - 2 B^{2} b^{2} c + \frac {2 B a^{2} b d^{2} \left (A - B\right )}{a d - b c} - \frac {4 B a b^{2} c d \left (A - B\right )}{a d - b c} + \frac {2 B b^{3} c^{2} \left (A - B\right )}{a d - b c}}{4 A B b^{2} d - 4 B^{2} b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} + \frac {\left (- 2 A B + 2 B^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{c d i^{2} + d^{2} i^{2} x} + \frac {\left (- B^{2} a - B^{2} b x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a c d i^{2} + a d^{2} i^{2} x - b c^{2} i^{2} - b c d i^{2} x} + \frac {- A^{2} + 2 A B - 2 B^{2}}{c d i^{2} + d^{2} i^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.60, size = 179, normalized size = 1.18 \begin {gather*} -{\left (\frac {{\left (b x e + a e\right )} B^{2} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + \frac {2 \, {\left (b x e + a e\right )} {\left (A B - B^{2}\right )} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {{\left (b x e + a e\right )} {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )}}{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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 5.62, size = 222, normalized size = 1.46 \begin {gather*} \frac {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (\frac {2\,B^2}{b\,d^2\,i^2}-\frac {2\,A\,B}{b\,d^2\,i^2}\right )}{\frac {x}{b}+\frac {c}{b\,d}}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2\,\left (\frac {B^2}{d^2\,i^2\,\left (x+\frac {c}{d}\right )}+\frac {B^2\,b}{d\,i^2\,\left (a\,d-b\,c\right )}\right )-\frac {A^2-2\,A\,B+2\,B^2}{x\,d^2\,i^2+c\,d\,i^2}+\frac {B\,b\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a\,d^2\,i^2+b\,c\,d\,i^2}{d\,i^2}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-B\right )\,4{}\mathrm {i}}{d\,i^2\,\left (a\,d-b\,c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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