3.1.70 \(\int (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [70]

Optimal. Leaf size=135 \[ \frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b}+\frac {2 B (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{b d}+\frac {2 B^2 (b c-a d) n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{b d} \]

[Out]

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

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Rubi [A]
time = 0.17, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2535, 2543, 2458, 2378, 2370, 2352} \begin {gather*} \frac {2 B^2 n^2 (b c-a d) \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b d}+\frac {2 B n (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b d}+\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2370

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*
x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Rule 2378

Int[((a_.) + Log[(c_.)*(x_)^(n_)]*(b_.))/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Dist[1/n, Subst[Int[(a
+ b*Log[c*x])/(x*(d + e*x^(r/n))), x], x, x^n], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IntegerQ[r/n]

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2535

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.), x_Symbol] :> Simp[(a +
b*x)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])^p/b), x] - Dist[B*n*p*((b*c - a*d)/b), Int[(A + B*Log[e*((a + b*
x)/(c + d*x))^n])^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[b*c - a*d, 0] && IGtQ
[p, 0]

Rule 2543

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))/((f_.) + (g_.)*(x_)), x_Symbo
l] :> Simp[(-Log[(b*c - a*d)/(b*(c + d*x))])*((A + B*Log[e*((a + b*x)/(c + d*x))^n])/g), x] + Dist[B*n*((b*c -
a*d)/g), Int[Log[(b*c - a*d)/(b*(c + d*x))]/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A, B
, n}, x] && NeQ[b*c - a*d, 0] && EqQ[d*f - c*g, 0]

Rubi steps

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

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Mathematica [A]
time = 0.12, size = 226, normalized size = 1.67 \begin {gather*} x \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B n \left (2 a d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 b c \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-a B d n \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 n \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 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to 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))^n))^2,x, algorithm="maxima")

[Out]

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

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Fricas [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))^n))^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + B \log {\left (e \left (\frac {a + b x}{c + d x}\right )^{n} \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*log(e*((a + b*x)/(c + d*x))**n))**2, x)

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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))^n))^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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