3.3.60 \(\int \frac {(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{a f h+b g h x^2+h (b f x+a g x)} \, dx\) [260]

Optimal. Leaf size=203 \[ -\frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \log \left (1-\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {2 B n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {2 B^2 n^2 \text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h} \]

[Out]

-(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2*ln(1-(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h+2*B*n*(A+B*ln(e*
(b*x+a)^n/((d*x+c)^n)))*polylog(2,(-a*g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h+2*B^2*n^2*polylog(3,(-a*
g+b*f)*(d*x+c)/(-c*g+d*f)/(b*x+a))/(-a*g+b*f)/h

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Rubi [A]
time = 0.34, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 51, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.137, Rules used = {2573, 2576, 3, 1607, 2379, 2421, 6724} \begin {gather*} \frac {2 B n \text {PolyLog}\left (2,\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{h (b f-a g)}+\frac {2 B^2 n^2 \text {PolyLog}\left (3,\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right )}{h (b f-a g)}-\frac {\log \left (1-\frac {(c+d x) (b f-a g)}{(a+b x) (d f-c g)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{h (b f-a g)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2*Log[1 - ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f
- a*g)*h)) + (2*B*n*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])*PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(
a + b*x))])/((b*f - a*g)*h) + (2*B^2*n^2*PolyLog[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])/((b*f -
a*g)*h)

Rule 3

Int[(u_.)*((a_) + (c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[u*(b*x^n + c*x^(2*n))^p, x] /;
FreeQ[{a, b, c, n, p}, x] && EqQ[j, 2*n] && EqQ[a, 0]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2379

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

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

Rule 2576

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*(P2x_)^(m_.), x_Symbol]
:> With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, Dist[b*c - a*d, Subst[Int[(b^2*f
- a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^2*h)*x^2)^m*((A + B*Log[e*x^n])^p
/(b - d*x)^(2*(m + 1))), x], x, (a + b*x)/(c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x,
x, 2] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{a f h+b g h x^2+h (b f x+a g x)} \, dx &=\int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{h (a+b x) (f+g x)} \, dx\\ &=\frac {\int \frac {\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{(a+b x) (f+g x)} \, dx}{h}\\ &=\frac {\int \left (\frac {A^2}{(a+b x) (f+g x)}+\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)}+\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)}\right ) \, dx}{h}\\ &=\frac {A^2 \int \frac {1}{(a+b x) (f+g x)} \, dx}{h}+\frac {(2 A B) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}+\frac {B^2 \int \frac {\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(a+b x) (f+g x)} \, dx}{h}\\ &=-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {\left (A^2 b\right ) \int \frac {1}{a+b x} \, dx}{(b f-a g) h}-\frac {\left (A^2 g\right ) \int \frac {1}{f+g x} \, dx}{(b f-a g) h}+\frac {(2 A B (b c-a d) n) \int \frac {\log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}+\frac {\left (2 B^2 (b c-a d) n\right ) \int \frac {\log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^2 \log (a+b x)}{(b f-a g) h}-\frac {A^2 \log (f+g x)}{(b f-a g) h}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {(2 A B (b c-a d) n) \text {Subst}\left (\int \frac {\log \left (-\frac {(b c-a d) x}{d f-c g}\right )}{1+\frac {(b c-a d) x}{d f-c g}} \, dx,x,\frac {f+g x}{a+b x}\right )}{(b f-a g) (d f-c g) h}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \int \frac {\text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(a+b x) (c+d x)} \, dx}{(b f-a g) h}\\ &=\frac {A^2 \log (a+b x)}{(b f-a g) h}-\frac {A^2 \log (f+g x)}{(b f-a g) h}-\frac {2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}-\frac {B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (-\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {2 A B n \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {2 B^2 n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \text {Li}_2\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}+\frac {2 B^2 n^2 \text {Li}_3\left (1+\frac {(b c-a d) (f+g x)}{(d f-c g) (a+b x)}\right )}{(b f-a g) h}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1415\) vs. \(2(203)=406\).
time = 0.40, size = 1415, normalized size = 6.97 \begin {gather*} \frac {3 \log (a+b x) \left (A+B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )^2-3 \left (A+B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )^2 \log (f+g x)+3 B n \left (A+B \left (-n \log (a+b x)+n \log (c+d x)+\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right ) \left (\log ^2(a+b x)-2 \left (\log (a+b x) \log \left (\frac {b (f+g x)}{b f-a g}\right )+\text {Li}_2\left (\frac {g (a+b x)}{-b f+a g}\right )\right )\right )-6 A B n \left (\log (c+d x) \left (\log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log \left (\frac {d (f+g x)}{d f-c g}\right )\right )+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-\text {Li}_2\left (\frac {g (c+d x)}{-d f+c g}\right )\right )+6 B^2 n \left (n \log (a+b x)-n \log (c+d x)-\log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \left (\log (c+d x) \left (\log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log \left (\frac {d (f+g x)}{d f-c g}\right )\right )+\text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-\text {Li}_2\left (\frac {g (c+d x)}{-d f+c g}\right )\right )+B^2 n^2 \left (\log ^2(a+b x) \left (\log (a+b x)-3 \log \left (\frac {b (f+g x)}{b f-a g}\right )\right )-6 \log (a+b x) \text {Li}_2\left (\frac {g (a+b x)}{-b f+a g}\right )+6 \text {Li}_3\left (\frac {g (a+b x)}{-b f+a g}\right )\right )+3 B^2 n^2 \left (\log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log ^2(c+d x)-\log ^2(c+d x) \log \left (\frac {d (f+g x)}{d f-c g}\right )+2 \log (c+d x) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )-2 \log (c+d x) \text {Li}_2\left (\frac {g (c+d x)}{-d f+c g}\right )-2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )+2 \text {Li}_3\left (\frac {g (c+d x)}{-d f+c g}\right )\right )-6 B^2 n^2 \left (\frac {1}{2} \log ^2(a+b x) \left (\log (c+d x)-\log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-\log (a+b x) \log (c+d x) \log \left (\frac {b (f+g x)}{b f-a g}\right )-\frac {1}{2} \log \left (\frac {g (c+d x)}{-d f+c g}\right ) \left (-2 \log (a+b x)+\log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \left (\log \left (\frac {b (f+g x)}{b f-a g}\right )-\log \left (\frac {d (f+g x)}{d f-c g}\right )\right )+\log \left (\frac {g (c+d x)}{-d f+c g}\right ) \log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (\log \left (\frac {b (f+g x)}{b f-a g}\right )-\log \left (\frac {d (f+g x)}{d f-c g}\right )\right )-\frac {1}{2} \log ^2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (\log \left (\frac {-b c+a d}{d (a+b x)}\right )+\log \left (\frac {b (f+g x)}{b f-a g}\right )-\log \left (\frac {(-b c+a d) (f+g x)}{(d f-c g) (a+b x)}\right )\right )-\log (a+b x) \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )-\left (\log (c+d x)-\log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {g (a+b x)}{-b f+a g}\right )-\left (\log (a+b x)+\log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right ) \text {Li}_2\left (\frac {g (c+d x)}{-d f+c g}\right )-\log \left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right ) \left (\text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )-\text {Li}_2\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right )+\text {Li}_3\left (\frac {d (a+b x)}{-b c+a d}\right )+\text {Li}_3\left (\frac {g (a+b x)}{-b f+a g}\right )+\text {Li}_3\left (\frac {g (c+d x)}{-d f+c g}\right )+\text {Li}_3\left (\frac {b (c+d x)}{d (a+b x)}\right )-\text {Li}_3\left (\frac {(b f-a g) (c+d x)}{(d f-c g) (a+b x)}\right )\right )}{3 (b f-a g) h} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Log[a + b*x]*(A + B*(-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^n]))^2 - 3*(A + B*(
-(n*Log[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^n]))^2*Log[f + g*x] + 3*B*n*(A + B*(-(n*Log
[a + b*x]) + n*Log[c + d*x] + Log[(e*(a + b*x)^n)/(c + d*x)^n]))*(Log[a + b*x]^2 - 2*(Log[a + b*x]*Log[(b*(f +
g*x))/(b*f - a*g)] + PolyLog[2, (g*(a + b*x))/(-(b*f) + a*g)])) - 6*A*B*n*(Log[c + d*x]*(Log[(d*(a + b*x))/(-
(b*c) + a*d)] - Log[(d*(f + g*x))/(d*f - c*g)]) + PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - PolyLog[2, (g*(c + d
*x))/(-(d*f) + c*g)]) + 6*B^2*n*(n*Log[a + b*x] - n*Log[c + d*x] - Log[(e*(a + b*x)^n)/(c + d*x)^n])*(Log[c +
d*x]*(Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[(d*(f + g*x))/(d*f - c*g)]) + PolyLog[2, (b*(c + d*x))/(b*c - a*
d)] - PolyLog[2, (g*(c + d*x))/(-(d*f) + c*g)]) + B^2*n^2*(Log[a + b*x]^2*(Log[a + b*x] - 3*Log[(b*(f + g*x))/
(b*f - a*g)]) - 6*Log[a + b*x]*PolyLog[2, (g*(a + b*x))/(-(b*f) + a*g)] + 6*PolyLog[3, (g*(a + b*x))/(-(b*f) +
a*g)]) + 3*B^2*n^2*(Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x]^2 - Log[c + d*x]^2*Log[(d*(f + g*x))/(d*f
- c*g)] + 2*Log[c + d*x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] - 2*Log[c + d*x]*PolyLog[2, (g*(c + d*x))/(-(d*
f) + c*g)] - 2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)] + 2*PolyLog[3, (g*(c + d*x))/(-(d*f) + c*g)]) - 6*B^2*n^2
*((Log[a + b*x]^2*(Log[c + d*x] - Log[(b*(c + d*x))/(b*c - a*d)]))/2 - Log[a + b*x]*Log[c + d*x]*Log[(b*(f + g
*x))/(b*f - a*g)] - (Log[(g*(c + d*x))/(-(d*f) + c*g)]*(-2*Log[a + b*x] + Log[(g*(c + d*x))/(-(d*f) + c*g)])*(
Log[(b*(f + g*x))/(b*f - a*g)] - Log[(d*(f + g*x))/(d*f - c*g)]))/2 + Log[(g*(c + d*x))/(-(d*f) + c*g)]*Log[((
b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]*(Log[(b*(f + g*x))/(b*f - a*g)] - Log[(d*(f + g*x))/(d*f - c*g)
]) - (Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]^2*(Log[(-(b*c) + a*d)/(d*(a + b*x))] + Log[(b*(f +
g*x))/(b*f - a*g)] - Log[((-(b*c) + a*d)*(f + g*x))/((d*f - c*g)*(a + b*x))]))/2 - Log[a + b*x]*PolyLog[2, (d*
(a + b*x))/(-(b*c) + a*d)] - (Log[c + d*x] - Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])*PolyLog[2,
(g*(a + b*x))/(-(b*f) + a*g)] - (Log[a + b*x] + Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))])*PolyLog[
2, (g*(c + d*x))/(-(d*f) + c*g)] - Log[((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]*(PolyLog[2, (b*(c + d*
x))/(d*(a + b*x))] - PolyLog[2, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]) + PolyLog[3, (d*(a + b*x))/(
-(b*c) + a*d)] + PolyLog[3, (g*(a + b*x))/(-(b*f) + a*g)] + PolyLog[3, (g*(c + d*x))/(-(d*f) + c*g)] + PolyLog
[3, (b*(c + d*x))/(d*(a + b*x))] - PolyLog[3, ((b*f - a*g)*(c + d*x))/((d*f - c*g)*(a + b*x))]))/(3*(b*f - a*g
)*h)

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Maple [F]
time = 0.55, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{2}}{a f h +b g h \,x^{2}+h \left (a g x +b f x \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)),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)^n/((d*x+c)^n)))^2/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)),x, algorithm="maxima")

[Out]

A^2*(log(b*x + a)/((b*f - a*g)*h) - log(g*x + f)/((b*f - a*g)*h)) + integrate((B^2*log((b*x + a)^n)^2 + B^2*lo
g((d*x + c)^n)^2 + 2*A*B + B^2 + 2*(A*B + B^2)*log((b*x + a)^n) - 2*(B^2*log((b*x + a)^n) + A*B + B^2)*log((d*
x + c)^n))/(b*g*h*x^2 + a*f*h + (b*f*h + a*g*h)*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)^n/((d*x+c)^n)))^2/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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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)^n/((d*x+c)^n)))^2/(a*f*h+b*g*h*x^2+h*(a*g*x+b*f*x)),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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