\(\int \frac {(c i+d i x) (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^3} \, dx\) [165]

Optimal result

Integrand size = 43, antiderivative size = 151 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {B^2 i n^2 (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {B i n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d) g^3 (a+b x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {2561, 2342, 2341} \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {i (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B i n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 (a+b x)^2 (b c-a d)}-\frac {B^2 i n^2 (c+d x)^2}{4 g^3 (a+b x)^2 (b c-a d)} \]

[In]

Int[((c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g*x)^3,x]

[Out]

-1/4*(B^2*i*n^2*(c + d*x)^2)/((b*c - a*d)*g^3*(a + b*x)^2) - (B*i*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d
*x))^n]))/(2*(b*c - a*d)*g^3*(a + b*x)^2) - (i*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c -
a*d)*g^3*(a + b*x)^2)

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(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] && 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*} \text {integral}& = \frac {i \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^n\right )\right )^2}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g^3} \\ & = -\frac {i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d) g^3 (a+b x)^2}+\frac {(B i n) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d) g^3} \\ & = -\frac {B^2 i n^2 (c+d x)^2}{4 (b c-a d) g^3 (a+b x)^2}-\frac {B i n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d) g^3 (a+b x)^2}-\frac {i (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d) g^3 (a+b x)^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.50 (sec) , antiderivative size = 801, normalized size of antiderivative = 5.30 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {i \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-4 d (-b c+a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+4 B d n (a+b x) \left (2 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 d (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+2 B n (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d n (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 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+B d n (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 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )+B n \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-4 B d n (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B n \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 n (a+b x)^2 \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 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 n (a+b x)^2 \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 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{4 b^2 (b c-a d) g^3 (a+b x)^2} \]

[In]

Integrate[((c*i + d*i*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g*x)^3,x]

[Out]

-1/4*(i*(2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 4*d*(-(b*c) + a*d)*(a + b*x)*(A + B*Log[e*
((a + b*x)/(c + d*x))^n])^2 + 4*B*d*n*(a + b*x)*(2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*d*(a
+ b*x)*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 2*d*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))
^n])*Log[c + d*x] + 2*B*n*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - B*d*n*(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*n*(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*n*(2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 4*d*(-(b*c) + a*d)*(a + b*x
)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 4*d^2*(a + b*x)^2*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]) + 4*d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 4*B*d*n*(a + b*x)*(b*c - a*d + d*
(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*n*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2
*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*d^2*n*(a + b*x)^2*(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)]) - 2*B*d^2*n*(a + b*x)^2*((
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^2*(b*c - a*d)*g^3*(a + b*x)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(529\) vs. \(2(145)=290\).

Time = 4.83 (sec) , antiderivative size = 530, normalized size of antiderivative = 3.51

method	result	size
parallelrisch	\(-\frac {-8 A B x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{2} i n -4 B^{2} x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c \,d^{2} i \,n^{2}+4 A B x a \,b^{3} d^{3} i \,n^{2}-4 A B x \,b^{4} c \,d^{2} i \,n^{2}-4 A B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{2} d i n -4 A B \,x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{3} i n -4 B^{2} x \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{4} c \,d^{2} i n +2 A B \,a^{2} b^{2} d^{3} i \,n^{2}+4 A^{2} x a \,b^{3} d^{3} i n -4 A^{2} x \,b^{4} c \,d^{2} i n -2 B^{2} x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{4} d^{3} i n -2 B^{2} x^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} d^{3} i \,n^{2}-2 A B \,b^{4} c^{2} d i \,n^{2}+2 B^{2} x a \,b^{3} d^{3} i \,n^{3}-2 B^{2} x \,b^{4} c \,d^{2} i \,n^{3}-2 B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{2} b^{4} c^{2} d i n -2 B^{2} \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) b^{4} c^{2} d i \,n^{2}+B^{2} a^{2} b^{2} d^{3} i \,n^{3}-B^{2} b^{4} c^{2} d i \,n^{3}+2 A^{2} a^{2} b^{2} d^{3} i n -2 A^{2} b^{4} c^{2} d i n}{4 g^{3} \left (b x +a \right )^{2} b^{4} d n \left (a d -c b \right )}\)	\(530\)

[In]

int((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3,x,method=_RETURNVERBOSE)

[Out]

-1/4*(-8*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c*d^2*i*n-4*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c*d^2*i*n^2+4*A*B
*x*a*b^3*d^3*i*n^2-4*A*B*x*b^4*c*d^2*i*n^2-4*A*B*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^2*d*i*n-4*A*B*x^2*ln(e*((b*x+
a)/(d*x+c))^n)*b^4*d^3*i*n-4*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)^2*b^4*c*d^2*i*n+2*A*B*a^2*b^2*d^3*i*n^2+4*A^2*x*a
*b^3*d^3*i*n-4*A^2*x*b^4*c*d^2*i*n-2*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*b^4*d^3*i*n-2*B^2*x^2*ln(e*((b*x+a)/(
d*x+c))^n)*b^4*d^3*i*n^2-2*A*B*b^4*c^2*d*i*n^2+2*B^2*x*a*b^3*d^3*i*n^3-2*B^2*x*b^4*c*d^2*i*n^3-2*B^2*ln(e*((b*
x+a)/(d*x+c))^n)^2*b^4*c^2*d*i*n-2*B^2*ln(e*((b*x+a)/(d*x+c))^n)*b^4*c^2*d*i*n^2+B^2*a^2*b^2*d^3*i*n^3-B^2*b^4
*c^2*d*i*n^3+2*A^2*a^2*b^2*d^3*i*n-2*A^2*b^4*c^2*d*i*n)/g^3/(b*x+a)^2/b^4/d/n/(a*d-b*c)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (145) = 290\).

Time = 0.35 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.97 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {{\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} i n^{2} + 2 \, {\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} i n + 2 \, {\left (2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} i x + {\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} i\right )} \log \left (e\right )^{2} + 2 \, {\left (B^{2} b^{2} d^{2} i n^{2} x^{2} + 2 \, B^{2} b^{2} c d i n^{2} x + B^{2} b^{2} c^{2} i n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left (A^{2} b^{2} c^{2} - A^{2} a^{2} d^{2}\right )} i + 2 \, {\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} i n^{2} + 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} i n + 2 \, {\left (A^{2} b^{2} c d - A^{2} a b d^{2}\right )} i\right )} x + 2 \, {\left ({\left (B^{2} b^{2} c^{2} - B^{2} a^{2} d^{2}\right )} i n + 2 \, {\left (A B b^{2} c^{2} - A B a^{2} d^{2}\right )} i + 2 \, {\left ({\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} i n + 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} i\right )} x + 2 \, {\left (B^{2} b^{2} d^{2} i n x^{2} + 2 \, B^{2} b^{2} c d i n x + B^{2} b^{2} c^{2} i n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 2 \, {\left (B^{2} b^{2} c^{2} i n^{2} + 2 \, A B b^{2} c^{2} i n + {\left (B^{2} b^{2} d^{2} i n^{2} + 2 \, A B b^{2} d^{2} i n\right )} x^{2} + 2 \, {\left (B^{2} b^{2} c d i n^{2} + 2 \, A B b^{2} c d i n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c - a b^{4} d\right )} g^{3} x^{2} + 2 \, {\left (a b^{4} c - a^{2} b^{3} d\right )} g^{3} x + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} g^{3}\right )}} \]

[In]

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3,x, algorithm="fricas")

[Out]

-1/4*((B^2*b^2*c^2 - B^2*a^2*d^2)*i*n^2 + 2*(A*B*b^2*c^2 - A*B*a^2*d^2)*i*n + 2*(2*(B^2*b^2*c*d - B^2*a*b*d^2)
*i*x + (B^2*b^2*c^2 - B^2*a^2*d^2)*i)*log(e)^2 + 2*(B^2*b^2*d^2*i*n^2*x^2 + 2*B^2*b^2*c*d*i*n^2*x + B^2*b^2*c^
2*i*n^2)*log((b*x + a)/(d*x + c))^2 + 2*(A^2*b^2*c^2 - A^2*a^2*d^2)*i + 2*((B^2*b^2*c*d - B^2*a*b*d^2)*i*n^2 +
2*(A*B*b^2*c*d - A*B*a*b*d^2)*i*n + 2*(A^2*b^2*c*d - A^2*a*b*d^2)*i)*x + 2*((B^2*b^2*c^2 - B^2*a^2*d^2)*i*n +
2*(A*B*b^2*c^2 - A*B*a^2*d^2)*i + 2*((B^2*b^2*c*d - B^2*a*b*d^2)*i*n + 2*(A*B*b^2*c*d - A*B*a*b*d^2)*i)*x + 2
*(B^2*b^2*d^2*i*n*x^2 + 2*B^2*b^2*c*d*i*n*x + B^2*b^2*c^2*i*n)*log((b*x + a)/(d*x + c)))*log(e) + 2*(B^2*b^2*c
^2*i*n^2 + 2*A*B*b^2*c^2*i*n + (B^2*b^2*d^2*i*n^2 + 2*A*B*b^2*d^2*i*n)*x^2 + 2*(B^2*b^2*c*d*i*n^2 + 2*A*B*b^2*
c*d*i*n)*x)*log((b*x + a)/(d*x + c)))/((b^5*c - a*b^4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a^2*b^3*c
- a^3*b^2*d)*g^3)

Sympy [F]

\[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx=\frac {i \left (\int \frac {A^{2} c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {A^{2} d x}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {B^{2} c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 A B c \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {B^{2} d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx + \int \frac {2 A B d x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\, dx\right )}{g^{3}} \]

[In]

integrate((d*i*x+c*i)*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**3,x)

[Out]

i*(Integral(A**2*c/(a**3 + 3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3), x) + Integral(A**2*d*x/(a**3 + 3*a**2*b*x
+ 3*a*b**2*x**2 + b**3*x**3), x) + Integral(B**2*c*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(a**3 + 3*a**2*b
*x + 3*a*b**2*x**2 + b**3*x**3), x) + Integral(2*A*B*c*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a**3 + 3*a**2*
b*x + 3*a*b**2*x**2 + b**3*x**3), x) + Integral(B**2*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)**2/(a**3 + 3*
a**2*b*x + 3*a*b**2*x**2 + b**3*x**3), x) + Integral(2*A*B*d*x*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(a**3 +
3*a**2*b*x + 3*a*b**2*x**2 + b**3*x**3), x))/g**3

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2017 vs. \(2 (145) = 290\).

Time = 0.31 (sec) , antiderivative size = 2017, normalized size of antiderivative = 13.36 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx=\text {Too large to display} \]

[In]

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3,x, algorithm="maxima")

[Out]

-1/2*A*B*d*i*n*((3*a*b*c - a^2*d + 2*(2*b^2*c - a*b*d)*x)/((b^5*c - a*b^4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)
*g^3*x + (a^2*b^3*c - a^3*b^2*d)*g^3) + 2*(2*b*c*d - a*d^2)*log(b*x + a)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2
)*g^3) - 2*(2*b*c*d - a*d^2)*log(d*x + c)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3)) + 1/2*A*B*c*i*n*((2*b*d
*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3*x + (a^2*b^2*c - a^3*b*d)*g^3) + 2*
d^2*log(b*x + a)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c)/((b^3*c^2 - 2*a*b^2*c*d + a^2*
b*d^2)*g^3)) - 1/2*(2*b*x + a)*B^2*d*i*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2/(b^4*g^3*x^2 + 2*a*b^3*g^3*x +
a^2*b^2*g^3) + 1/4*(2*n*((2*b*d*x - b*c + 3*a*d)/((b^4*c - a*b^3*d)*g^3*x^2 + 2*(a*b^3*c - a^2*b^2*d)*g^3*x +
(a^2*b^2*c - a^3*b*d)*g^3) + 2*d^2*log(b*x + a)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3) - 2*d^2*log(d*x + c
)/((b^3*c^2 - 2*a*b^2*c*d + a^2*b*d^2)*g^3))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - (b^2*c^2 - 8*a*b*c*d + 7
*a^2*d^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*lo
g(d*x + c)^2 - 6*(b^2*c*d - a*b*d^2)*x - 6*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a) + 2*(3*b^2*d^2*x
^2 + 6*a*b*d^2*x + 3*a^2*d^2 - 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*log(b*x + a))*log(d*x + c))*n^2/(a^2*b^
3*c^2*g^3 - 2*a^3*b^2*c*d*g^3 + a^4*b*d^2*g^3 + (b^5*c^2*g^3 - 2*a*b^4*c*d*g^3 + a^2*b^3*d^2*g^3)*x^2 + 2*(a*b
^4*c^2*g^3 - 2*a^2*b^3*c*d*g^3 + a^3*b^2*d^2*g^3)*x))*B^2*c*i - 1/4*(2*n*((3*a*b*c - a^2*d + 2*(2*b^2*c - a*b*
d)*x)/((b^5*c - a*b^4*d)*g^3*x^2 + 2*(a*b^4*c - a^2*b^3*d)*g^3*x + (a^2*b^3*c - a^3*b^2*d)*g^3) + 2*(2*b*c*d -
a*d^2)*log(b*x + a)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*g^3) - 2*(2*b*c*d - a*d^2)*log(d*x + c)/((b^4*c^2
- 2*a*b^3*c*d + a^2*b^2*d^2)*g^3))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + (7*a*b^2*c^2 - 8*a^2*b*c*d + a^3*d
^2 - 2*(2*a^2*b*c*d - a^3*d^2 + (2*b^3*c*d - a*b^2*d^2)*x^2 + 2*(2*a*b^2*c*d - a^2*b*d^2)*x)*log(b*x + a)^2 -
2*(2*a^2*b*c*d - a^3*d^2 + (2*b^3*c*d - a*b^2*d^2)*x^2 + 2*(2*a*b^2*c*d - a^2*b*d^2)*x)*log(d*x + c)^2 + 2*(4*
b^3*c^2 - 5*a*b^2*c*d + a^2*b*d^2)*x + 2*(4*a^2*b*c*d - a^3*d^2 + (4*b^3*c*d - a*b^2*d^2)*x^2 + 2*(4*a*b^2*c*d
- a^2*b*d^2)*x)*log(b*x + a) - 2*(4*a^2*b*c*d - a^3*d^2 + (4*b^3*c*d - a*b^2*d^2)*x^2 + 2*(4*a*b^2*c*d - a^2*
b*d^2)*x - 2*(2*a^2*b*c*d - a^3*d^2 + (2*b^3*c*d - a*b^2*d^2)*x^2 + 2*(2*a*b^2*c*d - a^2*b*d^2)*x)*log(b*x + a
))*log(d*x + c))*n^2/(a^2*b^4*c^2*g^3 - 2*a^3*b^3*c*d*g^3 + a^4*b^2*d^2*g^3 + (b^6*c^2*g^3 - 2*a*b^5*c*d*g^3 +
a^2*b^4*d^2*g^3)*x^2 + 2*(a*b^5*c^2*g^3 - 2*a^2*b^4*c*d*g^3 + a^3*b^3*d^2*g^3)*x))*B^2*d*i - (2*b*x + a)*A*B*
d*i*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^4*g^3*x^2 + 2*a*b^3*g^3*x + a^2*b^2*g^3) - 1/2*B^2*c*i*log(e*(b*
x/(d*x + c) + a/(d*x + c))^n)^2/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3) - 1/2*(2*b*x + a)*A^2*d*i/(b^4*g^3*x
^2 + 2*a*b^3*g^3*x + a^2*b^2*g^3) - A*B*c*i*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^3*g^3*x^2 + 2*a*b^2*g^3*
x + a^2*b*g^3) - 1/2*A^2*c*i/(b^3*g^3*x^2 + 2*a*b^2*g^3*x + a^2*b*g^3)

Giac [A] (verification not implemented)

none

Time = 1.70 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.29 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B^{2} i n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (b x + a\right )}^{2} g^{3}} + \frac {2 \, {\left (B^{2} i n^{2} + 2 \, B^{2} i n \log \left (e\right ) + 2 \, A B i n\right )} {\left (d x + c\right )}^{2} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{2} g^{3}} + \frac {{\left (B^{2} i n^{2} + 2 \, B^{2} i n \log \left (e\right ) + 2 \, B^{2} i \log \left (e\right )^{2} + 2 \, A B i n + 4 \, A B i \log \left (e\right ) + 2 \, A^{2} i\right )} {\left (d x + c\right )}^{2}}{{\left (b x + a\right )}^{2} g^{3}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

[In]

integrate((d*i*x+c*i)*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^3,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.31 (sec) , antiderivative size = 561, normalized size of antiderivative = 3.72 \[ \int \frac {(c i+d i x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^3} \, dx=-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {\frac {B^2\,c\,i}{2\,b}+\frac {B^2\,d\,i\,x}{b}+\frac {B^2\,a\,d\,i}{2\,b^2}}{a^2\,g^3+2\,a\,b\,g^3\,x+b^2\,g^3\,x^2}-\frac {B^2\,d^2\,i}{2\,b^2\,g^3\,\left (a\,d-b\,c\right )}\right )-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {A\,B\,a\,d\,i+A\,B\,b\,c\,i-B^2\,a\,d\,i\,n+B^2\,b\,c\,i\,n+2\,A\,B\,b\,d\,i\,x}{a^2\,b^2\,g^3+2\,a\,b^3\,g^3\,x+b^4\,g^3\,x^2}+\frac {B^2\,d^2\,i\,\left (\frac {a\,b^2\,g^3\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b^3\,g^3\,n\,x\,\left (a\,d-b\,c\right )}{d}+\frac {b^2\,g^3\,n\,\left (a\,d-b\,c\right )\,\left (2\,a\,d-b\,c\right )}{2\,d^2}\right )}{b^2\,g^3\,\left (a\,d-b\,c\right )\,\left (a^2\,b^2\,g^3+2\,a\,b^3\,g^3\,x+b^4\,g^3\,x^2\right )}\right )-\frac {x\,\left (2\,b\,d\,i\,A^2+2\,b\,d\,i\,A\,B\,n+b\,d\,i\,B^2\,n^2\right )+A^2\,a\,d\,i+A^2\,b\,c\,i+\frac {B^2\,a\,d\,i\,n^2}{2}+\frac {B^2\,b\,c\,i\,n^2}{2}+A\,B\,a\,d\,i\,n+A\,B\,b\,c\,i\,n}{2\,a^2\,b^2\,g^3+4\,a\,b^3\,g^3\,x+2\,b^4\,g^3\,x^2}-\frac {B\,d^2\,i\,n\,\mathrm {atan}\left (\frac {B\,d^2\,i\,n\,\left (2\,A+B\,n\right )\,\left (\frac {c\,b^3\,g^3+a\,d\,b^2\,g^3}{b^2\,g^3}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{\left (a\,d-b\,c\right )\,\left (i\,B^2\,d^2\,n^2+2\,A\,i\,B\,d^2\,n\right )}\right )\,\left (2\,A+B\,n\right )\,1{}\mathrm {i}}{b^2\,g^3\,\left (a\,d-b\,c\right )} \]

[In]

int(((c*i + d*i*x)*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2)/(a*g + b*g*x)^3,x)

[Out]

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