\(\int \frac {x^3 \log (e (\frac {a+b x}{c+d x})^n)}{f-g x^2} \, dx\) [76]

Optimal result

Integrand size = 32, antiderivative size = 560 \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=-\frac {a n x}{2 b g}+\frac {c n x}{2 d g}+\frac {a^2 n \log (a+b x)}{2 b^2 g}-\frac {n x^2 \log (a+b x)}{2 g}-\frac {c^2 n \log (c+d x)}{2 d^2 g}+\frac {n x^2 \log (c+d x)}{2 g}+\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 g}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^2}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^2}+\frac {f \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g^2}-\frac {f n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^2}-\frac {f n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^2}+\frac {f n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^2}+\frac {f n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {2593, 272, 45, 2463, 2442, 266, 2441, 2440, 2438} \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\frac {a^2 n \log (a+b x)}{2 b^2 g}+\frac {f \log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g^2}+\frac {x^2 \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}-\frac {f n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^2}-\frac {f n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{\sqrt {g} a+b \sqrt {f}}\right )}{2 g^2}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g^2}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^2}-\frac {n x^2 \log (a+b x)}{2 g}-\frac {a n x}{2 b g}-\frac {c^2 n \log (c+d x)}{2 d^2 g}+\frac {f n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^2}+\frac {f n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{\sqrt {g} c+d \sqrt {f}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^2}+\frac {n x^2 \log (c+d x)}{2 g}+\frac {c n x}{2 d g} \]

[In]

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

[Out]

-1/2*(a*n*x)/(b*g) + (c*n*x)/(2*d*g) + (a^2*n*Log[a + b*x])/(2*b^2*g) - (n*x^2*Log[a + b*x])/(2*g) - (c^2*n*Lo
g[c + d*x])/(2*d^2*g) + (n*x^2*Log[c + d*x])/(2*g) + (x^2*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n
*Log[c + d*x]))/(2*g) - (f*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^2) + (f
*n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^2) - (f*n*Log[a + b*x]*Log[(b*(Sq
rt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*g^2) + (f*n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqr
t[f] - c*Sqrt[g])])/(2*g^2) + (f*(n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f - g*
x^2])/(2*g^2) - (f*n*PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*g^2) - (f*n*PolyLog[2, (Sq
rt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])])/(2*g^2) + (f*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt
[g]))])/(2*g^2) + (f*n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^2)

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 266

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2438

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

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

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

Rule 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 2463

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2593

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

Rubi steps \begin{align*} \text {integral}& = n \int \frac {x^3 \log (a+b x)}{f-g x^2} \, dx-n \int \frac {x^3 \log (c+d x)}{f-g x^2} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac {x^3}{f-g x^2} \, dx \\ & = n \int \left (-\frac {x \log (a+b x)}{g}+\frac {f x \log (a+b x)}{g \left (f-g x^2\right )}\right ) \, dx-n \int \left (-\frac {x \log (c+d x)}{g}+\frac {f x \log (c+d x)}{g \left (f-g x^2\right )}\right ) \, dx-\frac {1}{2} \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \text {Subst}\left (\int \frac {x}{f-g x} \, dx,x,x^2\right ) \\ & = -\frac {n \int x \log (a+b x) \, dx}{g}+\frac {n \int x \log (c+d x) \, dx}{g}+\frac {(f n) \int \frac {x \log (a+b x)}{f-g x^2} \, dx}{g}-\frac {(f n) \int \frac {x \log (c+d x)}{f-g x^2} \, dx}{g}-\frac {1}{2} \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \text {Subst}\left (\int \left (-\frac {1}{g}-\frac {f}{g (-f+g x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {n x^2 \log (a+b x)}{2 g}+\frac {n x^2 \log (c+d x)}{2 g}+\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 g}+\frac {f \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g^2}+\frac {(b n) \int \frac {x^2}{a+b x} \, dx}{2 g}-\frac {(d n) \int \frac {x^2}{c+d x} \, dx}{2 g}+\frac {(f n) \int \left (\frac {\log (a+b x)}{2 \sqrt {g} \left (\sqrt {f}-\sqrt {g} x\right )}-\frac {\log (a+b x)}{2 \sqrt {g} \left (\sqrt {f}+\sqrt {g} x\right )}\right ) \, dx}{g}-\frac {(f n) \int \left (\frac {\log (c+d x)}{2 \sqrt {g} \left (\sqrt {f}-\sqrt {g} x\right )}-\frac {\log (c+d x)}{2 \sqrt {g} \left (\sqrt {f}+\sqrt {g} x\right )}\right ) \, dx}{g} \\ & = -\frac {n x^2 \log (a+b x)}{2 g}+\frac {n x^2 \log (c+d x)}{2 g}+\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 g}+\frac {f \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g^2}+\frac {(f n) \int \frac {\log (a+b x)}{\sqrt {f}-\sqrt {g} x} \, dx}{2 g^{3/2}}-\frac {(f n) \int \frac {\log (a+b x)}{\sqrt {f}+\sqrt {g} x} \, dx}{2 g^{3/2}}-\frac {(f n) \int \frac {\log (c+d x)}{\sqrt {f}-\sqrt {g} x} \, dx}{2 g^{3/2}}+\frac {(f n) \int \frac {\log (c+d x)}{\sqrt {f}+\sqrt {g} x} \, dx}{2 g^{3/2}}+\frac {(b n) \int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx}{2 g}-\frac {(d n) \int \left (-\frac {c}{d^2}+\frac {x}{d}+\frac {c^2}{d^2 (c+d x)}\right ) \, dx}{2 g} \\ & = -\frac {a n x}{2 b g}+\frac {c n x}{2 d g}+\frac {a^2 n \log (a+b x)}{2 b^2 g}-\frac {n x^2 \log (a+b x)}{2 g}-\frac {c^2 n \log (c+d x)}{2 d^2 g}+\frac {n x^2 \log (c+d x)}{2 g}+\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 g}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^2}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^2}+\frac {f \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g^2}+\frac {(b f n) \int \frac {\log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{a+b x} \, dx}{2 g^2}+\frac {(b f n) \int \frac {\log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{a+b x} \, dx}{2 g^2}-\frac {(d f n) \int \frac {\log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{c+d x} \, dx}{2 g^2}-\frac {(d f n) \int \frac {\log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{c+d x} \, dx}{2 g^2} \\ & = -\frac {a n x}{2 b g}+\frac {c n x}{2 d g}+\frac {a^2 n \log (a+b x)}{2 b^2 g}-\frac {n x^2 \log (a+b x)}{2 g}-\frac {c^2 n \log (c+d x)}{2 d^2 g}+\frac {n x^2 \log (c+d x)}{2 g}+\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 g}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^2}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^2}+\frac {f \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g^2}+\frac {(f n) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{b \sqrt {f}-a \sqrt {g}}\right )}{x} \, dx,x,a+b x\right )}{2 g^2}+\frac {(f n) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{b \sqrt {f}+a \sqrt {g}}\right )}{x} \, dx,x,a+b x\right )}{2 g^2}-\frac {(f n) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{d \sqrt {f}-c \sqrt {g}}\right )}{x} \, dx,x,c+d x\right )}{2 g^2}-\frac {(f n) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{d \sqrt {f}+c \sqrt {g}}\right )}{x} \, dx,x,c+d x\right )}{2 g^2} \\ & = -\frac {a n x}{2 b g}+\frac {c n x}{2 d g}+\frac {a^2 n \log (a+b x)}{2 b^2 g}-\frac {n x^2 \log (a+b x)}{2 g}-\frac {c^2 n \log (c+d x)}{2 d^2 g}+\frac {n x^2 \log (c+d x)}{2 g}+\frac {x^2 \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{2 g}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^2}-\frac {f n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^2}+\frac {f n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^2}+\frac {f \left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g^2}-\frac {f n \text {Li}_2\left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g^2}-\frac {f n \text {Li}_2\left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g^2}+\frac {f n \text {Li}_2\left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g^2}+\frac {f n \text {Li}_2\left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 461, normalized size of antiderivative = 0.82 \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\frac {-g x^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\frac {g n \left (a^2 d^2 \log (a+b x)-b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )}{b^2 d^2}-f \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )-f \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+f n \left (\left (\log \left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )-\log \left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )\right )+f n \left (\left (\log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )-\log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )-\operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )\right )}{2 g^2} \]

[In]

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

[Out]

(-(g*x^2*Log[e*((a + b*x)/(c + d*x))^n]) + (g*n*(a^2*d^2*Log[a + b*x] - b*(d*(-(b*c) + a*d)*x + b*c^2*Log[c +
d*x])))/(b^2*d^2) - f*Log[e*((a + b*x)/(c + d*x))^n]*Log[Sqrt[f] - Sqrt[g]*x] - f*Log[e*((a + b*x)/(c + d*x))^
n]*Log[Sqrt[f] + Sqrt[g]*x] + f*n*((Log[(Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])] - Log[(Sqrt[g]*(c + d*x))
/(d*Sqrt[f] + c*Sqrt[g])])*Log[Sqrt[f] - Sqrt[g]*x] + PolyLog[2, (b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt
[g])] - PolyLog[2, (d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])]) + f*n*((Log[-((Sqrt[g]*(a + b*x))/(b*Sq
rt[f] - a*Sqrt[g]))] - Log[-((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])*Log[Sqrt[f] + Sqrt[g]*x] + PolyLog
[2, (b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])] - PolyLog[2, (d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*S
qrt[g])]))/(2*g^2)

Maple [A] (verified)

Time = 1.52 (sec) , antiderivative size = 538, normalized size of antiderivative = 0.96

method	result	size
parts	\(-\frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) x^{2}}{2 g}-\frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) f \ln \left (-g \,x^{2}+f \right )}{2 g^{2}}-\frac {n \left (\frac {\left (a d -c b \right ) \left (\frac {x}{b d}+\frac {c^{2} \ln \left (d x +c \right )}{d^{2} \left (a d -c b \right )}-\frac {a^{2} \ln \left (b x +a \right )}{b^{2} \left (a d -c b \right )}\right )}{g}+\frac {f \left (a d -c b \right ) \left (\frac {\left (\frac {\ln \left (d x +c \right ) \ln \left (-g \,x^{2}+f \right )}{d}+\frac {2 g \left (-\frac {\ln \left (d x +c \right ) \left (\ln \left (\frac {d \sqrt {f g}-\left (d x +c \right ) g +c g}{d \sqrt {f g}+c g}\right )+\ln \left (\frac {d \sqrt {f g}+\left (d x +c \right ) g -c g}{d \sqrt {f g}-c g}\right )\right )}{2 g}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {f g}-\left (d x +c \right ) g +c g}{d \sqrt {f g}+c g}\right )+\operatorname {dilog}\left (\frac {d \sqrt {f g}+\left (d x +c \right ) g -c g}{d \sqrt {f g}-c g}\right )}{2 g}\right )}{d}\right ) d}{a d -c b}-\frac {\left (\frac {\ln \left (b x +a \right ) \ln \left (-g \,x^{2}+f \right )}{b}+\frac {2 g \left (-\frac {\ln \left (b x +a \right ) \left (\ln \left (\frac {b \sqrt {f g}-g \left (b x +a \right )+a g}{b \sqrt {f g}+a g}\right )+\ln \left (\frac {b \sqrt {f g}+g \left (b x +a \right )-a g}{b \sqrt {f g}-a g}\right )\right )}{2 g}-\frac {\operatorname {dilog}\left (\frac {b \sqrt {f g}-g \left (b x +a \right )+a g}{b \sqrt {f g}+a g}\right )+\operatorname {dilog}\left (\frac {b \sqrt {f g}+g \left (b x +a \right )-a g}{b \sqrt {f g}-a g}\right )}{2 g}\right )}{b}\right ) b}{a d -c b}\right )}{g^{2}}\right )}{2}\)	\(538\)

[In]

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

[Out]

-1/2*ln(e*((b*x+a)/(d*x+c))^n)*x^2/g-1/2*ln(e*((b*x+a)/(d*x+c))^n)*f/g^2*ln(-g*x^2+f)-1/2*n*((a*d-b*c)/g*(x/b/
d+1/d^2*c^2/(a*d-b*c)*ln(d*x+c)-1/b^2*a^2/(a*d-b*c)*ln(b*x+a))+f*(a*d-b*c)/g^2*((ln(d*x+c)/d*ln(-g*x^2+f)+2/d*
g*(-1/2*ln(d*x+c)*(ln((d*(f*g)^(1/2)-(d*x+c)*g+c*g)/(d*(f*g)^(1/2)+c*g))+ln((d*(f*g)^(1/2)+(d*x+c)*g-c*g)/(d*(
f*g)^(1/2)-c*g)))/g-1/2*(dilog((d*(f*g)^(1/2)-(d*x+c)*g+c*g)/(d*(f*g)^(1/2)+c*g))+dilog((d*(f*g)^(1/2)+(d*x+c)
*g-c*g)/(d*(f*g)^(1/2)-c*g)))/g))*d/(a*d-b*c)-(ln(b*x+a)/b*ln(-g*x^2+f)+2/b*g*(-1/2*ln(b*x+a)*(ln((b*(f*g)^(1/
2)-g*(b*x+a)+a*g)/(b*(f*g)^(1/2)+a*g))+ln((b*(f*g)^(1/2)+g*(b*x+a)-a*g)/(b*(f*g)^(1/2)-a*g)))/g-1/2*(dilog((b*
(f*g)^(1/2)-g*(b*x+a)+a*g)/(b*(f*g)^(1/2)+a*g))+dilog((b*(f*g)^(1/2)+g*(b*x+a)-a*g)/(b*(f*g)^(1/2)-a*g)))/g))*
b/(a*d-b*c)))

Fricas [F]

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

[In]

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

[Out]

integral(-x^3*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\text {Timed out} \]

[In]

integrate(x**3*ln(e*((b*x+a)/(d*x+c))**n)/(-g*x**2+f),x)

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

-integrate(x^3*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)

Giac [F]

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

[In]

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

[Out]

integrate(-x^3*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int \frac {x^3\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f-g\,x^2} \,d x \]

[In]

int((x^3*log(e*((a + b*x)/(c + d*x))^n))/(f - g*x^2),x)

[Out]

int((x^3*log(e*((a + b*x)/(c + d*x))^n))/(f - g*x^2), x)