3.3.74 \(\int \frac {a+b \log (c (d+e x)^n)}{x^2 (f+g x^2)^2} \, dx\) [274]

Optimal. Leaf size=560 \[ \frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}-\frac {b e \sqrt {g} n \log (d+e x)}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {b e \sqrt {g} n \log (d+e x)}{4 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{5/2}}+\frac {b e \sqrt {g} n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 f \left (e (-f)^{3/2}+d f \sqrt {g}\right )}+\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{5/2}}+\frac {3 b \sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{5/2}}-\frac {3 b \sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{5/2}} \]

[Out]

b*e*n*ln(x)/d/f^2-b*e*n*ln(e*x+d)/d/f^2+(-a-b*ln(c*(e*x+d)^n))/f^2/x-3/4*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2
)-x*g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))*g^(1/2)/(-f)^(5/2)+3/4*(a+b*ln(c*(e*x+d)^n))*ln(e*((-f)^(1/2)+x*g^(1/2)
)/(e*(-f)^(1/2)-d*g^(1/2)))*g^(1/2)/(-f)^(5/2)+3/4*b*n*polylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))*g^
(1/2)/(-f)^(5/2)-3/4*b*n*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))*g^(1/2)/(-f)^(5/2)-1/4*b*e*n*ln(e
*x+d)*g^(1/2)/f^2/(e*(-f)^(1/2)+d*g^(1/2))+1/4*b*e*n*ln((-f)^(1/2)-x*g^(1/2))*g^(1/2)/f^2/(e*(-f)^(1/2)+d*g^(1
/2))-1/4*b*e*n*ln(e*x+d)*g^(1/2)/f/(e*(-f)^(3/2)+d*f*g^(1/2))+1/4*b*e*n*ln((-f)^(1/2)+x*g^(1/2))*g^(1/2)/f/(e*
(-f)^(3/2)+d*f*g^(1/2))+1/4*(a+b*ln(c*(e*x+d)^n))*g^(1/2)/f^2/((-f)^(1/2)-x*g^(1/2))-1/4*(a+b*ln(c*(e*x+d)^n))
*g^(1/2)/f^2/((-f)^(1/2)+x*g^(1/2))

________________________________________________________________________________________

Rubi [A]
time = 0.65, antiderivative size = 560, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {296, 331, 211, 2463, 2442, 36, 29, 31, 2456, 2441, 2440, 2438} \begin {gather*} \frac {3 b \sqrt {g} n \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{5/2}}-\frac {3 b \sqrt {g} n \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{d \sqrt {g}+e \sqrt {-f}}\right )}{4 (-f)^{5/2}}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac {3 \sqrt {g} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 (-f)^{5/2}}+\frac {3 \sqrt {g} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 (-f)^{5/2}}-\frac {b e \sqrt {g} n \log (d+e x)}{4 f^2 \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e \sqrt {g} n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f^2 \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}-\frac {b e \sqrt {g} n \log (d+e x)}{4 f \left (d f \sqrt {g}+e (-f)^{3/2}\right )}+\frac {b e \sqrt {g} n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 f \left (d f \sqrt {g}+e (-f)^{3/2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*n*Log[x])/(d*f^2) - (b*e*n*Log[d + e*x])/(d*f^2) - (b*e*Sqrt[g]*n*Log[d + e*x])/(4*f^2*(e*Sqrt[-f] + d*Sq
rt[g])) - (b*e*Sqrt[g]*n*Log[d + e*x])/(4*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) - (a + b*Log[c*(d + e*x)^n])/(f^2*x)
+ (Sqrt[g]*(a + b*Log[c*(d + e*x)^n]))/(4*f^2*(Sqrt[-f] - Sqrt[g]*x)) - (Sqrt[g]*(a + b*Log[c*(d + e*x)^n]))/
(4*f^2*(Sqrt[-f] + Sqrt[g]*x)) + (b*e*Sqrt[g]*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*f^2*(e*Sqrt[-f] + d*Sqrt[g])) -
(3*Sqrt[g]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(5/2))
+ (b*e*Sqrt[g]*n*Log[Sqrt[-f] + Sqrt[g]*x])/(4*f*(e*(-f)^(3/2) + d*f*Sqrt[g])) + (3*Sqrt[g]*(a + b*Log[c*(d +
e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(4*(-f)^(5/2)) + (3*b*Sqrt[g]*n*PolyLog[2,
-((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(4*(-f)^(5/2)) - (3*b*Sqrt[g]*n*PolyLog[2, (Sqrt[g]*(d + e*
x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*(-f)^(5/2))

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 296

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-(c*x)^(m + 1))*((a + b*x^n)^(p + 1)/
(a*c*n*(p + 1))), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; Free
Q[{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 2456

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 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]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2 \left (f+g x^2\right )^2} \, dx &=\int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f \left (f+g x^2\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (f+g x^2\right )}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{x^2} \, dx}{f^2}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x^2} \, dx}{f^2}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (f+g x^2\right )^2} \, dx}{f}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}-\frac {g \int \left (\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f^2}-\frac {g \int \left (-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f \left (\sqrt {-f} \sqrt {g}-g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f \left (\sqrt {-f} \sqrt {g}+g x\right )^2}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f \left (-f g-g^2 x^2\right )}\right ) \, dx}{f}+\frac {(b e n) \int \frac {1}{x (d+e x)} \, dx}{f^2}\\ &=-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 (-f)^{5/2}}+\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 (-f)^{5/2}}+\frac {g^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (\sqrt {-f} \sqrt {g}-g x\right )^2} \, dx}{4 f^2}+\frac {g^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{\left (\sqrt {-f} \sqrt {g}+g x\right )^2} \, dx}{4 f^2}+\frac {g^2 \int \frac {a+b \log \left (c (d+e x)^n\right )}{-f g-g^2 x^2} \, dx}{2 f^2}+\frac {(b e n) \int \frac {1}{x} \, dx}{d f^2}-\frac {\left (b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{d f^2}\\ &=\frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {g^2 \int \left (-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{2 f^2}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{2 (-f)^{5/2}}-\frac {\left (b e \sqrt {g} n\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{2 (-f)^{5/2}}-\frac {(b e g n) \int \frac {1}{(d+e x) \left (\sqrt {-f} \sqrt {g}-g x\right )} \, dx}{4 f^2}+\frac {(b e g n) \int \frac {1}{(d+e x) \left (\sqrt {-f} \sqrt {g}+g x\right )} \, dx}{4 f^2}\\ &=\frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x} \, dx}{4 (-f)^{5/2}}+\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x} \, dx}{4 (-f)^{5/2}}-\frac {\left (b \sqrt {g} n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 (-f)^{5/2}}+\frac {\left (b \sqrt {g} n\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{2 (-f)^{5/2}}+\frac {\left (b e^2 \sqrt {g} n\right ) \int \frac {1}{d+e x} \, dx}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {\left (b e^2 \sqrt {g} n\right ) \int \frac {1}{d+e x} \, dx}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {\left (b e g^{3/2} n\right ) \int \frac {1}{\sqrt {-f} \sqrt {g}+g x} \, dx}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {\left (b e g^{3/2} n\right ) \int \frac {1}{\sqrt {-f} \sqrt {g}-g x} \, dx}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}\\ &=\frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}+\frac {b e \sqrt {g} n \log (d+e x)}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b e \sqrt {g} n \log (d+e x)}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{5/2}}-\frac {b e \sqrt {g} n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{5/2}}+\frac {b \sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {b \sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}+\frac {\left (b e \sqrt {g} n\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{d+e x} \, dx}{4 (-f)^{5/2}}-\frac {\left (b e \sqrt {g} n\right ) \int \frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{d+e x} \, dx}{4 (-f)^{5/2}}\\ &=\frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}+\frac {b e \sqrt {g} n \log (d+e x)}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b e \sqrt {g} n \log (d+e x)}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{5/2}}-\frac {b e \sqrt {g} n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{5/2}}+\frac {b \sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {b \sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 (-f)^{5/2}}-\frac {\left (b \sqrt {g} n\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {g} x}{e \sqrt {-f}-d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{4 (-f)^{5/2}}+\frac {\left (b \sqrt {g} n\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {g} x}{e \sqrt {-f}+d \sqrt {g}}\right )}{x} \, dx,x,d+e x\right )}{4 (-f)^{5/2}}\\ &=\frac {b e n \log (x)}{d f^2}-\frac {b e n \log (d+e x)}{d f^2}+\frac {b e \sqrt {g} n \log (d+e x)}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b e \sqrt {g} n \log (d+e x)}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{f^2 x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e \sqrt {g} n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{5/2}}-\frac {b e \sqrt {g} n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{5/2}}+\frac {3 b \sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 (-f)^{5/2}}-\frac {3 b \sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 (-f)^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 0.53, size = 476, normalized size = 0.85 \begin {gather*} \frac {1}{4} \left (\frac {4 b e n (\log (x)-\log (d+e x))}{d f^2}-\frac {4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 x}+\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {\sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right )}{f^2 \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e \sqrt {g} n \left (\log (d+e x)-\log \left (\sqrt {-f}-\sqrt {g} x\right )\right )}{f^2 \left (e \sqrt {-f}+d \sqrt {g}\right )}-\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}+\frac {b e \sqrt {g} n \left (\log (d+e x)-\log \left (\sqrt {-f}+\sqrt {g} x\right )\right )}{f^2 \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {3 \sqrt {g} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}+\frac {3 b \sqrt {g} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{5/2}}-\frac {3 b \sqrt {g} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{(-f)^{5/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((4*b*e*n*(Log[x] - Log[d + e*x]))/(d*f^2) - (4*(a + b*Log[c*(d + e*x)^n]))/(f^2*x) + (Sqrt[g]*(a + b*Log[c*(d
+ e*x)^n]))/(f^2*(Sqrt[-f] - Sqrt[g]*x)) - (Sqrt[g]*(a + b*Log[c*(d + e*x)^n]))/(f^2*(Sqrt[-f] + Sqrt[g]*x))
- (b*e*Sqrt[g]*n*(Log[d + e*x] - Log[Sqrt[-f] - Sqrt[g]*x]))/(f^2*(e*Sqrt[-f] + d*Sqrt[g])) - (3*Sqrt[g]*(a +
b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(-f)^(5/2) + (b*e*Sqrt[g]*n*(L
og[d + e*x] - Log[Sqrt[-f] + Sqrt[g]*x]))/(f^2*(e*Sqrt[-f] - d*Sqrt[g])) + (3*Sqrt[g]*(a + b*Log[c*(d + e*x)^n
])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(-f)^(5/2) + (3*b*Sqrt[g]*n*PolyLog[2, -((Sqrt[g]
*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(-f)^(5/2) - (3*b*Sqrt[g]*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f]
+ d*Sqrt[g])])/(-f)^(5/2))/4

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.63, size = 2032, normalized size = 3.63

method	result	size
risch	\(\text {Expression too large to display}\)	\(2032\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*b*e^4*n*g*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*
g)^(1/2)-d*g))-1/2*b*e^4*n/f*g*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)*x-1/2*b*e^3*n/f*g*ln(e*x+d)/(d^2*g+e^
2*f)/(e^2*g*x^2+e^2*f)*d-b*ln((e*x+d)^n)/f^2/x-3/4*b*n/f^2*g/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)
/(e*(-f*g)^(1/2)+d*g))-1/2*b*ln(c)/f^2*g*x/(g*x^2+f)-3/2*b*ln(c)/f^2*g/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-b*l
n(c)/f^2/x-a/f^2/x+1/4*b*e*n/f^2*g/(d^2*g+e^2*f)*d*ln(g*(e*x+d)^2-2*d*g*(e*x+d)+d^2*g+f*e^2)+1/2*b/f^2*g/(e^2*
g*x^2+e^2*f)*x*e^2*n*ln(e*x+d)+1/2*b*n/f^2*g*ln(e*x+d)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g
)^(1/2)-d*g))-1/2*b*n/f^2*g*ln(e*x+d)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))+3/2
*b/f^2*g/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))*n*ln(e*x+d)-1/2*a/f^2*g*x/(g*x^2+f)-3/2*a/f
^2*g/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/f^2*g*x/(g*x^2+f)+1/2*b*e^2*n/f*g/(d
^2*g+e^2*f)/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))+1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/f^2/x-1
/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/f^2/x-1/2*b*n/f^2*g^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2
*f)*d*x^2*e^3-1/2*b*n/f^2*g^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)*d^2*x*e^2+3/4*b*n/f^2*g/(-f*g)^(1/2)*d
ilog((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+b*e*n/f^2/d*ln(e*x)-3/2*b/f^2*g/(f*g)^(1/2)*arctan(1
/2*(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))*ln((e*x+d)^n)-1/2*b/f^2*g/(e^2*g*x^2+e^2*f)*x*e^2*ln((e*x+d)^n)-1/2*I*b*
Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/f^2/x-1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/f^2*g*x/(g*x^2+f)-
1/4*b*e^4*n*g*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*
g)^(1/2)+d*g))+1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/f^2/x+3/4*I*b*Pi*csgn(I*c*(e*x+d)^n)
^3/f^2*g/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-3/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/f^2*g/(f*g)^(1
/2)*arctan(x*g/(f*g)^(1/2))-3/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/f^2*g/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2
))+1/4*b*e^2*n/f^2*g^3*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g
)/(e*(-f*g)^(1/2)-d*g))*d^2*x^2-1/4*b*e^2*n/f^2*g^3*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln(
(e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))*d^2*x^2-b*e*n*ln(e*x+d)/d/f^2+3/4*I*b*Pi*csgn(I*c)*csgn(I
*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/f^2*g/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/4*b*e^4*n/f*g^2*ln(e*x+d)/(d^2*g+e
^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))*x^2+1/4*b*e^2*n/f
*g^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-
d*g))*d^2-1/4*b*e^4*n/f*g^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d
)+d*g)/(e*(-f*g)^(1/2)+d*g))*x^2-1/4*b*e^2*n/f*g^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((
e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))*d^2-1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/f^2*g*x/(g*
x^2+f)+1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/f^2*g*x/(g*x^2+f)

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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(c*(e*x+d)^n))/x^2/(g*x^2+f)^2,x, algorithm="maxima")

[Out]

-1/2*a*((3*g*x^2 + 2*f)/(f^2*g*x^3 + f^3*x) + 3*g*arctan(g*x/sqrt(f*g))/(sqrt(f*g)*f^2)) + b*integrate((log((x
*e + d)^n) + log(c))/(g^2*x^6 + 2*f*g*x^4 + f^2*x^2), 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(c*(e*x+d)^n))/x^2/(g*x^2+f)^2,x, algorithm="fricas")

[Out]

integral((b*log((x*e + d)^n*c) + a)/(g^2*x^6 + 2*f*g*x^4 + f^2*x^2), 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(c*(e*x+d)**n))/x**2/(g*x**2+f)**2,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(c*(e*x+d)^n))/x^2/(g*x^2+f)^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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