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

Optimal. Leaf size=430 \[ -\frac{b^2 e n^2 \left (d \sqrt{g}+e \sqrt{-f}\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \sqrt{-f} g \left (d^2 g+e^2 f\right )}-\frac{b^2 e n^2 \left (d \sqrt{-f} \sqrt{g}+e f\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 f g \left (d^2 g+e^2 f\right )}-\frac{b e n \left (d \sqrt{-f} \sqrt{g}+e f\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (d^2 g+e^2 f\right )}-\frac{b e n \left (e f-d \sqrt{-f} \sqrt{g}\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (d^2 g+e^2 f\right )}+\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (d^2 g+e^2 f\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.546976, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2413, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{b^2 e n^2 \left (d \sqrt{g}+e \sqrt{-f}\right ) \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 \sqrt{-f} g \left (d^2 g+e^2 f\right )}-\frac{b^2 e n^2 \left (d \sqrt{-f} \sqrt{g}+e f\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 f g \left (d^2 g+e^2 f\right )}-\frac{b e n \left (d \sqrt{-f} \sqrt{g}+e f\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (d^2 g+e^2 f\right )}-\frac{b e n \left (e f-d \sqrt{-f} \sqrt{g}\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 f g \left (d^2 g+e^2 f\right )}+\frac{e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (d^2 g+e^2 f\right )}-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2413

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2394

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 2393

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 2391

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]

Rubi steps

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

Mathematica [C]  time = 0.564844, size = 590, normalized size = 1.37 \[ \frac{\frac{i b^2 n^2 \left (\frac{2 e \left (\sqrt{g} x+i \sqrt{f}\right ) \text{PolyLog}\left (2,\frac{i \sqrt{g} (d+e x)}{e \sqrt{f}+i d \sqrt{g}}\right )+2 e \left (\sqrt{g} x+i \sqrt{f}\right ) \log (d+e x) \log \left (\frac{e \left (\sqrt{f}-i \sqrt{g} x\right )}{e \sqrt{f}+i d \sqrt{g}}\right )-\sqrt{g} (d+e x) \log ^2(d+e x)}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (e \sqrt{f}+i d \sqrt{g}\right )}+\frac{2 i e \left (\sqrt{f}+i \sqrt{g} x\right ) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )+\log (d+e x) \left (\sqrt{g} (d+e x) \log (d+e x)+2 i e \left (\sqrt{f}+i \sqrt{g} x\right ) \log \left (\frac{e \left (\sqrt{f}+i \sqrt{g} x\right )}{e \sqrt{f}-i d \sqrt{g}}\right )\right )}{\left (\sqrt{f}+i \sqrt{g} x\right ) \left (e \sqrt{f}-i d \sqrt{g}\right )}\right )}{\sqrt{f}}+\frac{2 b n \left (2 \sqrt{f} g \left (d^2-e^2 x^2\right ) \log (d+e x)+e \left (f+g x^2\right ) \left (\left (e \sqrt{f}+i d \sqrt{g}\right ) \log \left (-\sqrt{g} x+i \sqrt{f}\right )+\left (e \sqrt{f}-i d \sqrt{g}\right ) \log \left (\sqrt{g} x+i \sqrt{f}\right )\right )\right ) \left (-a-b \log \left (c (d+e x)^n\right )+b n \log (d+e x)\right )}{\sqrt{f} \left (f+g x^2\right ) \left (d^2 g+e^2 f\right )}-\frac{2 \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{f+g x^2}}{4 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2) + (2*b*n*(-a + b*n*Log[d + e*x] - b*Log[c*(d
+ e*x)^n])*(2*Sqrt[f]*g*(d^2 - e^2*x^2)*Log[d + e*x] + e*(f + g*x^2)*((e*Sqrt[f] + I*d*Sqrt[g])*Log[I*Sqrt[f]
- Sqrt[g]*x] + (e*Sqrt[f] - I*d*Sqrt[g])*Log[I*Sqrt[f] + Sqrt[g]*x])))/(Sqrt[f]*(e^2*f + d^2*g)*(f + g*x^2))
+ (I*b^2*n^2*((-(Sqrt[g]*(d + e*x)*Log[d + e*x]^2) + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*Log[d + e*x]*Log[(e*(Sqrt[f]
- I*Sqrt[g]*x))/(e*Sqrt[f] + I*d*Sqrt[g])] + 2*e*(I*Sqrt[f] + Sqrt[g]*x)*PolyLog[2, (I*Sqrt[g]*(d + e*x))/(e*S
qrt[f] + I*d*Sqrt[g])])/((e*Sqrt[f] + I*d*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x)) + (Log[d + e*x]*(Sqrt[g]*(d + e*x)
*Log[d + e*x] + (2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*Log[(e*(Sqrt[f] + I*Sqrt[g]*x))/(e*Sqrt[f] - I*d*Sqrt[g])]) +
(2*I)*e*(Sqrt[f] + I*Sqrt[g]*x)*PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])/((e*Sqrt[f] - I*d*S
qrt[g])*(Sqrt[f] + I*Sqrt[g]*x))))/Sqrt[f])/(4*g)

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Maple [C]  time = 0.64, size = 2134, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*I/g*n*e^2/(d^2*g+e^2*f)*ln(g*x^2+f)*b^2*Pi*csgn(I*c*(e*x+d)^n)^3+1/2*I/g/(g*x^2+f)*ln((e*x+d)^n)*b^2*Pi*cs
gn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/2*I/g*n*e^2/(d^2*g+e^2*f)*ln(e*x+d)*b^2*Pi*csgn(I*c*(e*x+d)^n)
^3-1/8*(-I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+I*b*Pi*
csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-I*b*Pi*csgn(I*c*(e*x+d)^n)^3+2*b*ln(c)+2*a)^2/g/(g*x^2+f)-1/2*I/g/(g*x
^2+f)*ln((e*x+d)^n)*b^2*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2+b^2*n*e/(d^2*g+e^2*f)*d/(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)+b*n*e/(d^2*g+e^2*f)*d/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))
*a+n*e/(d^2*g+e^2*f)*d/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*b^2*ln(c)+b^2/g*n*e^2/(d^2*g+e^2*f)*ln(e*x+d)*ln((e
*x+d)^n)-1/2*I/g/(g*x^2+f)*ln((e*x+d)^n)*b^2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-b^2*n^2*e/(d^2*g+e^2*f)*d/(f*g
)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))*ln(e*x+d)+1/2*b^2*n^2*e/(d^2*g+e^2*f)*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))*d-1/2*b^2*n^2*e/(d^2*g+e^2*f)*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))*d-1/4*I/g*n*e^2/(d^2*g+e^2*f)*ln(g*x^2+f)*b^2*Pi*c
sgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*b^2/g/(g*x^2+f)*ln((e*x+d)^n)^2-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*ln(
e*x+d)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*ln(e*x+d)*ln((e
*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/2*b^2*n^2*e/(d^2*g+e^2*f)/(-f*g)^(1/2)*dilog((e*(-f*g)^(1
/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))*d-1/2*b^2*n^2*e/(d^2*g+e^2*f)/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)+g*(e
*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))*d+1/2*I/g*n*e^2/(d^2*g+e^2*f)*ln(e*x+d)*b^2*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*
x+d)^n)^2-1/4*I/g*n*e^2/(d^2*g+e^2*f)*ln(g*x^2+f)*b^2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-b/g/(g*x^2+f)*ln((e*x
+d)^n)*a-1/g/(g*x^2+f)*ln((e*x+d)^n)*b^2*ln(c)-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*ln(e*x+d)^2-1/2*b^2/g*n^2*e^2/(
d^2*g+e^2*f)*dilog((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))-1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*dilog(
(e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/2*b^2/g*n^2*e^2/(d^2*g+e^2*f)*ln(g*(e*x+d)^2-2*d*g*(e*x
+d)+d^2*g+f*e^2)*ln(e*x+d)+b/g*n*e^2/(d^2*g+e^2*f)*ln(e*x+d)*a-1/2*b/g*n*e^2/(d^2*g+e^2*f)*ln(g*x^2+f)*a+1/g*n
*e^2/(d^2*g+e^2*f)*ln(e*x+d)*b^2*ln(c)-1/2*I*n*e/(d^2*g+e^2*f)*d/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*b^2*Pi*cs
gn(I*c*(e*x+d)^n)^3+1/2*I/g*n*e^2/(d^2*g+e^2*f)*ln(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2+1/2*I*n*e/(d^
2*g+e^2*f)*d/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*b^2*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2-1/2*I*n*e/(d^2*g+e^2*f
)*d/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*b^2*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/2*I/g*n*e^2/(
d^2*g+e^2*f)*ln(e*x+d)*b^2*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/4*I/g*n*e^2/(d^2*g+e^2*f)*ln(g
*x^2+f)*b^2*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)-1/2/g*n*e^2/(d^2*g+e^2*f)*ln(g*x^2+f)*b^2*ln(c)
+1/2*I/g/(g*x^2+f)*ln((e*x+d)^n)*b^2*Pi*csgn(I*c*(e*x+d)^n)^3-1/2*b^2/g*n*e^2/(d^2*g+e^2*f)*ln(g*(e*x+d)^2-2*d
*g*(e*x+d)+d^2*g+f*e^2)*ln((e*x+d)^n)+1/2*I*n*e/(d^2*g+e^2*f)*d/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))*b^2*Pi*csg
n(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b x \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} x}{g^{2} x^{4} + 2 \, f g x^{2} + f^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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