3.4.22 \(\int \frac {x (a+b \log (c (d+e x)^n))^2}{(f+g x^2)^2} \, dx\) [322]

Optimal. Leaf size=430 \[ \frac {e^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (e^2 f+d^2 g\right )}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g \left (f+g x^2\right )}-\frac {b e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )}-\frac {b e \left (e f-d \sqrt {-f} \sqrt {g}\right ) n \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (e \sqrt {-f}+d \sqrt {g}\right ) n^2 \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 \sqrt {-f} g \left (e^2 f+d^2 g\right )}-\frac {b^2 e \left (e f+d \sqrt {-f} \sqrt {g}\right ) n^2 \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 f g \left (e^2 f+d^2 g\right )} \]

[Out]

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

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Rubi [A]
time = 0.41, antiderivative size = 430, normalized size of antiderivative = 1.00, 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 = {2460, 2465, 2437, 2338, 2441, 2440, 2438} \begin {gather*} -\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 )} \end {gather*}

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 2338

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 2437

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 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 2460

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 2465

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]

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 ) \text {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 ) \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 \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 ) \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 \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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.43, size = 590, normalized size = 1.37 \begin {gather*} \frac {-\frac {2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2}+\frac {2 b n \left (-a+b n \log (d+e x)-b \log \left (c (d+e x)^n\right )\right ) \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 (i \sqrt {f}-\sqrt {g} x\right )+\left (e \sqrt {f}-i d \sqrt {g}\right ) \log \left (i \sqrt {f}+\sqrt {g} x\right )\right )\right )}{\sqrt {f} \left (e^2 f+d^2 g\right ) \left (f+g x^2\right )}+\frac {i b^2 n^2 \left (\frac {-\sqrt {g} (d+e x) \log ^2(d+e x)+2 e \left (i \sqrt {f}+\sqrt {g} x\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 )+2 e \left (i \sqrt {f}+\sqrt {g} x\right ) \text {Li}_2\left (\frac {i \sqrt {g} (d+e x)}{e \sqrt {f}+i d \sqrt {g}}\right )}{\left (e \sqrt {f}+i d \sqrt {g}\right ) \left (\sqrt {f}-i \sqrt {g} x\right )}+\frac {\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 )+2 i e \left (\sqrt {f}+i \sqrt {g} x\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )}{\left (e \sqrt {f}-i d \sqrt {g}\right ) \left (\sqrt {f}+i \sqrt {g} x\right )}\right )}{\sqrt {f}}}{4 g} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.68, size = 2134, normalized size = 4.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(2134\)

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,method=_RETURNVERBOSE)

[Out]

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

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

[Out]

-1/2*a*b*n*(e*log(g*x^2 + f)/(d^2*g^2 + f*g*e^2) - 2*e*log(x*e + d)/(d^2*g^2 + f*g*e^2) - 2*d*arctan(g*x/sqrt(
f*g))/((d^2*g + f*e^2)*sqrt(f*g)))*e - 1/2*b^2*(log((x*e + d)^n)^2/(g^2*x^2 + f*g) - 2*integrate((g*x^2*e*log(
c)^2 + d*g*x*log(c)^2 + ((g*n + 2*g*log(c))*x^2*e + 2*d*g*x*log(c) + f*n*e)*log((x*e + d)^n))/(g^3*x^5*e + d*g
^3*x^4 + 2*f*g^2*x^3*e + 2*d*f*g^2*x^2 + f^2*g*x*e + d*f^2*g), x)) - a*b*log((x*e + d)^n*c)/(g^2*x^2 + f*g) -
1/2*a^2/(g^2*x^2 + f*g)

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

[Out]

integral((b^2*x*log((x*e + d)^n*c)^2 + 2*a*b*x*log((x*e + 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)] 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(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 = 0.00, 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(x*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f)^2,x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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