3.28 \(\int x^2 (a+b \log (c x^n)) \log (d (\frac{1}{d}+f x^2)) \, dx\)

Optimal. Leaf size=241 \[ \frac{i b n \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}-\frac{i b n \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac{1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac{2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{9 d^{3/2} f^{3/2}}-\frac{1}{9} b n x^3 \log \left (d f x^2+1\right )-\frac{8 b n x}{9 d f}+\frac{4}{27} b n x^3 \]

[Out]

(-8*b*n*x)/(9*d*f) + (4*b*n*x^3)/27 + (2*b*n*ArcTan[Sqrt[d]*Sqrt[f]*x])/(9*d^(3/2)*f^(3/2)) + (2*x*(a + b*Log[
c*x^n]))/(3*d*f) - (2*x^3*(a + b*Log[c*x^n]))/9 - (2*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(3*d^(3/2)*
f^(3/2)) - (b*n*x^3*Log[1 + d*f*x^2])/9 + (x^3*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/3 + ((I/3)*b*n*PolyLog[2,
(-I)*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2)) - ((I/3)*b*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2))

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Rubi [A]  time = 0.179349, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {2455, 302, 205, 2376, 4848, 2391, 203} \[ \frac{i b n \text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}-\frac{i b n \text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac{1}{3} x^3 \log \left (d f x^2+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac{2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )+\frac{2 b n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{9 d^{3/2} f^{3/2}}-\frac{1}{9} b n x^3 \log \left (d f x^2+1\right )-\frac{8 b n x}{9 d f}+\frac{4}{27} b n x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)],x]

[Out]

(-8*b*n*x)/(9*d*f) + (4*b*n*x^3)/27 + (2*b*n*ArcTan[Sqrt[d]*Sqrt[f]*x])/(9*d^(3/2)*f^(3/2)) + (2*x*(a + b*Log[
c*x^n]))/(3*d*f) - (2*x^3*(a + b*Log[c*x^n]))/9 - (2*ArcTan[Sqrt[d]*Sqrt[f]*x]*(a + b*Log[c*x^n]))/(3*d^(3/2)*
f^(3/2)) - (b*n*x^3*Log[1 + d*f*x^2])/9 + (x^3*(a + b*Log[c*x^n])*Log[1 + d*f*x^2])/3 + ((I/3)*b*n*PolyLog[2,
(-I)*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2)) - ((I/3)*b*n*PolyLog[2, I*Sqrt[d]*Sqrt[f]*x])/(d^(3/2)*f^(3/2))

Rule 2455

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

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 205

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

Rule 2376

Int[Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((g_.)*(x_))^(q_.), x_Sym
bol] :> With[{u = IntHide[(g*x)^q*Log[d*(e + f*x^m)^r], x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[Dist
[1/x, u, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, r, m, n, q}, x] && (IntegerQ[(q + 1)/m] || (RationalQ[m] &
& RationalQ[q])) && NeQ[q, -1]

Rule 4848

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[(I*b)/2, Int[Log[1 - I*c*x
]/x, x], x] - Dist[(I*b)/2, Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

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]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int x^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac{1}{d}+f x^2\right )\right ) \, dx &=\frac{2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac{2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-(b n) \int \left (\frac{2}{3 d f}-\frac{2 x^2}{9}-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2} x}+\frac{1}{3} x^2 \log \left (1+d f x^2\right )\right ) \, dx\\ &=-\frac{2 b n x}{3 d f}+\frac{2}{27} b n x^3+\frac{2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac{2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )-\frac{1}{3} (b n) \int x^2 \log \left (1+d f x^2\right ) \, dx+\frac{(2 b n) \int \frac{\tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}\\ &=-\frac{2 b n x}{3 d f}+\frac{2}{27} b n x^3+\frac{2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac{2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac{1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{(i b n) \int \frac{\log \left (1-i \sqrt{d} \sqrt{f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}-\frac{(i b n) \int \frac{\log \left (1+i \sqrt{d} \sqrt{f} x\right )}{x} \, dx}{3 d^{3/2} f^{3/2}}+\frac{1}{9} (2 b d f n) \int \frac{x^4}{1+d f x^2} \, dx\\ &=-\frac{2 b n x}{3 d f}+\frac{2}{27} b n x^3+\frac{2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac{2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac{1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{i b n \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}-\frac{i b n \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}+\frac{1}{9} (2 b d f n) \int \left (-\frac{1}{d^2 f^2}+\frac{x^2}{d f}+\frac{1}{d^2 f^2 \left (1+d f x^2\right )}\right ) \, dx\\ &=-\frac{8 b n x}{9 d f}+\frac{4}{27} b n x^3+\frac{2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac{2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac{1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{i b n \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}-\frac{i b n \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}+\frac{(2 b n) \int \frac{1}{1+d f x^2} \, dx}{9 d f}\\ &=-\frac{8 b n x}{9 d f}+\frac{4}{27} b n x^3+\frac{2 b n \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{9 d^{3/2} f^{3/2}}+\frac{2 x \left (a+b \log \left (c x^n\right )\right )}{3 d f}-\frac{2}{9} x^3 \left (a+b \log \left (c x^n\right )\right )-\frac{2 \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right ) \left (a+b \log \left (c x^n\right )\right )}{3 d^{3/2} f^{3/2}}-\frac{1}{9} b n x^3 \log \left (1+d f x^2\right )+\frac{1}{3} x^3 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+d f x^2\right )+\frac{i b n \text{Li}_2\left (-i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}-\frac{i b n \text{Li}_2\left (i \sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0881363, size = 364, normalized size = 1.51 \[ -\frac{2}{3} b d f n \left (-\frac{i \left (\text{PolyLog}\left (2,-i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1+i \sqrt{d} \sqrt{f} x\right )\right )}{2 d^{5/2} f^{5/2}}+\frac{i \left (\text{PolyLog}\left (2,i \sqrt{d} \sqrt{f} x\right )+\log (x) \log \left (1-i \sqrt{d} \sqrt{f} x\right )\right )}{2 d^{5/2} f^{5/2}}-\frac{x (\log (x)-1)}{d^2 f^2}+\frac{\frac{1}{3} x^3 \log (x)-\frac{x^3}{9}}{d f}\right )-\frac{2 a \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{3 d^{3/2} f^{3/2}}+\frac{1}{3} a x^3 \log \left (d f x^2+1\right )+\frac{2 a x}{3 d f}-\frac{2 a x^3}{9}-\frac{2 b \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right ) \tan ^{-1}\left (\sqrt{d} \sqrt{f} x\right )}{9 d^{3/2} f^{3/2}}+\frac{1}{9} b x^3 \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )+3 n \log (x)-n\right ) \log \left (d f x^2+1\right )+\frac{2 b x \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right )}{9 d f}-\frac{2}{27} b x^3 \left (3 \left (\log \left (c x^n\right )-n \log (x)\right )-n\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*Log[c*x^n])*Log[d*(d^(-1) + f*x^2)],x]

[Out]

(2*a*x)/(3*d*f) - (2*a*x^3)/9 - (2*a*ArcTan[Sqrt[d]*Sqrt[f]*x])/(3*d^(3/2)*f^(3/2)) + (2*b*x*(-n + 3*(-(n*Log[
x]) + Log[c*x^n])))/(9*d*f) - (2*b*x^3*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/27 - (2*b*ArcTan[Sqrt[d]*Sqrt[f]*x
]*(-n + 3*(-(n*Log[x]) + Log[c*x^n])))/(9*d^(3/2)*f^(3/2)) + (a*x^3*Log[1 + d*f*x^2])/3 + (b*x^3*(-n + 3*n*Log
[x] + 3*(-(n*Log[x]) + Log[c*x^n]))*Log[1 + d*f*x^2])/9 - (2*b*d*f*n*(-((x*(-1 + Log[x]))/(d^2*f^2)) + (-x^3/9
+ (x^3*Log[x])/3)/(d*f) - ((I/2)*(Log[x]*Log[1 + I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, (-I)*Sqrt[d]*Sqrt[f]*x]))/
(d^(5/2)*f^(5/2)) + ((I/2)*(Log[x]*Log[1 - I*Sqrt[d]*Sqrt[f]*x] + PolyLog[2, I*Sqrt[d]*Sqrt[f]*x]))/(d^(5/2)*f
^(5/2))))/3

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Maple [F]  time = 0.178, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \ln \left ( d \left ({d}^{-1}+f{x}^{2} \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))*ln(d*(1/d+f*x^2)),x)

[Out]

int(x^2*(a+b*ln(c*x^n))*ln(d*(1/d+f*x^2)),x)

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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^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^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 (b x^{2} \log \left (d f x^{2} + 1\right ) \log \left (c x^{n}\right ) + a x^{2} \log \left (d f x^{2} + 1\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x, algorithm="fricas")

[Out]

integral(b*x^2*log(d*f*x^2 + 1)*log(c*x^n) + a*x^2*log(d*f*x^2 + 1), 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**2*(a+b*ln(c*x**n))*ln(d*(1/d+f*x**2)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2} \log \left ({\left (f x^{2} + \frac{1}{d}\right )} d\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))*log(d*(1/d+f*x^2)),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^2*log((f*x^2 + 1/d)*d), x)