3.216 \(\int \frac{x^4 (a+b \log (c x^n))}{d+e x^2} \, dx\)

Optimal. Leaf size=167 \[ -\frac{i b d^{3/2} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 e^{5/2}}+\frac{i b d^{3/2} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 e^{5/2}}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac{a d x}{e^2}-\frac{b d x \log \left (c x^n\right )}{e^2}+\frac{b d n x}{e^2}-\frac{b n x^3}{9 e} \]

[Out]

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

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Rubi [A]  time = 0.206553, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.391, Rules used = {302, 205, 2351, 2295, 2304, 2324, 12, 4848, 2391} \[ -\frac{i b d^{3/2} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 e^{5/2}}+\frac{i b d^{3/2} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 e^{5/2}}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}-\frac{a d x}{e^2}-\frac{b d x \log \left (c x^n\right )}{e^2}+\frac{b d n x}{e^2}-\frac{b n x^3}{9 e} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(a + b*Log[c*x^n]))/(d + e*x^2),x]

[Out]

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

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 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2324

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rubi steps

\begin{align*} \int \frac{x^4 \left (a+b \log \left (c x^n\right )\right )}{d+e x^2} \, dx &=\int \left (-\frac{d \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{e}+\frac{d^2 \left (a+b \log \left (c x^n\right )\right )}{e^2 \left (d+e x^2\right )}\right ) \, dx\\ &=-\frac{d \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^2}+\frac{d^2 \int \frac{a+b \log \left (c x^n\right )}{d+e x^2} \, dx}{e^2}+\frac{\int x^2 \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}\\ &=-\frac{a d x}{e^2}-\frac{b n x^3}{9 e}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac{(b d) \int \log \left (c x^n\right ) \, dx}{e^2}-\frac{\left (b d^2 n\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e} x} \, dx}{e^2}\\ &=-\frac{a d x}{e^2}+\frac{b d n x}{e^2}-\frac{b n x^3}{9 e}-\frac{b d x \log \left (c x^n\right )}{e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac{\left (b d^{3/2} n\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{e^{5/2}}\\ &=-\frac{a d x}{e^2}+\frac{b d n x}{e^2}-\frac{b n x^3}{9 e}-\frac{b d x \log \left (c x^n\right )}{e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac{\left (i b d^{3/2} n\right ) \int \frac{\log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 e^{5/2}}+\frac{\left (i b d^{3/2} n\right ) \int \frac{\log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{x} \, dx}{2 e^{5/2}}\\ &=-\frac{a d x}{e^2}+\frac{b d n x}{e^2}-\frac{b n x^3}{9 e}-\frac{b d x \log \left (c x^n\right )}{e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{3 e}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^{5/2}}-\frac{i b d^{3/2} n \text{Li}_2\left (-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 e^{5/2}}+\frac{i b d^{3/2} n \text{Li}_2\left (\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 e^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.149333, size = 208, normalized size = 1.25 \[ \frac{9 b (-d)^{3/2} n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )-9 b (-d)^{3/2} n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )+9 \sqrt{-d} d \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+9 (-d)^{3/2} \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+6 e^{3/2} x^3 \left (a+b \log \left (c x^n\right )\right )-18 a d \sqrt{e} x-18 b d \sqrt{e} x \log \left (c x^n\right )+18 b d \sqrt{e} n x-2 b e^{3/2} n x^3}{18 e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(a + b*Log[c*x^n]))/(d + e*x^2),x]

[Out]

(-18*a*d*Sqrt[e]*x + 18*b*d*Sqrt[e]*n*x - 2*b*e^(3/2)*n*x^3 - 18*b*d*Sqrt[e]*x*Log[c*x^n] + 6*e^(3/2)*x^3*(a +
b*Log[c*x^n]) + 9*Sqrt[-d]*d*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]] + 9*(-d)^(3/2)*(a + b*Log[c*x^n
])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)] + 9*b*(-d)^(3/2)*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]] - 9*b*(-d)^(3/2)*n*Po
lyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(18*e^(5/2))

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Maple [C]  time = 0.184, size = 693, normalized size = 4.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*ln(c*x^n))/(e*x^2+d),x)

[Out]

-b*ln(c)/e^2*d*x-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*d^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/
3*b*ln(c)/e*x^3-b*ln(x^n)/e^2*d*x-1/6*I*b*Pi*csgn(I*c*x^n)^3/e*x^3+b*d^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2
))*ln(x^n)+b*ln(c)*d^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/2*b*n*d^2/e^2/(-d*e)^(1/2)*dilog((-e*x+(-d*e)
^(1/2))/(-d*e)^(1/2))-1/2*b*n*d^2/e^2/(-d*e)^(1/2)*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-b*d^2/e^2/(d*e)^(1/2
)*arctan(x*e/(d*e)^(1/2))*n*ln(x)+1/2*b*n*d^2/e^2/(-d*e)^(1/2)*ln(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*
b*n*d^2/e^2/(-d*e)^(1/2)*ln(x)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e*x^
3+1/2*I*b*Pi*csgn(I*c*x^n)^3/e^2*d*x+1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*d^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)
^(1/2))+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/3*b*ln(x^n)/e*x^3
+1/6*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e*x^3+a*d^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/3*a/e*x^3+1/2*I*b*
Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^2*d*x-1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^2*d*x-1/2*I*b*Pi*csgn(I*
x^n)*csgn(I*c*x^n)^2/e^2*d*x-1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e*x^3-1/2*I*b*Pi*csgn(I*c*x^n)^3*d
^2/e^2/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+b*d*n*x/e^2-1/9*b*n*x^3/e-a*d*x/e^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^4*(a+b*log(c*x^n))/(e*x^2+d),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 x^{4} \log \left (c x^{n}\right ) + a x^{4}}{e x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*log(c*x^n))/(e*x^2+d),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*ln(c*x**n))/(e*x**2+d),x)

[Out]

Integral(x**4*(a + b*log(c*x**n))/(d + e*x**2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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