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

Optimal. Leaf size=369 \[ -\frac{b (-f)^{3/2} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^{5/2}}+\frac{b (-f)^{3/2} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g^{5/2}}+\frac{(-f)^{3/2} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^{5/2}}-\frac{(-f)^{3/2} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^{5/2}}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac{a f x}{g^2}-\frac{b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac{b d^2 n x}{3 e^2 g}+\frac{b d^3 n \log (d+e x)}{3 e^3 g}+\frac{b d n x^2}{6 e g}+\frac{b f n x}{g^2}-\frac{b n x^3}{9 g} \]

[Out]

-((a*f*x)/g^2) + (b*f*n*x)/g^2 - (b*d^2*n*x)/(3*e^2*g) + (b*d*n*x^2)/(6*e*g) - (b*n*x^3)/(9*g) + (b*d^3*n*Log[
d + e*x])/(3*e^3*g) - (b*f*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + (x^3*(a + b*Log[c*(d + e*x)^n]))/(3*g) + ((
-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2)) - (
(-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^(5/2)) -
(b*(-f)^(3/2)*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^(5/2)) + (b*(-f)^(3/2)*n*Pol
yLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.394707, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.407, Rules used = {302, 205, 2416, 2389, 2295, 2395, 43, 2409, 2394, 2393, 2391} \[ -\frac{b (-f)^{3/2} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^{5/2}}+\frac{b (-f)^{3/2} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{2 g^{5/2}}+\frac{(-f)^{3/2} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^{5/2}}-\frac{(-f)^{3/2} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 g^{5/2}}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac{a f x}{g^2}-\frac{b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac{b d^2 n x}{3 e^2 g}+\frac{b d^3 n \log (d+e x)}{3 e^3 g}+\frac{b d n x^2}{6 e g}+\frac{b f n x}{g^2}-\frac{b n x^3}{9 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*f*x)/g^2) + (b*f*n*x)/g^2 - (b*d^2*n*x)/(3*e^2*g) + (b*d*n*x^2)/(6*e*g) - (b*n*x^3)/(9*g) + (b*d^3*n*Log[
d + e*x])/(3*e^3*g) - (b*f*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) + (x^3*(a + b*Log[c*(d + e*x)^n]))/(3*g) + ((
-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(5/2)) - (
(-f)^(3/2)*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])])/(2*g^(5/2)) -
(b*(-f)^(3/2)*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/(2*g^(5/2)) + (b*(-f)^(3/2)*n*Pol
yLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^(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 2416

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]

Rule 2389

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

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 2395

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2409

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 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^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{f+g x^2} \, dx &=\int \left (-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac{f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac{f \int \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g^2}+\frac{f^2 \int \frac{a+b \log \left (c (d+e x)^n\right )}{f+g x^2} \, dx}{g^2}+\frac{\int x^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) \, dx}{g}\\ &=-\frac{a f x}{g^2}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac{(b f) \int \log \left (c (d+e x)^n\right ) \, dx}{g^2}+\frac{f^2 \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}{g^2}-\frac{(b e n) \int \frac{x^3}{d+e x} \, dx}{3 g}\\ &=-\frac{a f x}{g^2}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}-\frac{(-f)^{3/2} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 g^2}-\frac{(-f)^{3/2} \int \frac{a+b \log \left (c (d+e x)^n\right )}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 g^2}-\frac{(b f) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e g^2}-\frac{(b e n) \int \left (\frac{d^2}{e^3}-\frac{d x}{e^2}+\frac{x^2}{e}-\frac{d^3}{e^3 (d+e x)}\right ) \, dx}{3 g}\\ &=-\frac{a f x}{g^2}+\frac{b f n x}{g^2}-\frac{b d^2 n x}{3 e^2 g}+\frac{b d n x^2}{6 e g}-\frac{b n x^3}{9 g}+\frac{b d^3 n \log (d+e x)}{3 e^3 g}-\frac{b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac{(-f)^{3/2} \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 g^{5/2}}-\frac{(-f)^{3/2} \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 g^{5/2}}-\frac{\left (b e (-f)^{3/2} 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 g^{5/2}}+\frac{\left (b e (-f)^{3/2} 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 g^{5/2}}\\ &=-\frac{a f x}{g^2}+\frac{b f n x}{g^2}-\frac{b d^2 n x}{3 e^2 g}+\frac{b d n x^2}{6 e g}-\frac{b n x^3}{9 g}+\frac{b d^3 n \log (d+e x)}{3 e^3 g}-\frac{b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac{(-f)^{3/2} \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 g^{5/2}}-\frac{(-f)^{3/2} \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 g^{5/2}}+\frac{\left (b (-f)^{3/2} n\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 g^{5/2}}-\frac{\left (b (-f)^{3/2} n\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 g^{5/2}}\\ &=-\frac{a f x}{g^2}+\frac{b f n x}{g^2}-\frac{b d^2 n x}{3 e^2 g}+\frac{b d n x^2}{6 e g}-\frac{b n x^3}{9 g}+\frac{b d^3 n \log (d+e x)}{3 e^3 g}-\frac{b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac{(-f)^{3/2} \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 g^{5/2}}-\frac{(-f)^{3/2} \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 g^{5/2}}-\frac{b (-f)^{3/2} n \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^{5/2}}+\frac{b (-f)^{3/2} n \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.338764, size = 339, normalized size = 0.92 \[ \frac{-9 b (-f)^{3/2} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )+9 b (-f)^{3/2} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )+9 (-f)^{3/2} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+9 \sqrt{-f} f \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+6 g^{3/2} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )-18 a f \sqrt{g} x-\frac{18 b f \sqrt{g} (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac{b g^{3/2} n \left (e x \left (6 d^2-3 d e x+2 e^2 x^2\right )-6 d^3 \log (d+e x)\right )}{e^3}+18 b f \sqrt{g} n x}{18 g^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.5, size = 982, normalized size = 2.7 \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*(e*x+d)^n))/(g*x^2+f),x)

[Out]

-1/6*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g*x^3+1/3*b*ln(c)/g*x^3+b/e*n/g^2*d*f+1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^2*f
*x+a*f^2/g^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/3*b/e^3/g*d^3*ln((e*x+d)^n)+b*f^2/g^2/(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/e/g^2*f*d*ln((e*x+d)^n)-b*ln(c)/g^2*f*x+1/2*I*b*Pi*csgn(I
*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g^2*f*x+1/6*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/g*x^3-1/2
*I*b*Pi*csgn(I*c*(e*x+d)^n)^3*f^2/g^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/3*b*ln((e*x+d)^n)/g*x^3-b*ln((e*x+
d)^n)/g^2*f*x-11/18*b/e^3*n/g*d^3+1/6*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/g*x^3+1/3*a/g*x^3-1/2*I*b*Pi*csgn
(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/g^2*f*x-1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/g^2*f*x-1/6*I*b*Pi*csgn
(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g*x^3+b*ln(c)*f^2/g^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/2*b*n*
f^2/g^2/(-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*n*f^2/g^2/(-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*f^2/g^2/(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*b*n*f^2/g^2*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^2*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*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2*f^2/g^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/9*b*n*x^3/g+1/6
*b*d*n*x^2/e/g-a*f*x/g^2-1/3*b*d^2*n*x/e^2/g+b*f*n*x/g^2+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2*f^2/g^2/(f
*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)*f^2/g^2/(f*g)^(1/
2)*arctan(x*g/(f*g)^(1/2))

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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*(e*x+d)^n))/(g*x^2+f),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 ({\left (e x + d\right )}^{n} c\right ) + a x^{4}}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^4*log((e*x + d)^n*c) + a*x^4)/(g*x^2 + f), 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**4*(a+b*ln(c*(e*x+d)**n))/(g*x**2+f),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 )} x^{4}}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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