3.3.61 \(\int \frac {x^4 (a+b \log (c (d+e x)^n))}{f+g x^2} \, dx\) [261]

Optimal. Leaf size=369 \[ -\frac {a f x}{g^2}+\frac {b f n x}{g^2}-\frac {b d^2 n x}{3 e^2 g}+\frac {b d n x^2}{6 e g}-\frac {b n x^3}{9 g}+\frac {b d^3 n \log (d+e x)}{3 e^3 g}-\frac {b f (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 g}+\frac {(-f)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{5/2}}-\frac {(-f)^{3/2} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{5/2}}-\frac {b (-f)^{3/2} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^{5/2}}+\frac {b (-f)^{3/2} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^{5/2}} \]

[Out]

-a*f*x/g^2+b*f*n*x/g^2-1/3*b*d^2*n*x/e^2/g+1/6*b*d*n*x^2/e/g-1/9*b*n*x^3/g+1/3*b*d^3*n*ln(e*x+d)/e^3/g-b*f*(e*
x+d)*ln(c*(e*x+d)^n)/e/g^2+1/3*x^3*(a+b*ln(c*(e*x+d)^n))/g+1/2*(-f)^(3/2)*(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)))/g^(5/2)-1/2*(-f)^(3/2)*(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)))/g^(5/2)-1/2*b*(-f)^(3/2)*n*polylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))/g
^(5/2)+1/2*b*(-f)^(3/2)*n*polylog(2,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/g^(5/2)

________________________________________________________________________________________

Rubi [A]
time = 0.30, antiderivative size = 369, normalized size of antiderivative = 1.00, 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 = {308, 211, 2463, 2436, 2332, 2442, 45, 2456, 2441, 2440, 2438} \begin {gather*} -\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^3 n \log (d+e x)}{3 e^3 g}-\frac {b d^2 n x}{3 e^2 g}+\frac {b d n x^2}{6 e g}+\frac {b f n x}{g^2}-\frac {b n x^3}{9 g} \end {gather*}

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 45

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 211

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

Rule 308

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 2332

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

Rule 2436

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 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 2442

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 2456

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 2463

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]

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) \text {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 ) \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 g^{5/2}}-\frac {\left (b (-f)^{3/2} n\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 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 = 0.22, size = 339, normalized size = 0.92 \begin {gather*} \frac {-18 a f \sqrt {g} x+18 b f \sqrt {g} n x-\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}-\frac {18 b f \sqrt {g} (d+e x) \log \left (c (d+e x)^n\right )}{e}+6 g^{3/2} x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )+9 (-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 )+9 \sqrt {-f} f \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 )-9 b (-f)^{3/2} n \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+9 b (-f)^{3/2} n \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{18 g^{5/2}} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.42, size = 982, normalized size = 2.66

method	result	size
risch	\(-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) x^{3}}{6 g}-\frac {b \ln \left (c \right ) x f}{g^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) x f}{2 g^{2}}-\frac {11 b n \,d^{3}}{18 e^{3} g}+\frac {a \,f^{2} \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{g^{2} \sqrt {f g}}+\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} x^{3}}{6 g}+\frac {a \,x^{3}}{3 g}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} x f}{2 g^{2}}+\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x^{3}}{3 g}+\frac {b \ln \left (c \right ) x^{3}}{3 g}+\frac {b \,f^{2} \arctan \left (\frac {2 g \left (e x +d \right )-2 d g}{2 e \sqrt {f g}}\right ) \ln \left (\left (e x +d \right )^{n}\right )}{g^{2} \sqrt {f g}}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) x f}{g^{2}}-\frac {b f d \ln \left (\left (e x +d \right )^{n}\right )}{e \,g^{2}}-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} f^{2} \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{2 g^{2} \sqrt {f g}}+\frac {b n \,f^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 g^{2} \sqrt {-f g}}-\frac {b \,f^{2} \arctan \left (\frac {2 g \left (e x +d \right )-2 d g}{2 e \sqrt {f g}}\right ) n \ln \left (e x +d \right )}{g^{2} \sqrt {f g}}-\frac {b n \,f^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 g^{2} \sqrt {-f g}}+\frac {b \ln \left (c \right ) f^{2} \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{g^{2} \sqrt {f g}}+\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} x f}{2 g^{2}}+\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} x^{3}}{6 g}+\frac {b n d f}{e \,g^{2}}+\frac {b \,d^{3} \ln \left (\left (e x +d \right )^{n}\right )}{3 e^{3} g}-\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} x f}{2 g^{2}}+\frac {b n \,f^{2} \dilog \left (\frac {e \sqrt {-f g}-g \left (e x +d \right )+d g}{e \sqrt {-f g}+d g}\right )}{2 g^{2} \sqrt {-f g}}-\frac {b n \,f^{2} \dilog \left (\frac {e \sqrt {-f g}+g \left (e x +d \right )-d g}{e \sqrt {-f g}-d g}\right )}{2 g^{2} \sqrt {-f g}}-\frac {i b \pi \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} x^{3}}{6 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 {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} f^{2} \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{2 g^{2} \sqrt {f g}}+\frac {i b \pi \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} f^{2} \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{2 g^{2} \sqrt {f g}}-\frac {i b \pi \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) f^{2} \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{2 g^{2} \sqrt {f g}}\)	\(982\)

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,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{g\,x^2+f} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________