\(\int \frac {x^3 \log (c (a+b x^2)^p)}{d+e x} \, dx\) [226]

Optimal result

Integrand size = 23, antiderivative size = 394 \[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=-\frac {2 d^2 p x}{e^3}+\frac {2 a p x}{3 b e}+\frac {d p x^2}{2 e^2}-\frac {2 p x^3}{9 e}+\frac {2 \sqrt {a} d^2 p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^3}-\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2} e}+\frac {d^3 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^4}+\frac {d^2 x \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}-\frac {d \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e^2}-\frac {d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^4} \]

[Out]

-2*d^2*p*x/e^3+2/3*a*p*x/b/e+1/2*d*p*x^2/e^2-2/9*p*x^3/e-2/3*a^(3/2)*p*arctan(x*b^(1/2)/a^(1/2))/b^(3/2)/e+d^2
*x*ln(c*(b*x^2+a)^p)/e^3+1/3*x^3*ln(c*(b*x^2+a)^p)/e-1/2*d*(b*x^2+a)*ln(c*(b*x^2+a)^p)/b/e^2-d^3*ln(e*x+d)*ln(
c*(b*x^2+a)^p)/e^4+d^3*p*ln(e*x+d)*ln(e*((-a)^(1/2)-x*b^(1/2))/(e*(-a)^(1/2)+d*b^(1/2)))/e^4+d^3*p*ln(e*x+d)*l
n(-e*((-a)^(1/2)+x*b^(1/2))/(-e*(-a)^(1/2)+d*b^(1/2)))/e^4+d^3*p*polylog(2,(e*x+d)*b^(1/2)/(-e*(-a)^(1/2)+d*b^
(1/2)))/e^4+d^3*p*polylog(2,(e*x+d)*b^(1/2)/(e*(-a)^(1/2)+d*b^(1/2)))/e^4+2*d^2*p*arctan(x*b^(1/2)/a^(1/2))*a^
(1/2)/e^3/b^(1/2)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {2516, 2498, 327, 211, 2504, 2436, 2332, 2505, 308, 2512, 266, 2463, 2441, 2440, 2438} \[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=-\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2} e}+\frac {2 \sqrt {a} d^2 p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^3}-\frac {d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {d \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e^2}+\frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^4}+\frac {2 a p x}{3 b e}-\frac {2 d^2 p x}{e^3}+\frac {d p x^2}{2 e^2}-\frac {2 p x^3}{9 e} \]

[In]

Int[(x^3*Log[c*(a + b*x^2)^p])/(d + e*x),x]

[Out]

(-2*d^2*p*x)/e^3 + (2*a*p*x)/(3*b*e) + (d*p*x^2)/(2*e^2) - (2*p*x^3)/(9*e) + (2*Sqrt[a]*d^2*p*ArcTan[(Sqrt[b]*
x)/Sqrt[a]])/(Sqrt[b]*e^3) - (2*a^(3/2)*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*b^(3/2)*e) + (d^3*p*Log[(e*(Sqrt[-a]
- Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*
d - Sqrt[-a]*e))]*Log[d + e*x])/e^4 + (d^2*x*Log[c*(a + b*x^2)^p])/e^3 + (x^3*Log[c*(a + b*x^2)^p])/(3*e) - (d
*(a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*b*e^2) - (d^3*Log[d + e*x]*Log[c*(a + b*x^2)^p])/e^4 + (d^3*p*PolyLog[2,
(Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)])/e^4 + (d^3*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[
-a]*e)])/e^4

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 266

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

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 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 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]

Rule 2498

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

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2505

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 2512

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

Rule 2516

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_.) + (g_.)*(x_))^(r_.), x_S
ymbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e,
f, g, n, p, q}, x] && IntegerQ[m] && IntegerQ[r]

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\frac {-36 d^2 e p x+\frac {12 a e^3 p x}{b}-4 e^3 p x^3+\frac {36 \sqrt {a} d^2 e p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {12 a^{3/2} e^3 p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}+18 d^3 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+18 d^3 p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+18 d^2 e x \log \left (c \left (a+b x^2\right )^p\right )+6 e^3 x^3 \log \left (c \left (a+b x^2\right )^p\right )-18 d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+9 d e^2 \left (p x^2-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}\right )+18 d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+18 d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{18 e^4} \]

[In]

Integrate[(x^3*Log[c*(a + b*x^2)^p])/(d + e*x),x]

[Out]

(-36*d^2*e*p*x + (12*a*e^3*p*x)/b - 4*e^3*p*x^3 + (36*Sqrt[a]*d^2*e*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] - (
12*a^(3/2)*e^3*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2) + 18*d^3*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + S
qrt[-a]*e)]*Log[d + e*x] + 18*d^3*p*Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(-(Sqrt[b]*d) + Sqrt[-a]*e)]*Log[d + e*x] +
18*d^2*e*x*Log[c*(a + b*x^2)^p] + 6*e^3*x^3*Log[c*(a + b*x^2)^p] - 18*d^3*Log[d + e*x]*Log[c*(a + b*x^2)^p] +
9*d*e^2*(p*x^2 - ((a + b*x^2)*Log[c*(a + b*x^2)^p])/b) + 18*d^3*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d -
Sqrt[-a]*e)] + 18*d^3*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/(18*e^4)

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.05

method	result	size
parts	\(\frac {x^{3} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{3 e}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) d \,x^{2}}{2 e^{2}}+\frac {d^{2} x \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e^{4}}-\frac {2 p b \left (\frac {d^{3} \left (-\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{e^{2}}+\frac {-\frac {2 \left (e x +d \right ) a \,e^{2}-11 \left (e x +d \right ) b \,d^{2}+\frac {7 d \left (e x +d \right )^{2} b}{2}-\frac {2 \left (e x +d \right )^{3} b}{3}}{b^{2}}+\frac {a \,e^{2} \left (\frac {3 d \ln \left (\left (e x +d \right )^{2} b -2 d \left (e x +d \right ) b +a \,e^{2}+b \,d^{2}\right )}{2}+\frac {\left (2 a \,e^{2}-6 b \,d^{2}\right ) \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {a b}}\right )}{e \sqrt {a b}}\right )}{b^{2}}}{6 e^{2}}\right )}{e^{2}}\)	\(412\)
risch	\(\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) x^{3}}{3 e}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) d \,x^{2}}{2 e^{2}}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) x \,d^{2}}{e^{3}}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {2 p \,x^{3}}{9 e}+\frac {d p \,x^{2}}{2 e^{2}}-\frac {2 d^{2} p x}{e^{3}}-\frac {49 p \,d^{3}}{18 e^{4}}+\frac {2 a p x}{3 b e}+\frac {2 p a d}{3 b \,e^{2}}-\frac {p a d \ln \left (\left (e x +d \right )^{2} b -2 d \left (e x +d \right ) b +a \,e^{2}+b \,d^{2}\right )}{2 b \,e^{2}}-\frac {2 p \,a^{2} \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {a b}}\right )}{3 b e \sqrt {a b}}+\frac {2 p a \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {a b}}\right ) d^{2}}{e^{3} \sqrt {a b}}+\frac {p \,d^{3} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e^{4}}+\frac {p \,d^{3} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e^{4}}+\frac {p \,d^{3} \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e^{4}}+\frac {p \,d^{3} \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e^{4}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\frac {1}{3} e^{2} x^{3}-\frac {1}{2} d e \,x^{2}+d^{2} x}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right )}{e^{4}}\right )\)	\(610\)

[In]

int(x^3*ln(c*(b*x^2+a)^p)/(e*x+d),x,method=_RETURNVERBOSE)

[Out]

1/3*x^3*ln(c*(b*x^2+a)^p)/e-1/2*ln(c*(b*x^2+a)^p)/e^2*d*x^2+d^2*x*ln(c*(b*x^2+a)^p)/e^3-d^3*ln(e*x+d)*ln(c*(b*
x^2+a)^p)/e^4-2*p*b/e^2*(d^3/e^2*(-1/2*ln(e*x+d)*(ln((e*(-a*b)^(1/2)-(e*x+d)*b+b*d)/(e*(-a*b)^(1/2)+b*d))+ln((
e*(-a*b)^(1/2)+(e*x+d)*b-b*d)/(e*(-a*b)^(1/2)-b*d)))/b-1/2*(dilog((e*(-a*b)^(1/2)-(e*x+d)*b+b*d)/(e*(-a*b)^(1/
2)+b*d))+dilog((e*(-a*b)^(1/2)+(e*x+d)*b-b*d)/(e*(-a*b)^(1/2)-b*d)))/b)+1/6/e^2*(-1/b^2*(2*(e*x+d)*a*e^2-11*(e
*x+d)*b*d^2+7/2*d*(e*x+d)^2*b-2/3*(e*x+d)^3*b)+1/b^2*a*e^2*(3/2*d*ln((e*x+d)^2*b-2*d*(e*x+d)*b+a*e^2+b*d^2)+(2
*a*e^2-6*b*d^2)/e/(a*b)^(1/2)*arctan(1/2*(2*(e*x+d)*b-2*b*d)/e/(a*b)^(1/2)))))

Fricas [F]

\[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \]

[In]

integrate(x^3*log(c*(b*x^2+a)^p)/(e*x+d),x, algorithm="fricas")

[Out]

integral(x^3*log((b*x^2 + a)^p*c)/(e*x + d), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\text {Timed out} \]

[In]

integrate(x**3*ln(c*(b*x**2+a)**p)/(e*x+d),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \]

[In]

integrate(x^3*log(c*(b*x^2+a)^p)/(e*x+d),x, algorithm="maxima")

[Out]

integrate(x^3*log((b*x^2 + a)^p*c)/(e*x + d), x)

Giac [F]

\[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \]

[In]

integrate(x^3*log(c*(b*x^2+a)^p)/(e*x+d),x, algorithm="giac")

[Out]

integrate(x^3*log((b*x^2 + a)^p*c)/(e*x + d), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int \frac {x^3\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{d+e\,x} \,d x \]

[In]

int((x^3*log(c*(a + b*x^2)^p))/(d + e*x),x)

[Out]

int((x^3*log(c*(a + b*x^2)^p))/(d + e*x), x)