\(\int \frac {x^3 \log (c (a+\frac {b}{x^2})^p)}{d+e x} \, dx\) [247]

Optimal result

Integrand size = 23, antiderivative size = 421 \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx=\frac {2 b p x}{3 a e}+\frac {2 \sqrt {b} d^2 p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {2 b^{3/2} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}-\frac {d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^4}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e^4}-\frac {b d p \log \left (b+a x^2\right )}{2 a e^2}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^4}-\frac {2 d^3 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^4} \]

[Out]

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {2516, 2498, 269, 211, 2505, 266, 199, 327, 2512, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx=-\frac {2 b^{3/2} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{3 a^{3/2} e}+\frac {2 \sqrt {b} d^2 p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^3}-\frac {d^3 \log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^3}-\frac {d x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e^2}+\frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{3 e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^4}-\frac {b d p \log \left (a x^2+b\right )}{2 a e^2}+\frac {2 b p x}{3 a e}-\frac {2 d^3 p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^4}-\frac {2 d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^4} \]

[In]

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

[Out]

(2*b*p*x)/(3*a*e) + (2*Sqrt[b]*d^2*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[a]*e^3) - (2*b^(3/2)*p*ArcTan[(Sqrt[a]
*x)/Sqrt[b]])/(3*a^(3/2)*e) + (d^2*x*Log[c*(a + b/x^2)^p])/e^3 - (d*x^2*Log[c*(a + b/x^2)^p])/(2*e^2) + (x^3*L
og[c*(a + b/x^2)^p])/(3*e) - (d^3*Log[c*(a + b/x^2)^p]*Log[d + e*x])/e^4 - (2*d^3*p*Log[-((e*x)/d)]*Log[d + e*
x])/e^4 + (d^3*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*
(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/e^4 - (b*d*p*Log[b + a*x^2])/(2*a*e^2) + (d^3
*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/e^4 + (d^3*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sq
rt[-a]*d + Sqrt[b]*e)])/e^4 - (2*d^3*p*PolyLog[2, 1 + (e*x)/d])/e^4

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

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 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.16 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx=\frac {-12 \sqrt {a} \sqrt {b} d^2 e p \arctan \left (\frac {\sqrt {b}}{\sqrt {a} x}\right )+4 b e^3 p x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {b}{a x^2}\right )-3 b d e^2 p \log \left (a+\frac {b}{x^2}\right )+6 a d^2 e x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-3 a d e^2 x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+2 a e^3 x^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-6 b d e^2 p \log (x)-6 a d^3 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)-12 a d^3 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)+6 a d^3 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)+6 a d^3 p \log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)+6 a d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )+6 a d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )-12 a d^3 p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{6 a e^4} \]

[In]

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

[Out]

(-12*Sqrt[a]*Sqrt[b]*d^2*e*p*ArcTan[Sqrt[b]/(Sqrt[a]*x)] + 4*b*e^3*p*x*Hypergeometric2F1[-1/2, 1, 1/2, -(b/(a*
x^2))] - 3*b*d*e^2*p*Log[a + b/x^2] + 6*a*d^2*e*x*Log[c*(a + b/x^2)^p] - 3*a*d*e^2*x^2*Log[c*(a + b/x^2)^p] +
2*a*e^3*x^3*Log[c*(a + b/x^2)^p] - 6*b*d*e^2*p*Log[x] - 6*a*d^3*Log[c*(a + b/x^2)^p]*Log[d + e*x] - 12*a*d^3*p
*Log[-((e*x)/d)]*Log[d + e*x] + 6*a*d^3*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x
] + 6*a*d^3*p*Log[(e*(Sqrt[b] + Sqrt[-a]*x))/(-(Sqrt[-a]*d) + Sqrt[b]*e)]*Log[d + e*x] + 6*a*d^3*p*PolyLog[2,
(Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)] + 6*a*d^3*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[
b]*e)] - 12*a*d^3*p*PolyLog[2, 1 + (e*x)/d])/(6*a*e^4)

Maple [A] (verified)

Time = 1.82 (sec) , antiderivative size = 411, normalized size of antiderivative = 0.98

method	result	size
parts	\(\frac {x^{3} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{3 e}-\frac {d \,x^{2} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{2 e^{2}}+\frac {d^{2} x \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{e^{3}}-\frac {d^{3} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (e x +d \right )}{e^{4}}+2 p b \,e^{2} \left (\frac {d^{3} \left (-\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b \,e^{2}}-\frac {a \left (-\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\ln \left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )\right )}{2 a}-\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )}{2 a}\right )}{b \,e^{2}}\right )}{e^{4}}+\frac {\frac {2 e x +2 d}{a}+\frac {-\frac {3 d \ln \left (a \,d^{2}-2 a d \left (e x +d \right )+a \left (e x +d \right )^{2}+e^{2} b \right )}{2}+\frac {\left (6 a \,d^{2}-2 e^{2} b \right ) \arctan \left (\frac {-2 a d +2 a \left (e x +d \right )}{2 e \sqrt {a b}}\right )}{e \sqrt {a b}}}{a}}{6 e^{4}}\right )\)	\(411\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)