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

Optimal. Leaf size=394 \[ \frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )}{e^4}-\frac{2 a^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 b^{3/2} e}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^4}+\frac{d^2 x \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac{d \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e^2}+\frac{x^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}+\frac{d^3 p \log (d+e x) \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^4}+\frac{2 \sqrt{a} d^2 p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} e^3}+\frac{2 a p x}{3 b e}-\frac{2 d^2 p x}{e^3}+\frac{d p x^2}{2 e^2}-\frac{2 p x^3}{9 e} \]

[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

________________________________________________________________________________________

Rubi [A]  time = 0.426393, antiderivative size = 394, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 15, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.652, Rules used = {2466, 2448, 321, 205, 2454, 2389, 2295, 2455, 302, 2462, 260, 2416, 2394, 2393, 2391} \[ \frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )}{e^4}-\frac{2 a^{3/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 b^{3/2} e}-\frac{d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^4}+\frac{d^2 x \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac{d \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e^2}+\frac{x^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}+\frac{d^3 p \log (d+e x) \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right )}{e^4}+\frac{2 \sqrt{a} d^2 p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b} e^3}+\frac{2 a p x}{3 b e}-\frac{2 d^2 p x}{e^3}+\frac{d p x^2}{2 e^2}-\frac{2 p x^3}{9 e} \]

Antiderivative was successfully verified.

[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 2466

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]

Rule 2448

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 321

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 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 2454

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 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 2455

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 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 2462

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 260

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 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 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^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx &=\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 \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 0.319472, size = 338, normalized size = 0.86 \[ \frac{18 d^3 p \left (\text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )+\log (d+e x) \left (\log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )+\log \left (\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{-a} e-\sqrt{b} d}\right )\right )\right )-4 e^3 p \left (\frac{3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{3/2}}-\frac{3 a x}{b}+x^3\right )-18 d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+18 d^2 e x \log \left (c \left (a+b x^2\right )^p\right )+9 d e^2 \left (p x^2-\frac{\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}\right )+6 e^3 x^3 \log \left (c \left (a+b x^2\right )^p\right )-36 d^2 e p \left (x-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{b}}\right )}{18 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*e^3*p*((-3*a*x)/b + x^3 + (3*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(3/2)) - 36*d^2*e*p*(x - (Sqrt[a]*ArcT
an[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b]) + 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*((Log[(e
*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)] + Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(-(Sqrt[b]*d) + Sqrt[-a]*e
)])*Log[d + e*x] + PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)] + PolyLog[2, (Sqrt[b]*(d + e*x))/(
Sqrt[b]*d + Sqrt[-a]*e)]))/(18*e^4)

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Maple [C]  time = 0.541, size = 1083, normalized size = 2.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-ln((b*x^2+a)^p)*d^3/e^4*ln(e*x+d)-1/2*ln((b*x^2+a)^p)/e^2*x^2*d+2*p/e^3*a/(a*b)^(1/2)*arctan(1/2*(2*b*(e*x+d)
-2*b*d)/e/(a*b)^(1/2))*d^2+1/2*d*p*x^2/e^2+1/3*ln(c)/e*x^3-ln(c)*d^3/e^4*ln(e*x+d)-1/2*ln(c)/e^2*x^2*d+ln(c)/e
^3*x*d^2-49/18*p/e^4*d^3-1/2/b*p/e^2*a*d*ln(b*(e*x+d)^2-2*(e*x+d)*b*d+a*e^2+b*d^2)-2/3/b*p/e*a^2/(a*b)^(1/2)*a
rctan(1/2*(2*b*(e*x+d)-2*b*d)/e/(a*b)^(1/2))+1/4*I*Pi*csgn(I*c*(b*x^2+a)^p)^3/e^2*x^2*d+1/2*I*Pi*csgn(I*c*(b*x
^2+a)^p)^3*d^3/e^4*ln(e*x+d)+1/6*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2/e*x^3-1/2*I*Pi*csgn(I*c*(b*x
^2+a)^p)^3/e^3*x*d^2+p/e^4*d^3*ln(e*x+d)*ln((e*(-a*b)^(1/2)-b*(e*x+d)+b*d)/(e*(-a*b)^(1/2)+b*d))+p/e^4*d^3*ln(
e*x+d)*ln((e*(-a*b)^(1/2)+b*(e*x+d)-b*d)/(e*(-a*b)^(1/2)-b*d))+p/e^4*d^3*dilog((e*(-a*b)^(1/2)-b*(e*x+d)+b*d)/
(e*(-a*b)^(1/2)+b*d))+p/e^4*d^3*dilog((e*(-a*b)^(1/2)+b*(e*x+d)-b*d)/(e*(-a*b)^(1/2)-b*d))+1/4*I*Pi*csgn(I*(b*
x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)/e^2*x^2*d+1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*
c)*d^3/e^4*ln(e*x+d)-1/6*I*Pi*csgn(I*c*(b*x^2+a)^p)^3/e*x^3+1/3*ln((b*x^2+a)^p)/e*x^3+ln((b*x^2+a)^p)/e^3*x*d^
2+2/3*a*p*x/b/e+1/6*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)/e*x^3+2/3/b*p/e^2*a*d-2/9*p*x^3/e-1/2*I*Pi*csgn(I*(
b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)/e^3*x*d^2-2*d^2*p*x/e^3-1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)
*d^3/e^4*ln(e*x+d)-1/4*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)/e^2*x^2*d+1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*
(b*x^2+a)^p)^2/e^3*x*d^2+1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)/e^3*x*d^2-1/6*I*Pi*csgn(I*(b*x^2+a)^p)*csg
n(I*c*(b*x^2+a)^p)*csgn(I*c)/e*x^3-1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2*d^3/e^4*ln(e*x+d)-1/4*
I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2/e^2*x^2*d

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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^3*log(c*(b*x^2+a)^p)/(e*x+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{x^{3} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)