3.3.48 \(\int \frac {x^2 \log (c (a+\frac {b}{x^2})^p)}{d+e x} \, dx\) [248]

Optimal. Leaf size=353 \[ -\frac {2 \sqrt {b} d p \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^2}-\frac {d x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 e}+\frac {d^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^3}+\frac {2 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e^3}+\frac {b p \log \left (b+a x^2\right )}{2 a e}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^3}-\frac {d^2 p \text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^3}+\frac {2 d^2 p \text {Li}_2\left (1+\frac {e x}{d}\right )}{e^3} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 353, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 12, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {2516, 2498, 269, 211, 2505, 266, 2512, 2463, 2441, 2352, 2440, 2438} \begin {gather*} -\frac {d^2 p \text {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^3}-\frac {d^2 p \text {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^3}+\frac {2 d^2 p \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^3}-\frac {2 \sqrt {b} d p \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e^2}+\frac {d^2 \log (d+e x) \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 )}{2 e}-\frac {d^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^3}+\frac {b p \log \left (a x^2+b\right )}{2 a e}+\frac {2 d^2 p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]
time = 0.16, size = 319, normalized size = 0.90 \begin {gather*} \frac {\frac {4 \sqrt {b} d e p \tan ^{-1}\left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {a}}-2 d e x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+e^2 x^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )+\frac {b e^2 p \left (\log \left (a+\frac {b}{x^2}\right )+2 \log (x)\right )}{a}+2 d^2 \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)+2 d^2 p \left (2 \log \left (-\frac {e x}{d}\right ) \log (d+e x)-\log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)-\log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)-\text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )-\text {Li}_2\left (\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )+2 \text {Li}_2\left (1+\frac {e x}{d}\right )\right )}{2 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{e x +d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{d + e x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**2*log(c*(a + b/x**2)**p)/(d + e*x), x)

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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