\(\int \frac {x \log (c (a+\frac {b}{x^2})^p)}{d+e x} \, dx\) [249]

Optimal result

Integrand size = 21, antiderivative size = 291 \[ \int \frac {x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e}+\frac {x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e}-\frac {d \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)}{e^2}-\frac {2 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2}+\frac {d p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)}{e^2}+\frac {d p \log \left (-\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{\sqrt {-a} d-\sqrt {b} e}\right ) \log (d+e x)}{e^2}+\frac {d p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^2}+\frac {d p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^2}-\frac {2 d p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {2516, 2498, 269, 211, 2512, 266, 2463, 2441, 2352, 2440, 2438} \[ \int \frac {x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{\sqrt {a} e}-\frac {d \log (d+e x) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{e}+\frac {d p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^2}+\frac {d p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right )}{e^2}+\frac {d p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a} x+\sqrt {b}\right )}{\sqrt {-a} d-\sqrt {b} e}\right )}{e^2}-\frac {2 d p \operatorname {PolyLog}\left (2,\frac {e x}{d}+1\right )}{e^2}-\frac {2 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)}{e^2} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.93 \[ \int \frac {x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{d+e x} \, dx=\frac {-\frac {2 \sqrt {b} e p \arctan \left (\frac {\sqrt {b}}{\sqrt {a} x}\right )}{\sqrt {a}}+e x \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )-d \log \left (c \left (a+\frac {b}{x^2}\right )^p\right ) \log (d+e x)-2 d p \log \left (-\frac {e x}{d}\right ) \log (d+e x)+d p \log \left (\frac {e \left (\sqrt {b}-\sqrt {-a} x\right )}{\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)+d p \log \left (\frac {e \left (\sqrt {b}+\sqrt {-a} x\right )}{-\sqrt {-a} d+\sqrt {b} e}\right ) \log (d+e x)+d p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d-\sqrt {b} e}\right )+d p \operatorname {PolyLog}\left (2,\frac {\sqrt {-a} (d+e x)}{\sqrt {-a} d+\sqrt {b} e}\right )-2 d p \operatorname {PolyLog}\left (2,1+\frac {e x}{d}\right )}{e^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.02

method	result	size
parts	\(\frac {x \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right )}{e}-\frac {d \ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \ln \left (e x +d \right )}{e^{2}}+2 p b \,e^{2} \left (\frac {\arctan \left (\frac {-2 a d +2 a \left (e x +d \right )}{2 e \sqrt {a b}}\right )}{e^{3} \sqrt {a b}}+\frac {d \left (-\frac {\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{b \,e^{2}}-\frac {a \left (-\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\ln \left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )\right )}{2 a}-\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}+a d -a \left (e x +d \right )}{e \sqrt {-a b}+a d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}-a d +a \left (e x +d \right )}{e \sqrt {-a b}-a d}\right )}{2 a}\right )}{b \,e^{2}}\right )}{e^{2}}\right )\)	\(296\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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