\(\int \frac {\log (c (a+b x^2)^p)}{x^2 (d+e x)} \, dx\) [231]

Optimal result

Integrand size = 23, antiderivative size = 306 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {e p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {2516, 2505, 211, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=\frac {2 \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^2}+\frac {e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^2}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{d x}-\frac {e p \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2}-\frac {e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{d^2}-\frac {e p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{d^2} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.92 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x^2 (d+e x)} \, dx=-\frac {-\frac {4 \sqrt {b} d p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {a}}+2 e p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+2 e p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+\frac {2 d \log \left (c \left (a+b x^2\right )^p\right )}{x}+e \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )-2 e \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+2 e p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+e p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.16

method	result	size
parts	\(\frac {e \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d^{2}}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d x}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-2 p b \left (\frac {e \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{d^{2}}-\frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d \sqrt {a b}}-\frac {e \left (\frac {\ln \left (x \right ) \left (\ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 b}\right )}{d^{2}}\right )\)	\(354\)
risch	\(\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (e x +d \right )}{d^{2}}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{d x}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) e \ln \left (x \right )}{d^{2}}-\frac {p e \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{2}}-\frac {p e \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{2}}-\frac {p e \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d^{2}}-\frac {p e \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d^{2}}+\frac {2 p b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{d \sqrt {a b}}+\frac {p e \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\frac {p e \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d^{2}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {e \ln \left (e x +d \right )}{d^{2}}-\frac {1}{d x}-\frac {e \ln \left (x \right )}{d^{2}}\right )\)	\(514\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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