\(\int \frac {\log (c (a+b x^2)^p)}{x (d+e x)} \, dx\) [230]

Optimal result

Integrand size = 23, antiderivative size = 247 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x (d+e x)} \, dx=\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{d}+\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{d}+\frac {\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d}-\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2516, 2504, 2441, 2352, 2512, 266, 2463, 2440, 2438} \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x (d+e x)} \, dx=-\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d}+\frac {\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{d}+\frac {p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{d}+\frac {p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{d}+\frac {p \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 d} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.93 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{x (d+e x)} \, dx=\frac {2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+2 p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+\log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )-2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )+p \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 d} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.26

method	result	size
parts	\(-\frac {\ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{d}+\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{d}-2 p b \left (\frac {\frac {\ln \left (x \right ) \left (\ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )+\operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{2 b}}{d}-\frac {\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}}{d}\right )\)	\(311\)
risch	\(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (e x +d \right )}{d}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{d}-\frac {p \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {p \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {p \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}-\frac {p \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{d}+\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d}+\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{d}+\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{d}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (-\frac {\ln \left (e x +d \right )}{d}+\frac {\ln \left (x \right )}{d}\right )\)	\(455\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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