\(\int \frac {\log (c (a+b x^2)^p)}{d+e x} \, dx\) [229]

Optimal result

Integrand size = 20, antiderivative size = 201 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=-\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2512, 266, 2463, 2441, 2440, 2438} \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e}-\frac {p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{e}-\frac {p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{e} \]

[In]

Int[Log[c*(a + b*x^2)^p]/(d + e*x),x]

[Out]

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

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 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 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]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=-\frac {p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e}-\frac {p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e}+\frac {\log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.01 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.98

method	result	size
parts	\(\frac {\ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e}-\frac {2 p b \left (\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}+\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{e}\)	\(196\)
risch	\(\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (e x +d \right )}{e}-\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e}-\frac {p \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e}-\frac {p \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e}+\frac {\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \ln \left (e x +d \right )}{e}\)	\(331\)

[In]

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

[Out]

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

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(log(c*(a + b*x^2)^p)/(d + e*x),x)

[Out]

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