3.309 \(\int \frac{\log (a+b x)}{c+\frac{d}{x^2}} \, dx\)

Optimal. Leaf size=247 \[ \frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x)}{a \sqrt{-c}-b \sqrt{d}}\right )}{2 (-c)^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x)}{a \sqrt{-c}+b \sqrt{d}}\right )}{2 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{a \sqrt{-c}+b \sqrt{d}}\right )}{2 (-c)^{3/2}}+\frac{\sqrt{d} \log (a+b x) \log \left (-\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{a \sqrt{-c}-b \sqrt{d}}\right )}{2 (-c)^{3/2}}+\frac{(a+b x) \log (a+b x)}{b c}-\frac{x}{c} \]

[Out]

-(x/c) + ((a + b*x)*Log[a + b*x])/(b*c) - (Sqrt[d]*Log[a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/(a*Sqrt[-c] + b
*Sqrt[d])])/(2*(-c)^(3/2)) + (Sqrt[d]*Log[a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/(a*Sqrt[-c] - b*Sqrt[d]))]
)/(2*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] - b*Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[
d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2))

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Rubi [A]  time = 0.336548, antiderivative size = 247, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2409, 2389, 2295, 2394, 2393, 2391} \[ \frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x)}{a \sqrt{-c}-b \sqrt{d}}\right )}{2 (-c)^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x)}{a \sqrt{-c}+b \sqrt{d}}\right )}{2 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{a \sqrt{-c}+b \sqrt{d}}\right )}{2 (-c)^{3/2}}+\frac{\sqrt{d} \log (a+b x) \log \left (-\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{a \sqrt{-c}-b \sqrt{d}}\right )}{2 (-c)^{3/2}}+\frac{(a+b x) \log (a+b x)}{b c}-\frac{x}{c} \]

Antiderivative was successfully verified.

[In]

Int[Log[a + b*x]/(c + d/x^2),x]

[Out]

-(x/c) + ((a + b*x)*Log[a + b*x])/(b*c) - (Sqrt[d]*Log[a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/(a*Sqrt[-c] + b
*Sqrt[d])])/(2*(-c)^(3/2)) + (Sqrt[d]*Log[a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/(a*Sqrt[-c] - b*Sqrt[d]))]
)/(2*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] - b*Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[
d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2))

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2394

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 2393

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 2391

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]

Rubi steps

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

Mathematica [A]  time = 0.178514, size = 247, normalized size = 1. \[ \frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x)}{a \sqrt{-c}-b \sqrt{d}}\right )}{2 (-c)^{3/2}}-\frac{\sqrt{d} \text{PolyLog}\left (2,\frac{\sqrt{-c} (a+b x)}{a \sqrt{-c}+b \sqrt{d}}\right )}{2 (-c)^{3/2}}-\frac{\sqrt{d} \log (a+b x) \log \left (\frac{b \left (\sqrt{d}-\sqrt{-c} x\right )}{a \sqrt{-c}+b \sqrt{d}}\right )}{2 (-c)^{3/2}}+\frac{\sqrt{d} \log (a+b x) \log \left (-\frac{b \left (\sqrt{-c} x+\sqrt{d}\right )}{a \sqrt{-c}-b \sqrt{d}}\right )}{2 (-c)^{3/2}}+\frac{(a+b x) \log (a+b x)}{b c}-\frac{x}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[a + b*x]/(c + d/x^2),x]

[Out]

-(x/c) + ((a + b*x)*Log[a + b*x])/(b*c) - (Sqrt[d]*Log[a + b*x]*Log[(b*(Sqrt[d] - Sqrt[-c]*x))/(a*Sqrt[-c] + b
*Sqrt[d])])/(2*(-c)^(3/2)) + (Sqrt[d]*Log[a + b*x]*Log[-((b*(Sqrt[d] + Sqrt[-c]*x))/(a*Sqrt[-c] - b*Sqrt[d]))]
)/(2*(-c)^(3/2)) + (Sqrt[d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] - b*Sqrt[d])])/(2*(-c)^(3/2)) - (Sqrt[
d]*PolyLog[2, (Sqrt[-c]*(a + b*x))/(a*Sqrt[-c] + b*Sqrt[d])])/(2*(-c)^(3/2))

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Maple [A]  time = 0.07, size = 248, normalized size = 1. \begin{align*}{\frac{\ln \left ( bx+a \right ) x}{c}}+{\frac{\ln \left ( bx+a \right ) a}{bc}}-{\frac{x}{c}}-{\frac{a}{bc}}-{\frac{d\ln \left ( bx+a \right ) }{2\,c}\ln \left ({ \left ( b\sqrt{-cd}-c \left ( bx+a \right ) +ac \right ) \left ( b\sqrt{-cd}+ac \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}}+{\frac{d\ln \left ( bx+a \right ) }{2\,c}\ln \left ({ \left ( b\sqrt{-cd}+c \left ( bx+a \right ) -ac \right ) \left ( b\sqrt{-cd}-ac \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}}-{\frac{d}{2\,c}{\it dilog} \left ({ \left ( b\sqrt{-cd}-c \left ( bx+a \right ) +ac \right ) \left ( b\sqrt{-cd}+ac \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}}+{\frac{d}{2\,c}{\it dilog} \left ({ \left ( b\sqrt{-cd}+c \left ( bx+a \right ) -ac \right ) \left ( b\sqrt{-cd}-ac \right ) ^{-1}} \right ){\frac{1}{\sqrt{-cd}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{2} \log \left (b x + a\right )}{c x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (b x + a\right )}{c + \frac{d}{x^{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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