3.4.9 \(\int \frac {\log (a+b x)}{c+\frac {d}{x^2}} \, dx\) [309]

Optimal. Leaf size=247 \[ -\frac {x}{c}+\frac {(a+b x) \log (a+b x)}{b c}-\frac {\sqrt {d} \log (a+b x) \log \left (\frac {b \left (\sqrt {d}-\sqrt {-c} x\right )}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \log (a+b x) \log \left (-\frac {b \left (\sqrt {d}+\sqrt {-c} x\right )}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}+\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}-b \sqrt {d}}\right )}{2 (-c)^{3/2}}-\frac {\sqrt {d} \text {Li}_2\left (\frac {\sqrt {-c} (a+b x)}{a \sqrt {-c}+b \sqrt {d}}\right )}{2 (-c)^{3/2}} \]

[Out]

-x/c+(b*x+a)*ln(b*x+a)/b/c+1/2*ln(b*x+a)*ln(-b*(x*(-c)^(1/2)+d^(1/2))/(a*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(
3/2)-1/2*ln(b*x+a)*ln(b*(-x*(-c)^(1/2)+d^(1/2))/(a*(-c)^(1/2)+b*d^(1/2)))*d^(1/2)/(-c)^(3/2)+1/2*polylog(2,(b*
x+a)*(-c)^(1/2)/(a*(-c)^(1/2)-b*d^(1/2)))*d^(1/2)/(-c)^(3/2)-1/2*polylog(2,(b*x+a)*(-c)^(1/2)/(a*(-c)^(1/2)+b*
d^(1/2)))*d^(1/2)/(-c)^(3/2)

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Rubi [A]
time = 0.25, antiderivative size = 247, normalized size of antiderivative = 1.00, 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 = {2456, 2436, 2332, 2441, 2440, 2438} \begin {gather*} \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} \end {gather*}

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 2332

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

Rule 2436

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 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 2456

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]))

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 {\text {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} \text {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} \text {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*}

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Mathematica [A]
time = 0.13, size = 247, normalized size = 1.00 \begin {gather*} -\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 {gather*}

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 = 0.35, size = 220, normalized size = 0.89

method	result	size
derivativedivides	\(\frac {\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{c}-\frac {\left (\frac {\ln \left (b x +a \right ) \left (\ln \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )-\ln \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )\right )}{2 b \sqrt {-c d}}+\frac {\dilog \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )-\dilog \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 b \sqrt {-c d}}\right ) d \,b^{2}}{c}}{b}\)	\(220\)
default	\(\frac {\frac {\left (b x +a \right ) \ln \left (b x +a \right )-b x -a}{c}-\frac {\left (\frac {\ln \left (b x +a \right ) \left (\ln \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )-\ln \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )\right )}{2 b \sqrt {-c d}}+\frac {\dilog \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )-\dilog \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 b \sqrt {-c d}}\right ) d \,b^{2}}{c}}{b}\)	\(220\)
risch	\(\frac {\ln \left (b x +a \right ) x}{c}+\frac {\ln \left (b x +a \right ) a}{b c}-\frac {x}{c}-\frac {a}{b c}-\frac {d \ln \left (b x +a \right ) \ln \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )}{2 c \sqrt {-c d}}+\frac {d \ln \left (b x +a \right ) \ln \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 c \sqrt {-c d}}-\frac {d \dilog \left (\frac {b \sqrt {-c d}+c a -c \left (b x +a \right )}{b \sqrt {-c d}+c a}\right )}{2 c \sqrt {-c d}}+\frac {d \dilog \left (\frac {b \sqrt {-c d}-c a +c \left (b x +a \right )}{b \sqrt {-c d}-c a}\right )}{2 c \sqrt {-c d}}\)	\(248\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.56, size = 298, normalized size = 1.21 \begin {gather*} -{\left (\frac {d \arctan \left (\frac {c x}{\sqrt {c d}}\right )}{\sqrt {c d} c} - \frac {x}{c}\right )} \log \left (b x + a\right ) - \frac {2 \, b c x - 2 \, a c \log \left (b x + a\right ) + {\left (b \arctan \left (\frac {{\left (b^{2} x + a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c + b^{2} d}, \frac {a b c x + a^{2} c}{a^{2} c + b^{2} d}\right ) \log \left (c x^{2} + d\right ) - b \arctan \left (\frac {\sqrt {c} x}{\sqrt {d}}\right ) \log \left (\frac {b^{2} c x^{2} + 2 \, a b c x + a^{2} c}{a^{2} c + b^{2} d}\right ) + i \, b {\rm Li}_2\left (-\frac {a b c x + b^{2} d + {\left (i \, b^{2} x - i \, a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c + 2 i \, a b \sqrt {c} \sqrt {d} - b^{2} d}\right ) - i \, b {\rm Li}_2\left (-\frac {a b c x + b^{2} d - {\left (i \, b^{2} x - i \, a b\right )} \sqrt {c} \sqrt {d}}{a^{2} c - 2 i \, a b \sqrt {c} \sqrt {d} - b^{2} d}\right )\right )} \sqrt {c} \sqrt {d}}{2 \, b c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d*arctan(c*x/sqrt(c*d))/(sqrt(c*d)*c) - x/c)*log(b*x + a) - 1/2*(2*b*c*x - 2*a*c*log(b*x + a) + (b*arctan2((
b^2*x + a*b)*sqrt(c)*sqrt(d)/(a^2*c + b^2*d), (a*b*c*x + a^2*c)/(a^2*c + b^2*d))*log(c*x^2 + d) - b*arctan(sqr
t(c)*x/sqrt(d))*log((b^2*c*x^2 + 2*a*b*c*x + a^2*c)/(a^2*c + b^2*d)) + I*b*dilog(-(a*b*c*x + b^2*d + (I*b^2*x
- I*a*b)*sqrt(c)*sqrt(d))/(a^2*c + 2*I*a*b*sqrt(c)*sqrt(d) - b^2*d)) - I*b*dilog(-(a*b*c*x + b^2*d - (I*b^2*x
- I*a*b)*sqrt(c)*sqrt(d))/(a^2*c - 2*I*a*b*sqrt(c)*sqrt(d) - b^2*d)))*sqrt(c)*sqrt(d))/(b*c^2)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \log {\left (a + b x \right )}}{c x^{2} + d}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (a+b\,x\right )}{c+\frac {d}{x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(a + b*x)/(c + d/x^2),x)

[Out]

int(log(a + b*x)/(c + d/x^2), x)

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