Integrand size = 19, antiderivative size = 398 \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}+\frac {\sqrt [3]{b} \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^{4/3}}+\frac {\sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{4/3}} \] Output:
d*ln(x)/a/c-d*ln(d*x+c)/a/c-ln(d*x+c)/a/x+1/3*b^(1/3)*ln(-d*(a^(1/3)+b^(1/ 3)*x)/(b^(1/3)*c-a^(1/3)*d))*ln(d*x+c)/a^(4/3)-1/3*(-1)^(1/3)*b^(1/3)*ln(d *(a^(1/3)-(-1)^(1/3)*b^(1/3)*x)/((-1)^(1/3)*b^(1/3)*c+a^(1/3)*d))*ln(d*x+c )/a^(4/3)+1/3*(-1)^(2/3)*b^(1/3)*ln(-d*(a^(1/3)+(-1)^(2/3)*b^(1/3)*x)/((-1 )^(2/3)*b^(1/3)*c-a^(1/3)*d))*ln(d*x+c)/a^(4/3)+1/3*b^(1/3)*polylog(2,b^(1 /3)*(d*x+c)/(b^(1/3)*c-a^(1/3)*d))/a^(4/3)+1/3*(-1)^(2/3)*b^(1/3)*polylog( 2,(-1)^(2/3)*b^(1/3)*(d*x+c)/((-1)^(2/3)*b^(1/3)*c-a^(1/3)*d))/a^(4/3)-1/3 *(-1)^(1/3)*b^(1/3)*polylog(2,(-1)^(1/3)*b^(1/3)*(d*x+c)/((-1)^(1/3)*b^(1/ 3)*c+a^(1/3)*d))/a^(4/3)
Time = 0.21 (sec) , antiderivative size = 378, normalized size of antiderivative = 0.95 \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\frac {3 \sqrt [3]{a} d x \log (x)-3 \sqrt [3]{a} c \log (c+d x)-3 \sqrt [3]{a} d x \log (c+d x)+\sqrt [3]{b} c x \log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-\sqrt [3]{-1} \sqrt [3]{b} c x \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+(-1)^{2/3} \sqrt [3]{b} c x \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+\sqrt [3]{b} c x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+(-1)^{2/3} \sqrt [3]{b} c x \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )-\sqrt [3]{-1} \sqrt [3]{b} c x \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{4/3} c x} \] Input:
Integrate[Log[c + d*x]/(x^2*(a + b*x^3)),x]
Output:
(3*a^(1/3)*d*x*Log[x] - 3*a^(1/3)*c*Log[c + d*x] - 3*a^(1/3)*d*x*Log[c + d *x] + b^(1/3)*c*x*Log[(d*(a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*c) + a^(1/3)*d) ]*Log[c + d*x] - (-1)^(1/3)*b^(1/3)*c*x*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/ 3)*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x] + (-1)^(2/3)*b^(1/ 3)*c*x*Log[(d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(-((-1)^(2/3)*b^(1/3)*c) + a^(1/3)*d)]*Log[c + d*x] + b^(1/3)*c*x*PolyLog[2, (b^(1/3)*(c + d*x))/(b^ (1/3)*c - a^(1/3)*d)] + (-1)^(2/3)*b^(1/3)*c*x*PolyLog[2, ((-1)^(2/3)*b^(1 /3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)] - (-1)^(1/3)*b^(1/3)*c* x*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3 )*d)])/(3*a^(4/3)*c*x)
Time = 1.29 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2863, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {\log (c+d x)}{a x^2}-\frac {b x \log (c+d x)}{a \left (a+b x^3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {\sqrt [3]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{b} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 a^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^{4/3}}+\frac {d \log (x)}{a c}-\frac {d \log (c+d x)}{a c}-\frac {\log (c+d x)}{a x}\) |
Input:
Int[Log[c + d*x]/(x^2*(a + b*x^3)),x]
Output:
(d*Log[x])/(a*c) - (d*Log[c + d*x])/(a*c) - Log[c + d*x]/(a*x) + (b^(1/3)* Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3 *a^(4/3)) - ((-1)^(1/3)*b^(1/3)*Log[(d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/( (-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)]*Log[c + d*x])/(3*a^(4/3)) + ((-1)^(2/3) *b^(1/3)*Log[-((d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^(4/3)) + (b^(1/3)*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*a^(4/3)) + ((-1)^(2/3)*b^(1/3)*PolyL og[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/ (3*a^(4/3)) - ((-1)^(1/3)*b^(1/3)*PolyLog[2, ((-1)^(1/3)*b^(1/3)*(c + d*x) )/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*a^(4/3))
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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.61 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.31
| method | result | size |
| derivativedivides | \(d \left (\frac {\frac {\ln \left (-d x \right )}{c}-\frac {\ln \left (d x +c \right ) \left (d x +c \right )}{c d x}}{a}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}}{3 a}\right )\) | \(124\) |
| default | \(d \left (\frac {\frac {\ln \left (-d x \right )}{c}-\frac {\ln \left (d x +c \right ) \left (d x +c \right )}{c d x}}{a}+\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}}{3 a}\right )\) | \(124\) |
| risch | \(\frac {d \ln \left (-d x \right )}{a c}-\frac {d \ln \left (d x +c \right )}{a c}-\frac {\ln \left (d x +c \right )}{a x}+\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{3 a}\) | \(129\) |
Input:
int(ln(d*x+c)/x^2/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
d*((1/c*ln(-d*x)-ln(d*x+c)*(d*x+c)/c/d/x)/a+1/3*sum(1/(-_R1+c)*(ln(d*x+c)* ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1)),_R1=RootOf(_Z^3*b-3*_Z^2*b*c +3*_Z*b*c^2+a*d^3-b*c^3))/a)
\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}} \,d x } \] Input:
integrate(log(d*x+c)/x^2/(b*x^3+a),x, algorithm="fricas")
Output:
integral(log(d*x + c)/(b*x^5 + a*x^2), x)
Timed out. \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(ln(d*x+c)/x**2/(b*x**3+a),x)
Output:
Timed out
\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}} \,d x } \] Input:
integrate(log(d*x+c)/x^2/(b*x^3+a),x, algorithm="maxima")
Output:
integrate(log(d*x + c)/((b*x^3 + a)*x^2), x)
\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{2}} \,d x } \] Input:
integrate(log(d*x+c)/x^2/(b*x^3+a),x, algorithm="giac")
Output:
integrate(log(d*x + c)/((b*x^3 + a)*x^2), x)
Timed out. \[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x^2\,\left (b\,x^3+a\right )} \,d x \] Input:
int(log(c + d*x)/(x^2*(a + b*x^3)),x)
Output:
int(log(c + d*x)/(x^2*(a + b*x^3)), x)
\[ \int \frac {\log (c+d x)}{x^2 \left (a+b x^3\right )} \, dx=\int \frac {\mathrm {log}\left (d x +c \right )}{b \,x^{5}+a \,x^{2}}d x \] Input:
int(log(d*x+c)/x^2/(b*x^3+a),x)
Output:
int(log(c + d*x)/(a*x**2 + b*x**5),x)