Integrand size = 19, antiderivative size = 414 \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=-\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac {b \operatorname {PolyLog}\left (2,1+\frac {d x}{c}\right )}{a^2} \] Output:
-1/6*d/a/c/x^2+1/3*d^2/a/c^2/x+1/3*d^3*ln(x)/a/c^3-1/3*d^3*ln(d*x+c)/a/c^3 -1/3*ln(d*x+c)/a/x^3-b*ln(-d*x/c)*ln(d*x+c)/a^2+1/3*b*ln(-d*(a^(1/3)+b^(1/ 3)*x)/(b^(1/3)*c-a^(1/3)*d))*ln(d*x+c)/a^2+1/3*b*ln(-d*((-1)^(2/3)*a^(1/3) +b^(1/3)*x)/(b^(1/3)*c-(-1)^(2/3)*a^(1/3)*d))*ln(d*x+c)/a^2+1/3*b*ln((-1)^ (1/3)*d*(a^(1/3)+(-1)^(2/3)*b^(1/3)*x)/(b^(1/3)*c+(-1)^(1/3)*a^(1/3)*d))*l n(d*x+c)/a^2+1/3*b*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c-a^(1/3)*d))/a^2+1/ 3*b*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c+(-1)^(1/3)*a^(1/3)*d))/a^2+1/3*b* polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c-(-1)^(2/3)*a^(1/3)*d))/a^2-b*polylog( 2,1+d*x/c)/a^2
Time = 0.24 (sec) , antiderivative size = 405, normalized size of antiderivative = 0.98 \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {(-1)^{2/3} d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac {d \left (\frac {1}{c x^2}-\frac {2 d}{c^2 x}-\frac {2 d^2 \log (x)}{c^3}+\frac {2 d^2 \log (c+d x)}{c^3}\right )}{6 a}-\frac {b \operatorname {PolyLog}\left (2,\frac {c+d x}{c}\right )}{a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2} \] Input:
Integrate[Log[c + d*x]/(x^4*(a + b*x^3)),x]
Output:
-1/3*Log[c + d*x]/(a*x^3) - (b*Log[-((d*x)/c)]*Log[c + d*x])/a^2 + (b*Log[ -((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*a^2 ) + (b*Log[-(((-1)^(2/3)*d*(a^(1/3) - (-1)^(1/3)*b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/(3*a^2) + (b*Log[((-1)^(1/3)*d*(a^(1 /3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[c + d *x])/(3*a^2) - (d*(1/(c*x^2) - (2*d)/(c^2*x) - (2*d^2*Log[x])/c^3 + (2*d^2 *Log[c + d*x])/c^3))/(6*a) - (b*PolyLog[2, (c + d*x)/c])/a^2 + (b*PolyLog[ 2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, ( b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*a^2) + (b*PolyL og[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*a^2)
Time = 1.25 (sec) , antiderivative size = 414, 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.
| \(\displaystyle \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {b^2 x^2 \log (c+d x)}{a^2 \left (a+b x^3\right )}-\frac {b \log (c+d x)}{a^2 x}+\frac {\log (c+d x)}{a x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac {b \operatorname {PolyLog}\left (2,\frac {d x}{c}+1\right )}{a^2}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \log (c+d x) \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \log (c+d x) \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}+\frac {d^2}{3 a c^2 x}-\frac {\log (c+d x)}{3 a x^3}-\frac {d}{6 a c x^2}\) |
Input:
Int[Log[c + d*x]/(x^4*(a + b*x^3)),x]
Output:
-1/6*d/(a*c*x^2) + d^2/(3*a*c^2*x) + (d^3*Log[x])/(3*a*c^3) - (d^3*Log[c + d*x])/(3*a*c^3) - Log[c + d*x]/(3*a*x^3) - (b*Log[-((d*x)/c)]*Log[c + d*x ])/a^2 + (b*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[ c + d*x])/(3*a^2) + (b*Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3) *c - (-1)^(2/3)*a^(1/3)*d))]*Log[c + d*x])/(3*a^2) + (b*Log[((-1)^(1/3)*d* (a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[ c + d*x])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3) *d)])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)* a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1) ^(2/3)*a^(1/3)*d)])/(3*a^2) - (b*PolyLog[2, 1 + (d*x)/c])/a^2
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 = 2.42 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(\frac {d^{2}}{3 a \,c^{2} x}-\frac {d}{6 a c \,x^{2}}+\frac {d^{3} \ln \left (-d x \right )}{3 a \,c^{3}}-\frac {d^{3} \ln \left (d x +c \right )}{3 a \,c^{3}}-\frac {\ln \left (d x +c \right )}{3 a \,x^{3}}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2}}-\frac {b \ln \left (-\frac {d x}{c}\right ) \ln \left (d x +c \right )}{a^{2}}-\frac {b \operatorname {dilog}\left (-\frac {d x}{c}\right )}{a^{2}}\) | \(186\) |
| derivativedivides | \(d^{3} \left (\frac {\frac {1}{3 c^{2} d x}-\frac {1}{6 c \,d^{2} x^{2}}+\frac {\ln \left (-d x \right )}{3 c^{3}}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (3 c^{2}-3 c \left (d x +c \right )+\left (d x +c \right )^{2}\right )}{3 c^{3} d^{3} x^{3}}}{a}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2} d^{3}}-\frac {\left (\operatorname {dilog}\left (-\frac {d x}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {d x}{c}\right )\right ) b}{a^{2} d^{3}}\right )\) | \(199\) |
| default | \(d^{3} \left (\frac {\frac {1}{3 c^{2} d x}-\frac {1}{6 c \,d^{2} x^{2}}+\frac {\ln \left (-d x \right )}{3 c^{3}}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (3 c^{2}-3 c \left (d x +c \right )+\left (d x +c \right )^{2}\right )}{3 c^{3} d^{3} x^{3}}}{a}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2} d^{3}}-\frac {\left (\operatorname {dilog}\left (-\frac {d x}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {d x}{c}\right )\right ) b}{a^{2} d^{3}}\right )\) | \(199\) |
| parts | \(-\frac {\ln \left (d x +c \right )}{3 a \,x^{3}}-\frac {\ln \left (d x +c \right ) b \ln \left (x \right )}{a^{2}}+\frac {\ln \left (d x +c \right ) b \ln \left (b \,x^{3}+a \right )}{3 a^{2}}-\frac {d \left (\frac {b \left (\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{3}+a \right )}{d}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 b c \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{d}\right )}{a^{2}}-\frac {-\frac {d^{2} \ln \left (d x +c \right )}{c^{3}}-\frac {1}{2 c \,x^{2}}+\frac {d^{2} \ln \left (x \right )}{c^{3}}+\frac {d}{c^{2} x}}{a}-\frac {3 b \operatorname {dilog}\left (\frac {d x +c}{c}\right )}{a^{2} d}-\frac {3 b \ln \left (x \right ) \ln \left (\frac {d x +c}{c}\right )}{a^{2} d}\right )}{3}\) | \(239\) |
Input:
int(ln(d*x+c)/x^4/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/3*d^2/a/c^2/x-1/6*d/a/c/x^2+1/3*d^3/a/c^3*ln(-d*x)-1/3*d^3*ln(d*x+c)/a/c ^3-1/3*ln(d*x+c)/a/x^3+1/3*b*sum(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^2-b* ln(-d*x/c)*ln(d*x+c)/a^2-b/a^2*dilog(-d*x/c)
\[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{4}} \,d x } \] Input:
integrate(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="fricas")
Output:
integral(log(d*x + c)/(b*x^7 + a*x^4), x)
Timed out. \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(ln(d*x+c)/x**4/(b*x**3+a),x)
Output:
Timed out
\[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{4}} \,d x } \] Input:
integrate(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="maxima")
Output:
integrate(log(d*x + c)/((b*x^3 + a)*x^4), x)
\[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{3} + a\right )} x^{4}} \,d x } \] Input:
integrate(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="giac")
Output:
integrate(log(d*x + c)/((b*x^3 + a)*x^4), x)
Timed out. \[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x^4\,\left (b\,x^3+a\right )} \,d x \] Input:
int(log(c + d*x)/(x^4*(a + b*x^3)),x)
Output:
int(log(c + d*x)/(x^4*(a + b*x^3)), x)
\[ \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx=\frac {-6 \left (\int \frac {\mathrm {log}\left (d x +c \right )}{b \,x^{4}+a x}d x \right ) b \,c^{3} x^{3}-2 \,\mathrm {log}\left (d x +c \right ) c^{3}-2 \,\mathrm {log}\left (d x +c \right ) d^{3} x^{3}+2 \,\mathrm {log}\left (x \right ) d^{3} x^{3}-c^{2} d x +2 c \,d^{2} x^{2}}{6 a \,c^{3} x^{3}} \] Input:
int(log(d*x+c)/x^4/(b*x^3+a),x)
Output:
( - 6*int(log(c + d*x)/(a*x + b*x**4),x)*b*c**3*x**3 - 2*log(c + d*x)*c**3 - 2*log(c + d*x)*d**3*x**3 + 2*log(x)*d**3*x**3 - c**2*d*x + 2*c*d**2*x** 2)/(6*a*c**3*x**3)