Integrand size = 19, antiderivative size = 371 \[ \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx=-\frac {c^2 x}{3 b d^2}+\frac {c x^2}{6 b d}-\frac {x^3}{9 b}+\frac {c^3 \log (c+d x)}{3 b d^3}+\frac {x^3 \log (c+d x)}{3 b}-\frac {a \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2} \]
-1/3*c^2*x/b/d^2+1/6*c*x^2/b/d-1/9*x^3/b+1/3*c^3*ln(d*x+c)/b/d^3+1/3*x^3*l n(d*x+c)/b-1/3*a*ln(-d*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*c-a^(1/3)*d))*ln(d*x+c )/b^2-1/3*a*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)/b^2-1/3*a*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))*ln(d*x+c)/b^2-1/3*a*polylog(2,b^(1/3)* (d*x+c)/(b^(1/3)*c-a^(1/3)*d))/b^2-1/3*a*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3 )*c+(-1)^(1/3)*a^(1/3)*d))/b^2-1/3*a*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c- (-1)^(2/3)*a^(1/3)*d))/b^2
Time = 0.18 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.01 \[ \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx=-\frac {c^2 x}{3 b d^2}+\frac {c x^2}{6 b d}-\frac {x^3}{9 b}+\frac {c^3 \log (c+d x)}{3 b d^3}+\frac {x^3 \log (c+d x)}{3 b}-\frac {a \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \log \left (-\frac {(-1)^{2/3} d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2} \]
-1/3*(c^2*x)/(b*d^2) + (c*x^2)/(6*b*d) - x^3/(9*b) + (c^3*Log[c + d*x])/(3 *b*d^3) + (x^3*Log[c + d*x])/(3*b) - (a*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b ^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^2) - (a*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*b^2) - (a*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*b^2) - (a*PolyLog[2, ( b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1 /3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*b^2) - (a*PolyLog[2 , (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*b^2)
Time = 0.74 (sec) , antiderivative size = 371, 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.
| \(\displaystyle \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {x^2 \log (c+d x)}{b}-\frac {a x^2 \log (c+d x)}{b \left (a+b x^3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \log (c+d x) \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2}-\frac {a \log (c+d x) \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 b^2}+\frac {c^3 \log (c+d x)}{3 b d^3}-\frac {c^2 x}{3 b d^2}+\frac {x^3 \log (c+d x)}{3 b}+\frac {c x^2}{6 b d}-\frac {x^3}{9 b}\) |
-1/3*(c^2*x)/(b*d^2) + (c*x^2)/(6*b*d) - x^3/(9*b) + (c^3*Log[c + d*x])/(3 *b*d^3) + (x^3*Log[c + d*x])/(3*b) - (a*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b ^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^2) - (a*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* b^2) - (a*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*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d* x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1/3)* (c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*b^2)
3.3.83.3.1 Defintions of rubi rules used
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 = 0.65 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.41
| method | result | size |
| risch | \(-\frac {c^{2} x}{3 b \,d^{2}}-\frac {11 c^{3}}{18 d^{3} b}+\frac {c \,x^{2}}{6 b d}+\frac {x^{3} \ln \left (d x +c \right )}{3 b}+\frac {c^{3} \ln \left (d x +c \right )}{3 b \,d^{3}}-\frac {x^{3}}{9 b}-\frac {a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 b^{2}}\) | \(153\) |
| derivativedivides | \(\frac {\frac {d^{3} \left (c^{2} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )-2 c \left (\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}\right )+\frac {\left (d x +c \right )^{3} \ln \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{3}}{9}\right )}{b}-\frac {a \,d^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 b^{2}}}{d^{6}}\) | \(170\) |
| default | \(\frac {\frac {d^{3} \left (c^{2} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )-2 c \left (\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}\right )+\frac {\left (d x +c \right )^{3} \ln \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{3}}{9}\right )}{b}-\frac {a \,d^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 b^{2}}}{d^{6}}\) | \(170\) |
| parts | \(\frac {x^{3} \ln \left (d x +c \right )}{3 b}-\frac {\ln \left (d x +c \right ) a \ln \left (b \,x^{3}+a \right )}{3 b^{2}}-\frac {d \left (\frac {x^{3}}{3 b d}-\frac {x^{2} c}{2 b \,d^{2}}+\frac {x \,c^{2}}{b \,d^{3}}-\frac {c^{3} \ln \left (d x +c \right )}{b \,d^{4}}-\frac {a \ln \left (d x +c \right ) \ln \left (b \,x^{3}+a \right )}{b^{2} d}+\frac {a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{b^{2} d}\right )}{3}\) | \(193\) |
-1/3*c^2*x/b/d^2-11/18/d^3/b*c^3+1/6*c*x^2/b/d+1/3*x^3*ln(d*x+c)/b+1/3*c^3 *ln(d*x+c)/b/d^3-1/9*x^3/b-1/3*a/b^2*sum(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+di log((-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) )
\[ \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
Timed out. \[ \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
\[ \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
Timed out. \[ \int \frac {x^5 \log (c+d x)}{a+b x^3} \, dx=\int \frac {x^5\,\ln \left (c+d\,x\right )}{b\,x^3+a} \,d x \]