Integrand size = 19, antiderivative size = 383 \[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {\sqrt [3]{a} \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 b^{4/3}}-\frac {\sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 b^{4/3}} \]
-x/b+(d*x+c)*ln(d*x+c)/b/d-1/3*a^(1/3)*ln(-d*(a^(1/3)+b^(1/3)*x)/(b^(1/3)* c-a^(1/3)*d))*ln(d*x+c)/b^(4/3)-1/3*(-1)^(2/3)*a^(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)/b^(4/3)+1/3*( -1)^(1/3)*a^(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)/b^(4/3)-1/3*a^(1/3)*polylog(2,b^(1/3)*(d*x+c)/(b^ (1/3)*c-a^(1/3)*d))/b^(4/3)+1/3*(-1)^(1/3)*a^(1/3)*polylog(2,(-1)^(2/3)*b^ (1/3)*(d*x+c)/((-1)^(2/3)*b^(1/3)*c-a^(1/3)*d))/b^(4/3)-1/3*(-1)^(2/3)*a^( 1/3)*polylog(2,(-1)^(1/3)*b^(1/3)*(d*x+c)/((-1)^(1/3)*b^(1/3)*c+a^(1/3)*d) )/b^(4/3)
Time = 0.09 (sec) , antiderivative size = 369, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=\frac {-3 \sqrt [3]{b} d x+3 \sqrt [3]{b} c \log (c+d x)+3 \sqrt [3]{b} d x \log (c+d x)-\sqrt [3]{a} d \log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-(-1)^{2/3} \sqrt [3]{a} d \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)+\sqrt [3]{-1} \sqrt [3]{a} d \log \left (\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{-(-1)^{2/3} \sqrt [3]{b} c+\sqrt [3]{a} d}\right ) \log (c+d x)-\sqrt [3]{a} d \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+\sqrt [3]{-1} \sqrt [3]{a} d \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )-(-1)^{2/3} \sqrt [3]{a} d \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 b^{4/3} d} \]
(-3*b^(1/3)*d*x + 3*b^(1/3)*c*Log[c + d*x] + 3*b^(1/3)*d*x*Log[c + d*x] - a^(1/3)*d*Log[(d*(a^(1/3) + b^(1/3)*x))/(-(b^(1/3)*c) + a^(1/3)*d)]*Log[c + d*x] - (-1)^(2/3)*a^(1/3)*d*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)^(1/3)*a^(1/3)*d*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] - a^(1/3)*d*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1 /3)*d)] + (-1)^(1/3)*a^(1/3)*d*PolyLog[2, ((-1)^(2/3)*b^(1/3)*(c + d*x))/( (-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)] - (-1)^(2/3)*a^(1/3)*d*PolyLog[2, ((-1) ^(1/3)*b^(1/3)*(c + d*x))/((-1)^(1/3)*b^(1/3)*c + a^(1/3)*d)])/(3*b^(4/3)* d)
Time = 0.65 (sec) , antiderivative size = 383, 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.
| \(\displaystyle \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {\log (c+d x)}{b}-\frac {a \log (c+d x)}{b \left (a+b x^3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{b} (c+d x)}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{3 b^{4/3}}-\frac {\sqrt [3]{a} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{a} \log (c+d x) \log \left (\frac {d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{a} d+\sqrt [3]{-1} \sqrt [3]{b} c}\right )}{3 b^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{a} \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^{4/3}}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {x}{b}\) |
-(x/b) + ((c + d*x)*Log[c + d*x])/(b*d) - (a^(1/3)*Log[-((d*(a^(1/3) + b^( 1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^(4/3)) - ((-1)^(2/3) *a^(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*b^(4/3)) + ((-1)^(1/3)*a^(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*b^(4/3)) - (a^(1/3)*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a ^(1/3)*d)])/(3*b^(4/3)) + ((-1)^(1/3)*a^(1/3)*PolyLog[2, ((-1)^(2/3)*b^(1/ 3)*(c + d*x))/((-1)^(2/3)*b^(1/3)*c - a^(1/3)*d)])/(3*b^(4/3)) - ((-1)^(2/ 3)*a^(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*b^(4/3))
3.3.88.3.1 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 = 0.57 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.33
| method | result | size |
| derivativedivides | \(\frac {\frac {d^{3} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )}{b}-\frac {a \,d^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b^{2}}}{d^{4}}\) | \(127\) |
| default | \(\frac {\frac {d^{3} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )}{b}-\frac {a \,d^{6} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b^{2}}}{d^{4}}\) | \(127\) |
| risch | \(\frac {x \ln \left (d x +c \right )}{b}+\frac {\ln \left (d x +c \right ) c}{d b}-\frac {x}{b}-\frac {c}{d b}-\frac {d^{2} a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{3 b^{2}}\) | \(136\) |
1/d^4*(d^3/b*((d*x+c)*ln(d*x+c)-d*x-c)-1/3*a*d^6/b^2*sum(1/(_R1^2-2*_R1*c+ c^2)*(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)))
\[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x^{3} \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
Timed out. \[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x^{3} \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
\[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=\int { \frac {x^{3} \log \left (d x + c\right )}{b x^{3} + a} \,d x } \]
Timed out. \[ \int \frac {x^3 \log (c+d x)}{a+b x^3} \, dx=\int \frac {x^3\,\ln \left (c+d\,x\right )}{b\,x^3+a} \,d x \]