Integrand size = 19, antiderivative size = 498 \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}-\frac {a \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^2} \]
1/4*c^3*x/b/d^3-1/8*c^2*x^2/b/d^2+1/12*c*x^3/b/d-1/16*x^4/b-1/4*c^4*ln(d*x +c)/b/d^4+1/4*x^4*ln(d*x+c)/b-1/4*a*ln(d*((-a)^(1/4)-b^(1/4)*x)/(b^(1/4)*c +(-a)^(1/4)*d))*ln(d*x+c)/b^2-1/4*a*ln(-d*((-a)^(1/4)+b^(1/4)*x)/(b^(1/4)* c-(-a)^(1/4)*d))*ln(d*x+c)/b^2-1/4*a*ln(d*x+c)*ln(-d*(b^(1/4)*x+(-(-a)^(1/ 2))^(1/2))/(b^(1/4)*c-d*(-(-a)^(1/2))^(1/2)))/b^2-1/4*a*ln(d*x+c)*ln(d*(-b ^(1/4)*x+(-(-a)^(1/2))^(1/2))/(b^(1/4)*c+d*(-(-a)^(1/2))^(1/2)))/b^2-1/4*a *polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)^(1/4)*d))/b^2-1/4*a*polylog(2,b ^(1/4)*(d*x+c)/(b^(1/4)*c+(-a)^(1/4)*d))/b^2-1/4*a*polylog(2,b^(1/4)*(d*x+ c)/(b^(1/4)*c-d*(-(-a)^(1/2))^(1/2)))/b^2-1/4*a*polylog(2,b^(1/4)*(d*x+c)/ (b^(1/4)*c+d*(-(-a)^(1/2))^(1/2)))/b^2
Result contains complex when optimal does not.
Time = 0.15 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.96 \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}-\frac {a \log \left (\frac {d \left (i \sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (-\frac {d \left (i \sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^2} \]
(c^3*x)/(4*b*d^3) - (c^2*x^2)/(8*b*d^2) + (c*x^3)/(12*b*d) - x^4/(16*b) - (c^4*Log[c + d*x])/(4*b*d^4) + (x^4*Log[c + d*x])/(4*b) - (a*Log[(d*(I*(-a )^(1/4) - b^(1/4)*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]*Log[c + d*x])/(4*b^2) - (a*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*(I*(-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - I*(-a )^(1/4)*d))]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*((-a)^(1/4) + b^(1/4)*x)) /(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b^2) - (a*PolyLog[2, (b^(1/ 4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4 )*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/ 4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)])/(4*b^2) - (a*PolyLog[2, (b^(1 /4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*b^2)
Time = 0.98 (sec) , antiderivative size = 498, 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.
| \(\displaystyle \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \int \left (\frac {x^3 \log (c+d x)}{b}-\frac {a x^3 \log (c+d x)}{b \left (a+b x^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {x^4 \log (c+d x)}{4 b}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}\) |
(c^3*x)/(4*b*d^3) - (c^2*x^2)/(8*b*d^2) + (c*x^3)/(12*b*d) - x^4/(16*b) - (c^4*Log[c + d*x])/(4*b*d^4) + (x^4*Log[c + d*x])/(4*b) - (a*Log[(d*(Sqrt[ -Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4 *b^2) - (a*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Lo g[c + d*x])/(4*b^2) - (a*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)* c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b^2) - (a*PolyL og[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*b^2) - (a*P olyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*b^2) - ( a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*b^2)
3.3.93.3.1 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 = 0.75 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.35
| method | result | size |
| risch | \(-\frac {c^{4} \ln \left (d x +c \right )}{4 b \,d^{4}}+\frac {c^{3} x}{4 b \,d^{3}}+\frac {25 c^{4}}{48 d^{4} b}-\frac {c^{2} x^{2}}{8 b \,d^{2}}+\frac {c \,x^{3}}{12 b d}+\frac {x^{4} \ln \left (d x +c \right )}{4 b}-\frac {x^{4}}{16 b}-\frac {a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{4 b^{2}}\) | \(175\) |
| parts | \(\frac {x^{4} \ln \left (d x +c \right )}{4 b}-\frac {\ln \left (d x +c \right ) a \ln \left (b \,x^{4}+a \right )}{4 b^{2}}-\frac {d \left (\frac {\frac {\frac {1}{4} d^{3} x^{4}-\frac {1}{3} c \,x^{3} d^{2}+\frac {1}{2} d \,x^{2} c^{2}-x \,c^{3}}{d^{4}}+\frac {c^{4} \ln \left (d x +c \right )}{d^{5}}}{b}-\frac {a \ln \left (d x +c \right ) \ln \left (b \,x^{4}+a \right )}{b^{2} d}+\frac {a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{b^{2} d}\right )}{4}\) | \(205\) |
| derivativedivides | \(\frac {-\frac {d^{4} \left (c^{3} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )-3 c^{2} \left (\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}\right )+3 c \left (\frac {\left (d x +c \right )^{3} \ln \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{3}}{9}\right )-\frac {\left (d x +c \right )^{4} \ln \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{4}}{16}\right )}{b}-\frac {a \,d^{8} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{4 b^{2}}}{d^{8}}\) | \(209\) |
| default | \(\frac {-\frac {d^{4} \left (c^{3} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )-3 c^{2} \left (\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}\right )+3 c \left (\frac {\left (d x +c \right )^{3} \ln \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{3}}{9}\right )-\frac {\left (d x +c \right )^{4} \ln \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{4}}{16}\right )}{b}-\frac {a \,d^{8} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{4 b^{2}}}{d^{8}}\) | \(209\) |
-1/4*c^4*ln(d*x+c)/b/d^4+1/4*c^3*x/b/d^3+25/48/d^4/b*c^4-1/8*c^2*x^2/b/d^2 +1/12*c*x^3/b/d+1/4*x^4*ln(d*x+c)/b-1/16*x^4/b-1/4*a/b^2*sum(ln(d*x+c)*ln( (-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1),_R1=RootOf(_Z^4*b-4*_Z^3*b*c+6*_ Z^2*b*c^2-4*_Z*b*c^3+a*d^4+b*c^4))
\[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]
Timed out. \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \]
\[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]
\[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]
Timed out. \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\int \frac {x^7\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \]