\(\int \frac {\log (c+d x)}{x^3 (a+b x^4)} \, dx\) [298]

Optimal result

Integrand size = 19, antiderivative size = 537 \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=-\frac {d}{2 a c x}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {\sqrt {b} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac {\sqrt {b} \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}} \] Output:

-1/2*d/a/c/x-1/2*d^2*ln(x)/a/c^2+1/2*d^2*ln(d*x+c)/a/c^2-1/2*ln(d*x+c)/a/x 
^2-1/4*b^(1/2)*ln(d*((-(-a)^(1/2))^(1/2)-b^(1/4)*x)/(b^(1/4)*c+(-(-a)^(1/2 
))^(1/2)*d))*ln(d*x+c)/(-a)^(3/2)+1/4*b^(1/2)*ln(d*((-a)^(1/4)-b^(1/4)*x)/ 
(b^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)/(-a)^(3/2)-1/4*b^(1/2)*ln(-d*((-(-a)^( 
1/2))^(1/2)+b^(1/4)*x)/(b^(1/4)*c-(-(-a)^(1/2))^(1/2)*d))*ln(d*x+c)/(-a)^( 
3/2)+1/4*b^(1/2)*ln(-d*((-a)^(1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4)*d))*ln 
(d*x+c)/(-a)^(3/2)-1/4*b^(1/2)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-(-a) 
^(1/2))^(1/2)*d))/(-a)^(3/2)-1/4*b^(1/2)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4 
)*c+(-(-a)^(1/2))^(1/2)*d))/(-a)^(3/2)+1/4*b^(1/2)*polylog(2,b^(1/4)*(d*x+ 
c)/(b^(1/4)*c-(-a)^(1/4)*d))/(-a)^(3/2)+1/4*b^(1/2)*polylog(2,b^(1/4)*(d*x 
+c)/(b^(1/4)*c+(-a)^(1/4)*d))/(-a)^(3/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.94 \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=-\frac {\log (c+d x)}{2 a x^2}-\frac {\sqrt {b} \log \left (\frac {d \left (i \sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac {\sqrt {b} \log \left (-\frac {d \left (i \sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac {d \left (\frac {1}{c x}+\frac {d \log (x)}{c^2}-\frac {d \log (c+d x)}{c^2}\right )}{2 a}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}} \] Input:

Integrate[Log[c + d*x]/(x^3*(a + b*x^4)),x]
 

Output:

-1/2*Log[c + d*x]/(a*x^2) - (Sqrt[b]*Log[(d*(I*(-a)^(1/4) - b^(1/4)*x))/(b 
^(1/4)*c + I*(-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[(d 
*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(- 
a)^(3/2)) - (Sqrt[b]*Log[-((d*(I*(-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - I*( 
-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[-((d*((-a)^(1/4 
) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(3/2)) 
- (d*(1/(c*x) + (d*Log[x])/c^2 - (d*Log[c + d*x])/c^2))/(2*a) + (Sqrt[b]*P 
olyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*(-a)^(3/2)) 
- (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)])/( 
4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a 
)^(1/4)*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^( 
1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(3/2))
 

Rubi [A] (verified)

Time = 1.60 (sec) , antiderivative size = 537, 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.

\(\displaystyle \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx\)

\(\Big \downarrow \) 2863

\(\displaystyle \int \left (\frac {\log (c+d x)}{a x^3}-\frac {b x \log (c+d x)}{a \left (a+b x^4\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {d}{2 a c x}\)

Input:

Int[Log[c + d*x]/(x^3*(a + b*x^4)),x]
 

Output:

-1/2*d/(a*c*x) - (d^2*Log[x])/(2*a*c^2) + (d^2*Log[c + d*x])/(2*a*c^2) - L 
og[c + d*x]/(2*a*x^2) - (Sqrt[b]*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^ 
(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[ 
(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4* 
(-a)^(3/2)) - (Sqrt[b]*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c 
- Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[-((d*(( 
-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a) 
^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[ 
-a]]*d)])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4 
)*c + Sqrt[-Sqrt[-a]]*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2, (b^(1/4)*( 
c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2 
, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(3/2))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.70 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.29

method	result	size
derivativedivides	\(d^{2} \left (-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{4 a}+\frac {-\frac {\ln \left (-d x \right )}{2 c^{2}}-\frac {1}{2 c d x}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (-d x +c \right )}{2 c^{2} d^{2} x^{2}}}{a}\right )\)	\(158\)
default	\(d^{2} \left (-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{4 a}+\frac {-\frac {\ln \left (-d x \right )}{2 c^{2}}-\frac {1}{2 c d x}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (-d x +c \right )}{2 c^{2} d^{2} x^{2}}}{a}\right )\)	\(158\)
risch	\(-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{4 a}-\frac {d^{2} \ln \left (-d x \right )}{2 a \,c^{2}}-\frac {d}{2 a c x}+\frac {d^{2} \ln \left (d x +c \right )}{2 a \,c^{2}}-\frac {\ln \left (d x +c \right )}{2 a \,x^{2}}\)	\(162\)

Input:

int(ln(d*x+c)/x^3/(b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

d^2*(-1/4*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^4*b-4*_Z^3*b*c+6*_Z^2*b*c^2-4*_Z*b*c^3+a* 
d^4+b*c^4))/a+(-1/2/c^2*ln(-d*x)-1/2/c/d/x-1/2*ln(d*x+c)*(d*x+c)*(-d*x+c)/ 
c^2/d^2/x^2)/a)
 

Fricas [F]

\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{3}} \,d x } \] Input:

integrate(log(d*x+c)/x^3/(b*x^4+a),x, algorithm="fricas")
 

Output:

integral(log(d*x + c)/(b*x^7 + a*x^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\text {Timed out} \] Input:

integrate(ln(d*x+c)/x**3/(b*x**4+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{3}} \,d x } \] Input:

integrate(log(d*x+c)/x^3/(b*x^4+a),x, algorithm="maxima")
 

Output:

integrate(log(d*x + c)/((b*x^4 + a)*x^3), x)
 

Giac [F]

\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{3}} \,d x } \] Input:

integrate(log(d*x+c)/x^3/(b*x^4+a),x, algorithm="giac")
 

Output:

integrate(log(d*x + c)/((b*x^4 + a)*x^3), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x^3\,\left (b\,x^4+a\right )} \,d x \] Input:

int(log(c + d*x)/(x^3*(a + b*x^4)),x)
 

Output:

int(log(c + d*x)/(x^3*(a + b*x^4)), x)
 

Reduce [F]

\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int \frac {\mathrm {log}\left (d x +c \right )}{x^{3} \left (b \,x^{4}+a \right )}d x \] Input:

int(log(d*x+c)/x^3/(b*x^4+a),x)
 

Output:

int(log(d*x+c)/x^3/(b*x^4+a),x)