\(\int \frac {x^4 \log (c+d x)}{a+b x^4} \, dx\) [299]

Optimal result

Integrand size = 19, antiderivative size = 521 \[ \int \frac {x^4 \log (c+d x)}{a+b x^4} \, dx=-\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}+\frac {\sqrt {-\sqrt {-a}} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{5/4}} \] Output:

-x/b+(d*x+c)*ln(d*x+c)/b/d+1/4*(-(-a)^(1/2))^(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)/b^(5/4)+1/4*(-a 
)^(1/4)*ln(d*((-a)^(1/4)-b^(1/4)*x)/(b^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)/b^ 
(5/4)-1/4*(-(-a)^(1/2))^(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)/b^(5/4)-1/4*(-a)^(1/4)*ln(-d*((-a)^ 
(1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4)*d))*ln(d*x+c)/b^(5/4)-1/4*(-(-a)^(1 
/2))^(1/2)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-(-a)^(1/2))^(1/2)*d))/b^ 
(5/4)+1/4*(-(-a)^(1/2))^(1/2)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-(-a)^ 
(1/2))^(1/2)*d))/b^(5/4)-1/4*(-a)^(1/4)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4) 
*c-(-a)^(1/4)*d))/b^(5/4)+1/4*(-a)^(1/4)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4 
)*c+(-a)^(1/4)*d))/b^(5/4)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 458, normalized size of antiderivative = 0.88 \[ \int \frac {x^4 \log (c+d x)}{a+b x^4} \, dx=\frac {-4 \sqrt [4]{b} d x+4 \sqrt [4]{b} c \log (c+d x)+4 \sqrt [4]{b} d x \log (c+d x)+\sqrt [4]{-a} d \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-i \sqrt [4]{-a} d \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+i \sqrt [4]{-a} d \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{-i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt [4]{-a} d \log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt [4]{-a} d \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-i \sqrt [4]{-a} d \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )+i \sqrt [4]{-a} d \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\sqrt [4]{-a} d \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{5/4} d} \] Input:

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

Output:

(-4*b^(1/4)*d*x + 4*b^(1/4)*c*Log[c + d*x] + 4*b^(1/4)*d*x*Log[c + d*x] + 
(-a)^(1/4)*d*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]* 
Log[c + d*x] - I*(-a)^(1/4)*d*Log[(d*((-a)^(1/4) - I*b^(1/4)*x))/(I*b^(1/4 
)*c + (-a)^(1/4)*d)]*Log[c + d*x] + I*(-a)^(1/4)*d*Log[(d*((-a)^(1/4) + I* 
b^(1/4)*x))/((-I)*b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x] - (-a)^(1/4)*d*L 
og[(d*((-a)^(1/4) + b^(1/4)*x))/(-(b^(1/4)*c) + (-a)^(1/4)*d)]*Log[c + d*x 
] - (-a)^(1/4)*d*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d) 
] - I*(-a)^(1/4)*d*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4 
)*d)] + I*(-a)^(1/4)*d*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^ 
(1/4)*d)] + (-a)^(1/4)*d*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^ 
(1/4)*d)])/(4*b^(5/4)*d)
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 521, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\sqrt {-\sqrt {-a}} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {x}{b}\)

Input:

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

Output:

-(x/b) + ((c + d*x)*Log[c + d*x])/(b*d) + (Sqrt[-Sqrt[-a]]*Log[(d*(Sqrt[-S 
qrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*b 
^(5/4)) + ((-a)^(1/4)*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^( 
1/4)*d)]*Log[c + d*x])/(4*b^(5/4)) - (Sqrt[-Sqrt[-a]]*Log[-((d*(Sqrt[-Sqrt 
[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b^( 
5/4)) - ((-a)^(1/4)*Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^( 
1/4)*d))]*Log[c + d*x])/(4*b^(5/4)) - (Sqrt[-Sqrt[-a]]*PolyLog[2, (b^(1/4) 
*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*b^(5/4)) + (Sqrt[-Sqrt[-a 
]]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*b^( 
5/4)) - ((-a)^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4) 
*d)])/(4*b^(5/4)) + ((-a)^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c 
+ (-a)^(1/4)*d)])/(4*b^(5/4))
 

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 = 1.50 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.28

method	result	size
derivativedivides	\(\frac {\frac {\left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right ) d^{4}}{b}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 c \,\textit {\_R1}^{2}-3 c^{2} \textit {\_R1} +c^{3}}\right ) a \,d^{8}}{4 b^{2}}}{d^{5}}\)	\(145\)
default	\(\frac {\frac {\left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right ) d^{4}}{b}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 c \,\textit {\_R1}^{2}-3 c^{2} \textit {\_R1} +c^{3}}\right ) a \,d^{8}}{4 b^{2}}}{d^{5}}\)	\(145\)
risch	\(\frac {x \ln \left (d x +c \right )}{b}+\frac {\ln \left (d x +c \right ) c}{d b}-\frac {x}{b}-\frac {c}{b d}+\frac {d^{3} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 c \,\textit {\_R1}^{2}-3 c^{2} \textit {\_R1} +c^{3}}\right ) a}{4 b^{2}}\)	\(154\)

Input:

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

Output:

1/d^5*(((d*x+c)*ln(d*x+c)-d*x-c)*d^4/b+1/4/b^2*sum(1/(-_R1^3+3*_R1^2*c-3*_ 
R1*c^2+c^3)*(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1)),_R1=R 
ootOf(_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*d^8)
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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