\(\int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx\) [296]

Optimal result

Integrand size = 19, antiderivative size = 530 \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\frac {c x}{2 b d}-\frac {x^2}{4 b}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}-\frac {\sqrt {-a} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{3/2}}+\frac {\sqrt {-a} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{3/2}}-\frac {\sqrt {-a} \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{3/2}}+\frac {\sqrt {-a} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{3/2}}-\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}-\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.91 \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\frac {2 \sqrt {b} c d x-\sqrt {b} d^2 x^2-2 \sqrt {b} c^2 \log (c+d x)+2 \sqrt {b} d^2 x^2 \log (c+d x)+\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{-i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+\sqrt {-a} d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-\sqrt {-a} d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )-\sqrt {-a} d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\sqrt {-a} d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2} d^2} \] Input:

Integrate[(x^5*Log[c + d*x])/(a + b*x^4),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 530, 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.

\(\displaystyle \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx\)

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^5*Log[c + d*x])/(a + b*x^4),x]
 

Output:

(c*x)/(2*b*d) - x^2/(4*b) - (c^2*Log[c + d*x])/(2*b*d^2) + (x^2*Log[c + d* 
x])/(2*b) - (Sqrt[-a]*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + S 
qrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*b^(3/2)) + (Sqrt[-a]*Log[(d*((-a)^(1/4 
) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b^(3/2)) - (S 
qrt[-a]*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a 
]]*d))]*Log[c + d*x])/(4*b^(3/2)) + (Sqrt[-a]*Log[-((d*((-a)^(1/4) + b^(1/ 
4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b^(3/2)) - (Sqrt[-a]* 
PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*b^(3/2 
)) - (Sqrt[-a]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]] 
*d)])/(4*b^(3/2)) + (Sqrt[-a]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - 
(-a)^(1/4)*d)])/(4*b^(3/2)) + (Sqrt[-a]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^ 
(1/4)*c + (-a)^(1/4)*d)])/(4*b^(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 = 1.54 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.31

method	result	size
derivativedivides	\(\frac {-\frac {\left (-\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}+\frac {\left (d x +c \right )^{2}}{4}+c \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )\right ) d^{4}}{b}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right ) a \,d^{8}}{4 b^{2}}}{d^{6}}\)	\(163\)
default	\(\frac {-\frac {\left (-\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}+\frac {\left (d x +c \right )^{2}}{4}+c \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )\right ) d^{4}}{b}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right ) a \,d^{8}}{4 b^{2}}}{d^{6}}\)	\(163\)
risch	\(\frac {x^{2} \ln \left (d x +c \right )}{2 b}-\frac {c^{2} \ln \left (d x +c \right )}{2 b \,d^{2}}-\frac {x^{2}}{4 b}+\frac {c x}{2 d b}+\frac {3 c^{2}}{4 d^{2} b}-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right ) a}{4 b^{2}}\)	\(164\)

Input:

int(x^5*ln(d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

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

\[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:

integrate(x^5*log(d*x+c)/(b*x^4+a),x, algorithm="fricas")
 

Output:

integral(x^5*log(d*x + c)/(b*x^4 + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \] Input:

integrate(x**5*ln(d*x+c)/(b*x**4+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:

integrate(x^5*log(d*x+c)/(b*x^4+a),x, algorithm="maxima")
 

Output:

integrate(x^5*log(d*x + c)/(b*x^4 + a), x)
 

Giac [F]

\[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \] Input:

integrate(x^5*log(d*x+c)/(b*x^4+a),x, algorithm="giac")
 

Output:

integrate(x^5*log(d*x + c)/(b*x^4 + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\int \frac {x^5\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \] Input:

int((x^5*log(c + d*x))/(a + b*x^4),x)
 

Output:

int((x^5*log(c + d*x))/(a + b*x^4), x)
 

Reduce [F]

\[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\frac {-4 \left (\int \frac {\mathrm {log}\left (d x +c \right ) x}{b \,x^{4}+a}d x \right ) a \,d^{2}-2 \,\mathrm {log}\left (d x +c \right ) c^{2}+2 \,\mathrm {log}\left (d x +c \right ) d^{2} x^{2}+2 c d x -d^{2} x^{2}}{4 b \,d^{2}} \] Input:

int(x^5*log(d*x+c)/(b*x^4+a),x)
 

Output:

( - 4*int((log(c + d*x)*x)/(a + b*x**4),x)*a*d**2 - 2*log(c + d*x)*c**2 + 
2*log(c + d*x)*d**2*x**2 + 2*c*d*x - d**2*x**2)/(4*b*d**2)