\(\int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx\) [300]

Optimal result

Integrand size = 19, antiderivative size = 497 \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 \sqrt [4]{-a} b^{3/4}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 \sqrt {-\sqrt {-a}} b^{3/4}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt [4]{-a} b^{3/4}} \] Output:

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

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.93 \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\frac {\sqrt [4]{-a} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)+\sqrt {-\sqrt {-a}} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt [4]{-a} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)-\sqrt {-\sqrt {-a}} \log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt [4]{-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )+\sqrt [4]{-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )-\sqrt {-\sqrt {-a}} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )+\sqrt {-\sqrt {-a}} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 \sqrt {-\sqrt {-a}} \sqrt [4]{-a} b^{3/4}} \] Input:

Integrate[(x^2*Log[c + d*x])/(a + b*x^4),x]
 

Output:

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

Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 497, 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.

\(\displaystyle \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx\)

\(\Big \downarrow \) 2863

\(\displaystyle \int \left (\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}+\sqrt {b} x^2\right )}-\frac {\log (c+d x)}{2 \sqrt {b} \left (\sqrt {-a}-\sqrt {b} x^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(x^2*Log[c + d*x])/(a + b*x^4),x]
 

Output:

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

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 = 1.44 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.19

method	result	size
derivativedivides	\(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{4 b}\)	\(94\)
default	\(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{4 b}\)	\(94\)
risch	\(-\frac {d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1} +c}\right )}{4 b}\)	\(94\)

Input:

int(x^2*ln(d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

-1/4*d/b*sum(1/(-_R1+c)*(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))
 

Fricas [F]

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

integrate(x^2*log(d*x+c)/(b*x^4+a),x, algorithm="fricas")
 

Output:

integral(x^2*log(d*x + c)/(b*x^4 + a), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \] Input:

integrate(x**2*ln(d*x+c)/(b*x**4+a),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^2*log(d*x+c)/(b*x^4+a),x, algorithm="maxima")
 

Output:

integrate(x^2*log(d*x + c)/(b*x^4 + a), x)
 

Giac [F]

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

integrate(x^2*log(d*x+c)/(b*x^4+a),x, algorithm="giac")
 

Output:

integrate(x^2*log(d*x + c)/(b*x^4 + a), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \log (c+d x)}{a+b x^4} \, dx=\int \frac {x^2\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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