3.3.94 \(\int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx\) [294]

3.3.94.1 Optimal result

Integrand size = 19, antiderivative size = 401 \[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b} \]

output

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

3.3.94.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.05 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx=\frac {\log \left (\frac {d \left (i \sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (-\frac {d \left (i \sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 b}+\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b} \]

input

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

output

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

3.3.94.3 Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 401, 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.

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

\(\Big \downarrow \) 2863

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

\(\Big \downarrow \) 2009

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

input

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

3.3.94.3.1 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]
 

3.3.94.4 Maple [C] (verified)

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

Time = 0.60 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.21

method	result	size
derivativedivides	\(\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 b}\)	\(85\)
default	\(\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 b}\)	\(85\)
risch	\(\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{4 b}\)	\(85\)
parts	\(\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{4}+a \right )}{4 b}-\frac {d \left (\frac {\ln \left (d x +c \right ) \ln \left (b \,x^{4}+a \right )}{d}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )}{d}\right )}{4 b}\)	\(130\)

input

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

output

1/4/b*sum(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1),_R1=RootO 
f(_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))
 

3.3.94.5 Fricas [F]

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

input

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

output

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

3.3.94.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.3.94.7 Maxima [F]

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

input

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

output

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

3.3.94.8 Giac [F]

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

input

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

output

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

3.3.94.9 Mupad [F(-1)]

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

input

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

output

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