Integrand size = 23, antiderivative size = 463 \[ \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx=-\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{f}+\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{\left (d \sqrt [3]{e}+(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{3 f}+\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d \left ((-1)^{2/3} \sqrt [3]{e}+\sqrt [3]{f} x\right )}{\left ((-1)^{2/3} d \sqrt [3]{e}+(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{3 f}+\frac {(a+b \text {arctanh}(c+d x)) \log \left (\frac {2 \sqrt [3]{-1} d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{\left (\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{3 f}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{2 f}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{\left (d \sqrt [3]{e}+(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{6 f}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left ((-1)^{2/3} \sqrt [3]{e}+\sqrt [3]{f} x\right )}{\left ((-1)^{2/3} d \sqrt [3]{e}+(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{6 f}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt [3]{-1} d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{\left (\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}\right ) (1+c+d x)}\right )}{6 f} \] Output:
-(a+b*arctanh(d*x+c))*ln(2/(d*x+c+1))/f+1/3*(a+b*arctanh(d*x+c))*ln(2*d*(e ^(1/3)+f^(1/3)*x)/(d*e^(1/3)+(1-c)*f^(1/3))/(d*x+c+1))/f+1/3*(a+b*arctanh( d*x+c))*ln(2*d*((-1)^(2/3)*e^(1/3)+f^(1/3)*x)/((-1)^(2/3)*d*e^(1/3)+(1-c)* f^(1/3))/(d*x+c+1))/f+1/3*(a+b*arctanh(d*x+c))*ln(2*(-1)^(1/3)*d*(e^(1/3)+ (-1)^(2/3)*f^(1/3)*x)/((-1)^(1/3)*d*e^(1/3)-(1-c)*f^(1/3))/(d*x+c+1))/f+1/ 2*b*polylog(2,1-2/(d*x+c+1))/f-1/6*b*polylog(2,1-2*d*(e^(1/3)+f^(1/3)*x)/( d*e^(1/3)+(1-c)*f^(1/3))/(d*x+c+1))/f-1/6*b*polylog(2,1-2*d*((-1)^(2/3)*e^ (1/3)+f^(1/3)*x)/((-1)^(2/3)*d*e^(1/3)+(1-c)*f^(1/3))/(d*x+c+1))/f-1/6*b*p olylog(2,1-2*(-1)^(1/3)*d*(e^(1/3)+(-1)^(2/3)*f^(1/3)*x)/((-1)^(1/3)*d*e^( 1/3)-(1-c)*f^(1/3))/(d*x+c+1))/f
Time = 0.38 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.47 \[ \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx=-\frac {b \log (1-c-d x) \log \left (\frac {d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{d \sqrt [3]{e}+(1-c) \sqrt [3]{f}}\right )}{6 f}+\frac {b \log (1+c+d x) \log \left (\frac {d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{d \sqrt [3]{e}-(1+c) \sqrt [3]{f}}\right )}{6 f}-\frac {b \log (1-c-d x) \log \left (\frac {d \left ((-1)^{2/3} \sqrt [3]{e}+\sqrt [3]{f} x\right )}{(-1)^{2/3} d \sqrt [3]{e}+(1-c) \sqrt [3]{f}}\right )}{6 f}+\frac {b \log (1+c+d x) \log \left (\frac {(-1)^{2/3} d \left (\sqrt [3]{e}-\sqrt [3]{-1} \sqrt [3]{f} x\right )}{(-1)^{2/3} d \sqrt [3]{e}-(1+c) \sqrt [3]{f}}\right )}{6 f}-\frac {b \log (1-c-d x) \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}}\right )}{6 f}+\frac {b \log (1+c+d x) \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{\sqrt [3]{-1} d \sqrt [3]{e}+(1+c) \sqrt [3]{f}}\right )}{6 f}+\frac {a \log \left (e+f x^3\right )}{3 f}-\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{f} (1-c-d x)}{\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}}\right )}{6 f}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{f} (1-c-d x)}{d \sqrt [3]{e}+(1-c) \sqrt [3]{f}}\right )}{6 f}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{f} (1-c-d x)}{(-1)^{2/3} d \sqrt [3]{e}+(1-c) \sqrt [3]{f}}\right )}{6 f}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{f} (1+c+d x)}{d \sqrt [3]{e}-(1+c) \sqrt [3]{f}}\right )}{6 f}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{f} (1+c+d x)}{(-1)^{2/3} d \sqrt [3]{e}-(1+c) \sqrt [3]{f}}\right )}{6 f}+\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{f} (1+c+d x)}{\sqrt [3]{-1} d \sqrt [3]{e}+(1+c) \sqrt [3]{f}}\right )}{6 f} \] Input:
Integrate[(x^2*(a + b*ArcTanh[c + d*x]))/(e + f*x^3),x]
Output:
-1/6*(b*Log[1 - c - d*x]*Log[(d*(e^(1/3) + f^(1/3)*x))/(d*e^(1/3) + (1 - c )*f^(1/3))])/f + (b*Log[1 + c + d*x]*Log[(d*(e^(1/3) + f^(1/3)*x))/(d*e^(1 /3) - (1 + c)*f^(1/3))])/(6*f) - (b*Log[1 - c - d*x]*Log[(d*((-1)^(2/3)*e^ (1/3) + f^(1/3)*x))/((-1)^(2/3)*d*e^(1/3) + (1 - c)*f^(1/3))])/(6*f) + (b* Log[1 + c + d*x]*Log[((-1)^(2/3)*d*(e^(1/3) - (-1)^(1/3)*f^(1/3)*x))/((-1) ^(2/3)*d*e^(1/3) - (1 + c)*f^(1/3))])/(6*f) - (b*Log[1 - c - d*x]*Log[((-1 )^(1/3)*d*(e^(1/3) + (-1)^(2/3)*f^(1/3)*x))/((-1)^(1/3)*d*e^(1/3) - (1 - c )*f^(1/3))])/(6*f) + (b*Log[1 + c + d*x]*Log[((-1)^(1/3)*d*(e^(1/3) + (-1) ^(2/3)*f^(1/3)*x))/((-1)^(1/3)*d*e^(1/3) + (1 + c)*f^(1/3))])/(6*f) + (a*L og[e + f*x^3])/(3*f) - (b*PolyLog[2, -((f^(1/3)*(1 - c - d*x))/((-1)^(1/3) *d*e^(1/3) - (1 - c)*f^(1/3)))])/(6*f) - (b*PolyLog[2, (f^(1/3)*(1 - c - d *x))/(d*e^(1/3) + (1 - c)*f^(1/3))])/(6*f) - (b*PolyLog[2, (f^(1/3)*(1 - c - d*x))/((-1)^(2/3)*d*e^(1/3) + (1 - c)*f^(1/3))])/(6*f) + (b*PolyLog[2, -((f^(1/3)*(1 + c + d*x))/(d*e^(1/3) - (1 + c)*f^(1/3)))])/(6*f) + (b*Poly Log[2, -((f^(1/3)*(1 + c + d*x))/((-1)^(2/3)*d*e^(1/3) - (1 + c)*f^(1/3))) ])/(6*f) + (b*PolyLog[2, (f^(1/3)*(1 + c + d*x))/((-1)^(1/3)*d*e^(1/3) + ( 1 + c)*f^(1/3))])/(6*f)
Time = 1.25 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {7276, 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.
| \(\displaystyle \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {a x^2}{e+f x^3}+\frac {b x^2 \text {arctanh}(c+d x)}{e+f x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a \log \left (e+f x^3\right )}{3 f}+\frac {b \text {arctanh}(c+d x) \log \left (\frac {2 d \left (\sqrt [3]{e}+\sqrt [3]{f} x\right )}{(c+d x+1) \left ((1-c) \sqrt [3]{f}+d \sqrt [3]{e}\right )}\right )}{3 f}+\frac {b \text {arctanh}(c+d x) \log \left (\frac {2 d \left ((-1)^{2/3} \sqrt [3]{e}+\sqrt [3]{f} x\right )}{(c+d x+1) \left ((1-c) \sqrt [3]{f}+(-1)^{2/3} d \sqrt [3]{e}\right )}\right )}{3 f}+\frac {b \text {arctanh}(c+d x) \log \left (\frac {2 \sqrt [3]{-1} d \left (\sqrt [3]{e}+(-1)^{2/3} \sqrt [3]{f} x\right )}{(c+d x+1) \left (\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}\right )}\right )}{3 f}-\frac {b \text {arctanh}(c+d x) \log \left (\frac {2}{c+d x+1}\right )}{f}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt [3]{f} x+\sqrt [3]{e}\right )}{\left (\sqrt [3]{f} (1-c)+d \sqrt [3]{e}\right ) (c+d x+1)}\right )}{6 f}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 d \left (\sqrt [3]{f} x+(-1)^{2/3} \sqrt [3]{e}\right )}{\left (\sqrt [3]{f} (1-c)+(-1)^{2/3} d \sqrt [3]{e}\right ) (c+d x+1)}\right )}{6 f}-\frac {b \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt [3]{-1} d \left ((-1)^{2/3} \sqrt [3]{f} x+\sqrt [3]{e}\right )}{\left (\sqrt [3]{-1} d \sqrt [3]{e}-(1-c) \sqrt [3]{f}\right ) (c+d x+1)}\right )}{6 f}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{2 f}\) |
Input:
Int[(x^2*(a + b*ArcTanh[c + d*x]))/(e + f*x^3),x]
Output:
-((b*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/f) + (b*ArcTanh[c + d*x]*Log[( 2*d*(e^(1/3) + f^(1/3)*x))/((d*e^(1/3) + (1 - c)*f^(1/3))*(1 + c + d*x))]) /(3*f) + (b*ArcTanh[c + d*x]*Log[(2*d*((-1)^(2/3)*e^(1/3) + f^(1/3)*x))/(( (-1)^(2/3)*d*e^(1/3) + (1 - c)*f^(1/3))*(1 + c + d*x))])/(3*f) + (b*ArcTan h[c + d*x]*Log[(2*(-1)^(1/3)*d*(e^(1/3) + (-1)^(2/3)*f^(1/3)*x))/(((-1)^(1 /3)*d*e^(1/3) - (1 - c)*f^(1/3))*(1 + c + d*x))])/(3*f) + (a*Log[e + f*x^3 ])/(3*f) + (b*PolyLog[2, 1 - 2/(1 + c + d*x)])/(2*f) - (b*PolyLog[2, 1 - ( 2*d*(e^(1/3) + f^(1/3)*x))/((d*e^(1/3) + (1 - c)*f^(1/3))*(1 + c + d*x))]) /(6*f) - (b*PolyLog[2, 1 - (2*d*((-1)^(2/3)*e^(1/3) + f^(1/3)*x))/(((-1)^( 2/3)*d*e^(1/3) + (1 - c)*f^(1/3))*(1 + c + d*x))])/(6*f) - (b*PolyLog[2, 1 - (2*(-1)^(1/3)*d*(e^(1/3) + (-1)^(2/3)*f^(1/3)*x))/(((-1)^(1/3)*d*e^(1/3 ) - (1 - c)*f^(1/3))*(1 + c + d*x))])/(6*f)
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.33 (sec) , antiderivative size = 327, normalized size of antiderivative = 0.71
| method | result | size |
| risch | \(-\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} +c^{3} f -d^{3} e -3 c^{2} f +3 f c -f \right )}{\sum }\left (\ln \left (-d x -c +1\right ) \ln \left (\frac {d x +\textit {\_R1} +c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {d x +\textit {\_R1} +c -1}{\textit {\_R1}}\right )\right )\right )}{6 f}+\frac {a \ln \left (\left (-d x -c +1\right )^{3} f +3 \left (-d x -c +1\right )^{2} c f +3 \left (-d x -c +1\right ) c^{2} f +c^{3} f -d^{3} e -3 f \left (-d x -c +1\right )^{2}-6 \left (-d x -c +1\right ) c f -3 c^{2} f +3 \left (-d x -c +1\right ) f +3 f c -f \right )}{3 f}+\frac {b \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\left (\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )\right )\right )}{6 f}\) | \(327\) |
| parts | \(\frac {a \ln \left (f \,x^{3}+e \right )}{3 f}+\frac {b \left (\frac {d^{3} \ln \left (f \left (d x +c \right )^{3}-3 c f \left (d x +c \right )^{2}+3 c^{2} f \left (d x +c \right )-c^{3} f +d^{3} e \right ) \operatorname {arctanh}\left (d x +c \right )}{3 f}-\frac {d^{3} \left (\frac {\ln \left (d x +c +1\right ) \ln \left (f \left (d x +c \right )^{3}-3 c f \left (d x +c \right )^{2}+3 c^{2} f \left (d x +c \right )-c^{3} f +d^{3} e \right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\left (\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )\right )\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (f \left (d x +c \right )^{3}-3 c f \left (d x +c \right )^{2}+3 c^{2} f \left (d x +c \right )-c^{3} f +d^{3} e \right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c +3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e +3 c^{2} f -3 f c +f \right )}{\sum }\left (\ln \left (d x +c -1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )\right )\right )}{2}\right )}{3 f}\right )}{d^{3}}\) | \(399\) |
| derivativedivides | \(\frac {\frac {a \,d^{3} \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{3 f}-b \,d^{3} \left (-\frac {\ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right ) \operatorname {arctanh}\left (d x +c \right )}{3 f}+\frac {-\frac {\ln \left (d x +c -1\right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c +3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e +3 c^{2} f -3 f c +f \right )}{\sum }\left (\ln \left (d x +c -1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )\right )\right )}{2}+\frac {\ln \left (d x +c +1\right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\left (\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )\right )\right )}{2}}{3 f}\right )}{d^{3}}\) | \(441\) |
| default | \(\frac {\frac {a \,d^{3} \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{3 f}-b \,d^{3} \left (-\frac {\ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right ) \operatorname {arctanh}\left (d x +c \right )}{3 f}+\frac {-\frac {\ln \left (d x +c -1\right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c +3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f -6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e +3 c^{2} f -3 f c +f \right )}{\sum }\left (\ln \left (d x +c -1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c +1}{\textit {\_R1}}\right )\right )\right )}{2}+\frac {\ln \left (d x +c +1\right ) \ln \left (c^{3} f -3 c^{2} f \left (d x +c \right )+3 c f \left (d x +c \right )^{2}-d^{3} e -f \left (d x +c \right )^{3}\right )}{2}-\frac {\left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (f \,\textit {\_Z}^{3}+\left (-3 f c -3 f \right ) \textit {\_Z}^{2}+\left (3 c^{2} f +6 f c +3 f \right ) \textit {\_Z} -c^{3} f +d^{3} e -3 c^{2} f -3 f c -f \right )}{\sum }\left (\ln \left (d x +c +1\right ) \ln \left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c -1}{\textit {\_R1}}\right )\right )\right )}{2}}{3 f}\right )}{d^{3}}\) | \(441\) |
Input:
int(x^2*(a+b*arctanh(d*x+c))/(f*x^3+e),x,method=_RETURNVERBOSE)
Output:
-1/6*b/f*sum(ln(-d*x-c+1)*ln((d*x+_R1+c-1)/_R1)+dilog((d*x+_R1+c-1)/_R1),_ R1=RootOf(f*_Z^3+(3*c*f-3*f)*_Z^2+(3*c^2*f-6*c*f+3*f)*_Z+c^3*f-d^3*e-3*c^2 *f+3*f*c-f))+1/3*a/f*ln((-d*x-c+1)^3*f+3*(-d*x-c+1)^2*c*f+3*(-d*x-c+1)*c^2 *f+c^3*f-d^3*e-3*f*(-d*x-c+1)^2-6*(-d*x-c+1)*c*f-3*c^2*f+3*(-d*x-c+1)*f+3* f*c-f)+1/6*b/f*sum(ln(d*x+c+1)*ln((-d*x+_R1-c-1)/_R1)+dilog((-d*x+_R1-c-1) /_R1),_R1=RootOf(f*_Z^3+(-3*c*f-3*f)*_Z^2+(3*c^2*f+6*c*f+3*f)*_Z-c^3*f+d^3 *e-3*c^2*f-3*f*c-f))
\[ \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )} x^{2}}{f x^{3} + e} \,d x } \] Input:
integrate(x^2*(a+b*arctanh(d*x+c))/(f*x^3+e),x, algorithm="fricas")
Output:
integral((b*x^2*arctanh(d*x + c) + a*x^2)/(f*x^3 + e), x)
Timed out. \[ \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx=\text {Timed out} \] Input:
integrate(x**2*(a+b*atanh(d*x+c))/(f*x**3+e),x)
Output:
Timed out
\[ \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )} x^{2}}{f x^{3} + e} \,d x } \] Input:
integrate(x^2*(a+b*arctanh(d*x+c))/(f*x^3+e),x, algorithm="maxima")
Output:
1/2*b*integrate(x^2*(log(d*x + c + 1) - log(-d*x - c + 1))/(f*x^3 + e), x) + 1/3*a*log(f*x^3 + e)/f
\[ \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )} x^{2}}{f x^{3} + e} \,d x } \] Input:
integrate(x^2*(a+b*arctanh(d*x+c))/(f*x^3+e),x, algorithm="giac")
Output:
integrate((b*arctanh(d*x + c) + a)*x^2/(f*x^3 + e), x)
Timed out. \[ \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}{f\,x^3+e} \,d x \] Input:
int((x^2*(a + b*atanh(c + d*x)))/(e + f*x^3),x)
Output:
int((x^2*(a + b*atanh(c + d*x)))/(e + f*x^3), x)
\[ \int \frac {x^2 (a+b \text {arctanh}(c+d x))}{e+f x^3} \, dx=\frac {3 \left (\int \frac {\mathit {atanh} \left (d x +c \right ) x^{2}}{f \,x^{3}+e}d x \right ) b f +\mathrm {log}\left (e^{\frac {2}{3}}-f^{\frac {1}{3}} e^{\frac {1}{3}} x +f^{\frac {2}{3}} x^{2}\right ) a +\mathrm {log}\left (e^{\frac {1}{3}}+f^{\frac {1}{3}} x \right ) a}{3 f} \] Input:
int(x^2*(a+b*atanh(d*x+c))/(f*x^3+e),x)
Output:
(3*int((atanh(c + d*x)*x**2)/(e + f*x**3),x)*b*f + log(e**(2/3) - f**(1/3) *e**(1/3)*x + f**(2/3)*x**2)*a + log(e**(1/3) + f**(1/3)*x)*a)/(3*f)