\(\int \frac {2 b+a x^3}{x^4 (c-2 b^2 x^3+a x^6)} \, dx\) [24]

Optimal result

Integrand size = 30, antiderivative size = 117 \[ \int \frac {2 b+a x^3}{x^4 \left (c-2 b^2 x^3+a x^6\right )} \, dx=-\frac {2 b}{3 c x^3}+\frac {b \left (4 b^4-2 a c+a b c\right ) \text {arctanh}\left (\frac {b^2-a x^3}{\sqrt {b^4-a c}}\right )}{3 c^2 \sqrt {b^4-a c}}+\frac {\left (4 b^3+a c\right ) \log (x)}{c^2}-\frac {\left (4 b^3+a c\right ) \log \left (c-2 b^2 x^3+a x^6\right )}{6 c^2} \] Output:

-2/3*b/c/x^3+1/3*b*(4*b^4+a*b*c-2*a*c)*arctanh((-a*x^3+b^2)/(b^4-a*c)^(1/2 
))/c^2/(b^4-a*c)^(1/2)+(4*b^3+a*c)*ln(x)/c^2-1/6*(4*b^3+a*c)*ln(a*x^6-2*b^ 
2*x^3+c)/c^2
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.15 \[ \int \frac {2 b+a x^3}{x^4 \left (c-2 b^2 x^3+a x^6\right )} \, dx=-\frac {\frac {4 b c}{x^3}-6 \left (4 b^3+a c\right ) \log (x)+\text {RootSum}\left [c-2 b^2 \text {$\#$1}^3+a \text {$\#$1}^6\&,\frac {8 b^5 \log (x-\text {$\#$1})-2 a b c \log (x-\text {$\#$1})+2 a b^2 c \log (x-\text {$\#$1})-4 a b^3 \log (x-\text {$\#$1}) \text {$\#$1}^3-a^2 c \log (x-\text {$\#$1}) \text {$\#$1}^3}{b^2-a \text {$\#$1}^3}\&\right ]}{6 c^2} \] Input:

Integrate[(2*b + a*x^3)/(x^4*(c - 2*b^2*x^3 + a*x^6)),x]
 

Output:

-1/6*((4*b*c)/x^3 - 6*(4*b^3 + a*c)*Log[x] + RootSum[c - 2*b^2*#1^3 + a*#1 
^6 & , (8*b^5*Log[x - #1] - 2*a*b*c*Log[x - #1] + 2*a*b^2*c*Log[x - #1] - 
4*a*b^3*Log[x - #1]*#1^3 - a^2*c*Log[x - #1]*#1^3)/(b^2 - a*#1^3) & ])/c^2
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.01, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1802, 1200, 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.

\(\displaystyle \int \frac {a x^3+2 b}{x^4 \left (a x^6-2 b^2 x^3+c\right )} \, dx\)

\(\Big \downarrow \) 1802

\(\displaystyle \frac {1}{3} \int \frac {a x^3+2 b}{x^6 \left (a x^6-2 b^2 x^3+c\right )}dx^3\)

\(\Big \downarrow \) 1200

\(\displaystyle \frac {1}{3} \int \left (\frac {2 b}{c x^6}+\frac {2 b \left (4 b^4+a c b-a c\right )-a \left (4 b^3+a c\right ) x^3}{c^2 \left (a x^6-2 b^2 x^3+c\right )}+\frac {4 b^3+a c}{c^2 x^3}\right )dx^3\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{3} \left (\frac {b \left (a b c-2 a c+4 b^4\right ) \text {arctanh}\left (\frac {b^2-a x^3}{\sqrt {b^4-a c}}\right )}{c^2 \sqrt {b^4-a c}}+\frac {\log \left (x^3\right ) \left (a c+4 b^3\right )}{c^2}-\frac {\left (a c+4 b^3\right ) \log \left (a x^6-2 b^2 x^3+c\right )}{2 c^2}-\frac {2 b}{c x^3}\right )\)

Input:

Int[(2*b + a*x^3)/(x^4*(c - 2*b^2*x^3 + a*x^6)),x]
 

Output:

((-2*b)/(c*x^3) + (b*(4*b^4 - 2*a*c + a*b*c)*ArcTanh[(b^2 - a*x^3)/Sqrt[b^ 
4 - a*c]])/(c^2*Sqrt[b^4 - a*c]) + ((4*b^3 + a*c)*Log[x^3])/c^2 - ((4*b^3 
+ a*c)*Log[c - 2*b^2*x^3 + a*x^6])/(2*c^2))/3
 

Defintions of rubi rules used

rule 1200

Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_.) + (b_.)* 
(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g* 
x)^n/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && In 
tegersQ[n]
 

rule 1802

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( 
e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1 
)/n] - 1)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, 
c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\left (4 b^{3}+a c \right ) \ln \left (x \right )}{c^{2}}-\frac {2 b}{3 c \,x^{3}}-\frac {\frac {\left (4 a \,b^{3}+a^{2} c \right ) \ln \left (a \,x^{6}-2 b^{2} x^{3}+c \right )}{2 a}-\frac {\left (-8 b^{5}-2 a \,b^{2} c +2 a b c +\frac {\left (4 a \,b^{3}+a^{2} c \right ) b^{2}}{a}\right ) \operatorname {arctanh}\left (\frac {2 a \,x^{3}-2 b^{2}}{2 \sqrt {b^{4}-a c}}\right )}{\sqrt {b^{4}-a c}}}{3 c^{2}}\)	\(139\)
risch	\(-\frac {2 b}{3 c \,x^{3}}+\frac {4 \ln \left (x \right ) b^{3}}{c^{2}}+\frac {\ln \left (x \right ) a}{c}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (-c^{2} b^{4}+a \,c^{3}\right ) \textit {\_Z}^{2}+\left (-8 b^{7}-2 a \,b^{4} c +8 a \,b^{3} c +2 a^{2} c^{2}\right ) \textit {\_Z} +4 a^{2} b^{3}+a^{3} c +4 a^{2} b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (8 c^{2} b^{4}-7 a \,c^{3}\right ) \textit {\_R}^{2}+\left (-24 a \,b^{3} c -7 a^{2} c^{2}\right ) \textit {\_R} -24 a^{2} b^{2}\right ) x^{3}-b^{2} c^{3} \textit {\_R}^{2}+\left (24 b^{5} c +6 c^{2} b^{2} a -2 a b \,c^{2}\right ) \textit {\_R} +48 a \,b^{4}+12 a^{2} b c \right )\right )}{6}\)	\(207\)

Input:

int((a*x^3+2*b)/x^4/(a*x^6-2*b^2*x^3+c),x,method=_RETURNVERBOSE)
 

Output:

(4*b^3+a*c)*ln(x)/c^2-2/3*b/c/x^3-1/3/c^2*(1/2*(4*a*b^3+a^2*c)/a*ln(a*x^6- 
2*b^2*x^3+c)-(-8*b^5-2*a*b^2*c+2*a*b*c+(4*a*b^3+a^2*c)*b^2/a)/(b^4-a*c)^(1 
/2)*arctanh(1/2*(2*a*x^3-2*b^2)/(b^4-a*c)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 407, normalized size of antiderivative = 3.48 \[ \int \frac {2 b+a x^3}{x^4 \left (c-2 b^2 x^3+a x^6\right )} \, dx=\left [-\frac {4 \, b^{5} c - {\left (4 \, b^{5} + {\left (a b^{2} - 2 \, a b\right )} c\right )} \sqrt {b^{4} - a c} x^{3} \log \left (\frac {a^{2} x^{6} - 2 \, a b^{2} x^{3} + 2 \, b^{4} - a c - 2 \, \sqrt {b^{4} - a c} {\left (a x^{3} - b^{2}\right )}}{a x^{6} - 2 \, b^{2} x^{3} + c}\right ) + {\left (4 \, b^{7} - a^{2} c^{2} + {\left (a b^{4} - 4 \, a b^{3}\right )} c\right )} x^{3} \log \left (a x^{6} - 2 \, b^{2} x^{3} + c\right ) - 6 \, {\left (4 \, b^{7} - a^{2} c^{2} + {\left (a b^{4} - 4 \, a b^{3}\right )} c\right )} x^{3} \log \left (x\right ) - 4 \, a b c^{2}}{6 \, {\left (b^{4} c^{2} - a c^{3}\right )} x^{3}}, -\frac {4 \, b^{5} c + 2 \, {\left (4 \, b^{5} + {\left (a b^{2} - 2 \, a b\right )} c\right )} \sqrt {-b^{4} + a c} x^{3} \arctan \left (-\frac {\sqrt {-b^{4} + a c} {\left (a x^{3} - b^{2}\right )}}{b^{4} - a c}\right ) + {\left (4 \, b^{7} - a^{2} c^{2} + {\left (a b^{4} - 4 \, a b^{3}\right )} c\right )} x^{3} \log \left (a x^{6} - 2 \, b^{2} x^{3} + c\right ) - 6 \, {\left (4 \, b^{7} - a^{2} c^{2} + {\left (a b^{4} - 4 \, a b^{3}\right )} c\right )} x^{3} \log \left (x\right ) - 4 \, a b c^{2}}{6 \, {\left (b^{4} c^{2} - a c^{3}\right )} x^{3}}\right ] \] Input:

integrate((a*x^3+2*b)/x^4/(a*x^6-2*b^2*x^3+c),x, algorithm="fricas")
 

Output:

[-1/6*(4*b^5*c - (4*b^5 + (a*b^2 - 2*a*b)*c)*sqrt(b^4 - a*c)*x^3*log((a^2* 
x^6 - 2*a*b^2*x^3 + 2*b^4 - a*c - 2*sqrt(b^4 - a*c)*(a*x^3 - b^2))/(a*x^6 
- 2*b^2*x^3 + c)) + (4*b^7 - a^2*c^2 + (a*b^4 - 4*a*b^3)*c)*x^3*log(a*x^6 
- 2*b^2*x^3 + c) - 6*(4*b^7 - a^2*c^2 + (a*b^4 - 4*a*b^3)*c)*x^3*log(x) - 
4*a*b*c^2)/((b^4*c^2 - a*c^3)*x^3), -1/6*(4*b^5*c + 2*(4*b^5 + (a*b^2 - 2* 
a*b)*c)*sqrt(-b^4 + a*c)*x^3*arctan(-sqrt(-b^4 + a*c)*(a*x^3 - b^2)/(b^4 - 
 a*c)) + (4*b^7 - a^2*c^2 + (a*b^4 - 4*a*b^3)*c)*x^3*log(a*x^6 - 2*b^2*x^3 
 + c) - 6*(4*b^7 - a^2*c^2 + (a*b^4 - 4*a*b^3)*c)*x^3*log(x) - 4*a*b*c^2)/ 
((b^4*c^2 - a*c^3)*x^3)]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {2 b+a x^3}{x^4 \left (c-2 b^2 x^3+a x^6\right )} \, dx=\text {Timed out} \] Input:

integrate((a*x**3+2*b)/x**4/(a*x**6-2*b**2*x**3+c),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {2 b+a x^3}{x^4 \left (c-2 b^2 x^3+a x^6\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((a*x^3+2*b)/x^4/(a*x^6-2*b^2*x^3+c),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*b^4-4*a*c>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.12 \[ \int \frac {2 b+a x^3}{x^4 \left (c-2 b^2 x^3+a x^6\right )} \, dx=-\frac {{\left (4 \, b^{3} + a c\right )} \log \left (a x^{6} - 2 \, b^{2} x^{3} + c\right )}{6 \, c^{2}} + \frac {{\left (4 \, b^{3} + a c\right )} \log \left ({\left | x \right |}\right )}{c^{2}} + \frac {{\left (4 \, b^{5} + a b^{2} c - 2 \, a b c\right )} \arctan \left (\frac {a x^{3} - b^{2}}{\sqrt {-b^{4} + a c}}\right )}{3 \, \sqrt {-b^{4} + a c} c^{2}} - \frac {4 \, b^{3} x^{3} + a c x^{3} + 2 \, b c}{3 \, c^{2} x^{3}} \] Input:

integrate((a*x^3+2*b)/x^4/(a*x^6-2*b^2*x^3+c),x, algorithm="giac")
 

Output:

-1/6*(4*b^3 + a*c)*log(a*x^6 - 2*b^2*x^3 + c)/c^2 + (4*b^3 + a*c)*log(abs( 
x))/c^2 + 1/3*(4*b^5 + a*b^2*c - 2*a*b*c)*arctan((a*x^3 - b^2)/sqrt(-b^4 + 
 a*c))/(sqrt(-b^4 + a*c)*c^2) - 1/3*(4*b^3*x^3 + a*c*x^3 + 2*b*c)/(c^2*x^3 
)
 

Mupad [B] (verification not implemented)

Time = 19.63 (sec) , antiderivative size = 7097, normalized size of antiderivative = 60.66 \[ \int \frac {2 b+a x^3}{x^4 \left (c-2 b^2 x^3+a x^6\right )} \, dx=\text {Too large to display} \] Input:

int((2*b + a*x^3)/(x^4*(c + a*x^6 - 2*b^2*x^3)),x)
 

Output:

(log(x)*(a*c + 4*b^3))/c^2 - (2*b)/(3*c*x^3) + (log((((((((a*c + 4*b^3 + c 
^2*(-(b^2*(4*b^4 - 2*a*c + a*b*c)^2)/(c^4*(a*c - b^4)))^(1/2))*((216*a^3*b 
^5*(4*b^4 - a*c + a*b*c))/c + (36*a^3*b^4*(a*c + 4*b^3 + c^2*(-(b^2*(4*b^4 
 - 2*a*c + a*b*c)^2)/(c^4*(a*c - b^4)))^(1/2))*(b^2*c - 8*b^4*x^3 + 7*a*c* 
x^3))/c^2 - (36*a^4*b^3*x^3*(28*a*c - 8*b^4 + 7*a*b*c))/c))/(6*c^2) - (72* 
a^4*b^4*(12*b^4 - a*c + 3*a*b*c))/c^2 + (24*a^5*b^2*x^3*(7*a*c + 22*b^4 + 
7*a*b*c))/c^2)*(a*c + 4*b^3 + c^2*(-(b^2*(4*b^4 - 2*a*c + a*b*c)^2)/(c^4*( 
a*c - b^4)))^(1/2)))/(6*c^2) + (8*a^5*b^3*(36*b^4 - a*c + 9*a*b*c))/c^3 - 
(4*a^6*b^2*x^3*(7*a*c + 48*b^3))/c^3)*(a*c + 4*b^3 + c^2*(-(b^2*(4*b^4 - 2 
*a*c + a*b*c)^2)/(c^4*(a*c - b^4)))^(1/2)))/(6*c^2) - (8*a^6*b^3*(a*c + 4* 
b^3))/c^4 + (16*a^7*b^4*x^3)/c^4)*(((((((a*c + 4*b^3 - c^2*(-(b^2*(4*b^4 - 
 2*a*c + a*b*c)^2)/(c^4*(a*c - b^4)))^(1/2))*((216*a^3*b^5*(4*b^4 - a*c + 
a*b*c))/c + (36*a^3*b^4*(a*c + 4*b^3 - c^2*(-(b^2*(4*b^4 - 2*a*c + a*b*c)^ 
2)/(c^4*(a*c - b^4)))^(1/2))*(b^2*c - 8*b^4*x^3 + 7*a*c*x^3))/c^2 - (36*a^ 
4*b^3*x^3*(28*a*c - 8*b^4 + 7*a*b*c))/c))/(6*c^2) - (72*a^4*b^4*(12*b^4 - 
a*c + 3*a*b*c))/c^2 + (24*a^5*b^2*x^3*(7*a*c + 22*b^4 + 7*a*b*c))/c^2)*(a* 
c + 4*b^3 - c^2*(-(b^2*(4*b^4 - 2*a*c + a*b*c)^2)/(c^4*(a*c - b^4)))^(1/2) 
))/(6*c^2) + (8*a^5*b^3*(36*b^4 - a*c + 9*a*b*c))/c^3 - (4*a^6*b^2*x^3*(7* 
a*c + 48*b^3))/c^3)*(a*c + 4*b^3 - c^2*(-(b^2*(4*b^4 - 2*a*c + a*b*c)^2)/( 
c^4*(a*c - b^4)))^(1/2)))/(6*c^2) - (8*a^6*b^3*(a*c + 4*b^3))/c^4 + (16...
 

Reduce [F]

\[ \int \frac {2 b+a x^3}{x^4 \left (c-2 b^2 x^3+a x^6\right )} \, dx=\frac {-6 \left (\int \frac {1}{-2 a \,b^{4} x^{10}+a^{2} c \,x^{10}+4 b^{6} x^{7}-2 a \,b^{2} c \,x^{7}-2 b^{4} c \,x^{4}+a \,c^{2} x^{4}}d x \right ) a^{2} b^{2} c^{2} x^{3}+12 \left (\int \frac {1}{-2 a \,b^{4} x^{10}+a^{2} c \,x^{10}+4 b^{6} x^{7}-2 a \,b^{2} c \,x^{7}-2 b^{4} c \,x^{4}+a \,c^{2} x^{4}}d x \right ) a^{2} b \,c^{2} x^{3}+12 \left (\int \frac {1}{-2 a \,b^{4} x^{10}+a^{2} c \,x^{10}+4 b^{6} x^{7}-2 a \,b^{2} c \,x^{7}-2 b^{4} c \,x^{4}+a \,c^{2} x^{4}}d x \right ) a \,b^{6} c \,x^{3}-48 \left (\int \frac {1}{-2 a \,b^{4} x^{10}+a^{2} c \,x^{10}+4 b^{6} x^{7}-2 a \,b^{2} c \,x^{7}-2 b^{4} c \,x^{4}+a \,c^{2} x^{4}}d x \right ) a \,b^{5} c \,x^{3}+48 \left (\int \frac {1}{-2 a \,b^{4} x^{10}+a^{2} c \,x^{10}+4 b^{6} x^{7}-2 a \,b^{2} c \,x^{7}-2 b^{4} c \,x^{4}+a \,c^{2} x^{4}}d x \right ) b^{9} x^{3}-\mathrm {log}\left (a \,x^{6}-2 b^{2} x^{3}+c \right ) a^{2} x^{3}+6 \,\mathrm {log}\left (x \right ) a^{2} x^{3}-2 a \,b^{2}}{6 x^{3} \left (-2 b^{4}+a c \right )} \] Input:

int((a*x^3+2*b)/x^4/(a*x^6-2*b^2*x^3+c),x)
 

Output:

( - 6*int(1/(a**2*c*x**10 - 2*a*b**4*x**10 - 2*a*b**2*c*x**7 + a*c**2*x**4 
 + 4*b**6*x**7 - 2*b**4*c*x**4),x)*a**2*b**2*c**2*x**3 + 12*int(1/(a**2*c* 
x**10 - 2*a*b**4*x**10 - 2*a*b**2*c*x**7 + a*c**2*x**4 + 4*b**6*x**7 - 2*b 
**4*c*x**4),x)*a**2*b*c**2*x**3 + 12*int(1/(a**2*c*x**10 - 2*a*b**4*x**10 
- 2*a*b**2*c*x**7 + a*c**2*x**4 + 4*b**6*x**7 - 2*b**4*c*x**4),x)*a*b**6*c 
*x**3 - 48*int(1/(a**2*c*x**10 - 2*a*b**4*x**10 - 2*a*b**2*c*x**7 + a*c**2 
*x**4 + 4*b**6*x**7 - 2*b**4*c*x**4),x)*a*b**5*c*x**3 + 48*int(1/(a**2*c*x 
**10 - 2*a*b**4*x**10 - 2*a*b**2*c*x**7 + a*c**2*x**4 + 4*b**6*x**7 - 2*b* 
*4*c*x**4),x)*b**9*x**3 - log(a*x**6 - 2*b**2*x**3 + c)*a**2*x**3 + 6*log( 
x)*a**2*x**3 - 2*a*b**2)/(6*x**3*(a*c - 2*b**4))