3.16.67 \(\int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} (b+2 a^6 x^6)} \, dx\) [1567]

3.16.67.1 Optimal result

Integrand size = 32, antiderivative size = 107 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+2 a^6 x^6\right )} \, dx=-\frac {\text {RootSum}\left [a^{18}+2 a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b} \]

output

Unintegrable
 

3.16.67.2 Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.35 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+2 a^6 x^6\right )} \, dx=-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \text {RootSum}\left [a^{18}+2 a^6 b^{11}-6 a^{15} \text {$\#$1}^3+15 a^{12} \text {$\#$1}^6-20 a^9 \text {$\#$1}^9+15 a^6 \text {$\#$1}^{12}-6 a^3 \text {$\#$1}^{15}+\text {$\#$1}^{18}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]

input

Integrate[1/((b^2*x^2 + a^3*x^3)^(1/3)*(b + 2*a^6*x^6)),x]
 

output

-1/6*(x^(2/3)*(b^2 + a^3*x)^(1/3)*RootSum[a^18 + 2*a^6*b^11 - 6*a^15*#1^3 
+ 15*a^12*#1^6 - 20*a^9*#1^9 + 15*a^6*#1^12 - 6*a^3*#1^15 + #1^18 & , (-Lo 
g[x^(1/3)] + Log[(b^2 + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(b*(x^2*(b^2 + 
 a^3*x))^(1/3))
 

3.16.67.3 Rubi [F]

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.

\(\displaystyle \int \frac {1}{\sqrt [3]{a^3 x^3+b^2 x^2} \left (2 a^6 x^6+b\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{x^{2/3} \sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 2035

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {-b}-\sqrt {2} a^3 x^3\right )}+\frac {\sqrt {-b}}{2 b \sqrt [3]{x a^3+b^2} \left (\sqrt {2} a^3 x^3+\sqrt {-b}\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {1}{\sqrt [3]{x a^3+b^2} \left (2 a^6 x^6+b\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\)

input

Int[1/((b^2*x^2 + a^3*x^3)^(1/3)*(b + 2*a^6*x^6)),x]
 

output

$Aborted
 

3.16.67.3.1 Defintions of rubi rules used

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7276

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]
 

3.16.67.4 Maple [N/A] (verified)

Time = 0.45 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{18}-6 a^{3} \textit {\_Z}^{15}+15 a^{6} \textit {\_Z}^{12}-20 a^{9} \textit {\_Z}^{9}+15 a^{12} \textit {\_Z}^{6}-6 a^{15} \textit {\_Z}^{3}+a^{18}+2 a^{6} b^{11}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{6 b}\)	\(94\)

input

int(1/(a^3*x^3+b^2*x^2)^(1/3)/(2*a^6*x^6+b),x,method=_RETURNVERBOSE)
 

output

-1/6*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=RootOf(_Z^18-6*_Z^15* 
a^3+15*_Z^12*a^6-20*_Z^9*a^9+15*_Z^6*a^12-6*_Z^3*a^15+a^18+2*a^6*b^11))/b
 

3.16.67.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+2 a^6 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \]

input

integrate(1/(a^3*x^3+b^2*x^2)^(1/3)/(2*a^6*x^6+b),x, algorithm="fricas")
 

output

Exception raised: RuntimeError >>  System error:   Heap exhausted (no more 
 space for allocation).3014656 bytes available, 45670544 requested.PROCEED 
 WITH CAUTION.
 

3.16.67.6 Sympy [N/A]

Not integrable

Time = 6.93 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.25 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+2 a^6 x^6\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a^{6} x^{6} + b\right )}\, dx \]

input

integrate(1/(a**3*x**3+b**2*x**2)**(1/3)/(2*a**6*x**6+b),x)
 

output

Integral(1/((x**2*(a**3*x + b**2))**(1/3)*(2*a**6*x**6 + b)), x)
 

3.16.67.7 Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+2 a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (2 \, a^{6} x^{6} + b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate(1/(a^3*x^3+b^2*x^2)^(1/3)/(2*a^6*x^6+b),x, algorithm="maxima")
 

output

integrate(1/((2*a^6*x^6 + b)*(a^3*x^3 + b^2*x^2)^(1/3)), x)
 

3.16.67.8 Giac [N/A]

Not integrable

Time = 0.76 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+2 a^6 x^6\right )} \, dx=\int { \frac {1}{{\left (2 \, a^{6} x^{6} + b\right )} {\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

input

integrate(1/(a^3*x^3+b^2*x^2)^(1/3)/(2*a^6*x^6+b),x, algorithm="giac")
 

output

integrate(1/((2*a^6*x^6 + b)*(a^3*x^3 + b^2*x^2)^(1/3)), x)
 

3.16.67.9 Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt [3]{b^2 x^2+a^3 x^3} \left (b+2 a^6 x^6\right )} \, dx=\int \frac {1}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a^6\,x^6+b\right )} \,d x \]

input

int(1/((a^3*x^3 + b^2*x^2)^(1/3)*(b + 2*a^6*x^6)),x)
 

output

int(1/((a^3*x^3 + b^2*x^2)^(1/3)*(b + 2*a^6*x^6)), x)