Integrand size = 41, antiderivative size = 204 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a}-\frac {1}{4} \text {RootSum}\left [a^6-2 a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Output:
Unintegrable
Time = 0.00 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.11 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{b^2+a^3 x}}\right )-2 \log \left (a \left (a \sqrt [3]{x}-\sqrt [3]{b^2+a^3 x}\right )\right )+\log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{b^2+a^3 x}+\left (b^2+a^3 x\right )^{2/3}\right )-a \text {RootSum}\left [a^6-2 a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{4 a \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \] Input:
Integrate[(-b + a*x^2)/((-b + 2*a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]
Output:
(x^(2/3)*(b^2 + a^3*x)^(1/3)*(2*Sqrt[3]*ArcTan[(Sqrt[3]*a*x^(1/3))/(a*x^(1 /3) + 2*(b^2 + a^3*x)^(1/3))] - 2*Log[a*(a*x^(1/3) - (b^2 + a^3*x)^(1/3))] + Log[a^2*x^(2/3) + a*x^(1/3)*(b^2 + a^3*x)^(1/3) + (b^2 + a^3*x)^(2/3)] - a*RootSum[a^6 - 2*a*b^3 - 2*a^3*#1^3 + #1^6 & , (-Log[x^(1/3)] + Log[(b^ 2 + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ]))/(4*a*(x^2*(b^2 + a^3*x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(559\) vs. \(2(204)=408\).
Time = 1.14 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.74, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {2467, 2035, 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.
\(\displaystyle \int \frac {a x^2-b}{\left (2 a x^2-b\right ) \sqrt [3]{a^3 x^3+b^2 x^2}} \, dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {b-a x^2}{x^{2/3} \sqrt [3]{x a^3+b^2} \left (b-2 a x^2\right )}dx}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \frac {b-a x^2}{\sqrt [3]{x a^3+b^2} \left (b-2 a x^2\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \int \left (\frac {b}{2 \sqrt [3]{x a^3+b^2} \left (b-2 a x^2\right )}+\frac {1}{2 \sqrt [3]{x a^3+b^2}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a^3 x+b^2} \left (\frac {\log \left (\sqrt {2} \sqrt {a} \sqrt {b}-2 a x\right )}{24 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}}+\frac {\log \left (\sqrt {2} \sqrt {a} \sqrt {b}+2 a x\right )}{24 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}}+\frac {\arctan \left (\frac {\frac {2 a \sqrt [3]{x}}{\sqrt [3]{a^3 x+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a}-\frac {\log \left (\sqrt [3]{a^3 x+b^2}-a \sqrt [3]{x}\right )}{4 a}+\frac {\arctan \left (\frac {\frac {2 \sqrt [6]{a} \sqrt [3]{x} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}}{\sqrt [3]{a^3 x+b^2}}+1}{\sqrt {3}}\right )}{4 \sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [6]{a} \sqrt [3]{x} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}}{\sqrt [3]{a^3 x+b^2}}+1}{\sqrt {3}}\right )}{4 \sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}}-\frac {\log \left (\sqrt [6]{a} \sqrt [3]{x} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}-\sqrt [3]{a^3 x+b^2}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}}-\frac {\log \left (\sqrt [6]{a} \sqrt [3]{x} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}-\sqrt [3]{a^3 x+b^2}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}}\right )}{\sqrt [3]{a^3 x^3+b^2 x^2}}\) |
Input:
Int[(-b + a*x^2)/((-b + 2*a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]
Output:
(3*x^(2/3)*(b^2 + a^3*x)^(1/3)*(ArcTan[(1 + (2*a*x^(1/3))/(b^2 + a^3*x)^(1 /3))/Sqrt[3]]/(2*Sqrt[3]*a) + ArcTan[(1 + (2*a^(1/6)*(a^(5/2) - Sqrt[2]*b^ (3/2))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[3]]/(4*Sqrt[3]*a^(1/6)*(a^ (5/2) - Sqrt[2]*b^(3/2))^(1/3)) + ArcTan[(1 + (2*a^(1/6)*(a^(5/2) + Sqrt[2 ]*b^(3/2))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[3]]/(4*Sqrt[3]*a^(1/6) *(a^(5/2) + Sqrt[2]*b^(3/2))^(1/3)) + Log[Sqrt[2]*Sqrt[a]*Sqrt[b] - 2*a*x] /(24*a^(1/6)*(a^(5/2) + Sqrt[2]*b^(3/2))^(1/3)) + Log[Sqrt[2]*Sqrt[a]*Sqrt [b] + 2*a*x]/(24*a^(1/6)*(a^(5/2) - Sqrt[2]*b^(3/2))^(1/3)) - Log[a^(1/6)* (a^(5/2) - Sqrt[2]*b^(3/2))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)]/(8*a^(1/6 )*(a^(5/2) - Sqrt[2]*b^(3/2))^(1/3)) - Log[a^(1/6)*(a^(5/2) + Sqrt[2]*b^(3 /2))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)]/(8*a^(1/6)*(a^(5/2) + Sqrt[2]*b^ (3/2))^(1/3)) - Log[-(a*x^(1/3)) + (b^2 + a^3*x)^(1/3)]/(4*a)))/(b^2*x^2 + a^3*x^3)^(1/3)
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]
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]
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]
Time = 0.00 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.86
method | result | size |
pseudoelliptic | \(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}-2 a \,b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) a +\ln \left (\frac {a^{2} x^{2}+\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{4 a}\) | \(175\) |
Input:
int((a*x^2-b)/(2*a*x^2-b)/(a^3*x^3+b^2*x^2)^(1/3),x,method=_RETURNVERBOSE)
Output:
1/4*(-2*3^(1/2)*arctan(1/3*(a*x+2*(x^2*(a^3*x+b^2))^(1/3))*3^(1/2)/a/x)-su m(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/3))/x)/_R,_R=RootOf(_Z^6-2*_Z^3*a^3+a^6-2 *a*b^3))*a+ln((a^2*x^2+(x^2*(a^3*x+b^2))^(1/3)*a*x+(x^2*(a^3*x+b^2))^(2/3) )/x^2)-2*ln((-a*x+(x^2*(a^3*x+b^2))^(1/3))/x))/a
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.12 (sec) , antiderivative size = 1444, normalized size of antiderivative = 7.08 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \] Input:
integrate((a*x^2-b)/(2*a*x^2-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="fric as")
Output:
1/8*((sqrt(-3)*a - a)*(-(2*sqrt(1/2)*(a^5 - 2*b^3)*sqrt(b^3/(a^11 - 4*a^6* b^3 + 4*a*b^6)) + a^2)/(a^5 - 2*b^3))^(1/3)*log(1/2*((sqrt(-3)*a^3*x + a^3 *x - 2*sqrt(1/2)*(sqrt(-3)*(a^6 - 2*a*b^3)*x + (a^6 - 2*a*b^3)*x)*sqrt(b^3 /(a^11 - 4*a^6*b^3 + 4*a*b^6)))*(-(2*sqrt(1/2)*(a^5 - 2*b^3)*sqrt(b^3/(a^1 1 - 4*a^6*b^3 + 4*a*b^6)) + a^2)/(a^5 - 2*b^3))^(2/3) + 2*(a^3*x^3 + b^2*x ^2)^(1/3))/x) - (sqrt(-3)*a + a)*(-(2*sqrt(1/2)*(a^5 - 2*b^3)*sqrt(b^3/(a^ 11 - 4*a^6*b^3 + 4*a*b^6)) + a^2)/(a^5 - 2*b^3))^(1/3)*log(-1/2*((sqrt(-3) *a^3*x - a^3*x - 2*sqrt(1/2)*(sqrt(-3)*(a^6 - 2*a*b^3)*x - (a^6 - 2*a*b^3) *x)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6)))*(-(2*sqrt(1/2)*(a^5 - 2*b^3)*s qrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6)) + a^2)/(a^5 - 2*b^3))^(2/3) - 2*(a^3 *x^3 + b^2*x^2)^(1/3))/x) + 2*a*(-(2*sqrt(1/2)*(a^5 - 2*b^3)*sqrt(b^3/(a^1 1 - 4*a^6*b^3 + 4*a*b^6)) + a^2)/(a^5 - 2*b^3))^(1/3)*log(-((a^3*x - 2*sqr t(1/2)*(a^6 - 2*a*b^3)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6))*x)*(-(2*sqrt (1/2)*(a^5 - 2*b^3)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6)) + a^2)/(a^5 - 2 *b^3))^(2/3) - (a^3*x^3 + b^2*x^2)^(1/3))/x) + (sqrt(-3)*a - a)*((2*sqrt(1 /2)*(a^5 - 2*b^3)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6)) - a^2)/(a^5 - 2*b ^3))^(1/3)*log(1/2*((sqrt(-3)*a^3*x + a^3*x + 2*sqrt(1/2)*(sqrt(-3)*(a^6 - 2*a*b^3)*x + (a^6 - 2*a*b^3)*x)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6)))*( (2*sqrt(1/2)*(a^5 - 2*b^3)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6)) - a^2)/( a^5 - 2*b^3))^(2/3) + 2*(a^3*x^3 + b^2*x^2)^(1/3))/x) - (sqrt(-3)*a + a...
Not integrable
Time = 2.56 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.15 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a x^{2} - b}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{2} - b\right )}\, dx \] Input:
integrate((a*x**2-b)/(2*a*x**2-b)/(a**3*x**3+b**2*x**2)**(1/3),x)
Output:
Integral((a*x**2 - b)/((x**2*(a**3*x + b**2))**(1/3)*(2*a*x**2 - b)), x)
Not integrable
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.20 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{2} - b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{2} - b\right )}} \,d x } \] Input:
integrate((a*x^2-b)/(2*a*x^2-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="maxi ma")
Output:
integrate((a*x^2 - b)/((a^3*x^3 + b^2*x^2)^(1/3)*(2*a*x^2 - b)), x)
Not integrable
Time = 3.43 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.20 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{2} - b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{2} - b\right )}} \,d x } \] Input:
integrate((a*x^2-b)/(2*a*x^2-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="giac ")
Output:
integrate((a*x^2 - b)/((a^3*x^3 + b^2*x^2)^(1/3)*(2*a*x^2 - b)), x)
Not integrable
Time = 0.00 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.19 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {b-a\,x^2}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (b-2\,a\,x^2\right )} \,d x \] Input:
int((b - a*x^2)/((a^3*x^3 + b^2*x^2)^(1/3)*(b - 2*a*x^2)),x)
Output:
int((b - a*x^2)/((a^3*x^3 + b^2*x^2)^(1/3)*(b - 2*a*x^2)), x)
Not integrable
Time = 0.51 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.43 \[ \int \frac {-b+a x^2}{\left (-b+2 a x^2\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\left (\int \frac {x^{2}}{2 x^{\frac {8}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} a -x^{\frac {2}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} b}d x \right ) a -\left (\int \frac {1}{2 x^{\frac {8}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} a -x^{\frac {2}{3}} \left (a^{3} x +b^{2}\right )^{\frac {1}{3}} b}d x \right ) b \] Input:
int((a*x^2-b)/(2*a*x^2-b)/(a^3*x^3+b^2*x^2)^(1/3),x)
Output:
int(x**2/(2*x**(2/3)*(a**3*x + b**2)**(1/3)*a*x**2 - x**(2/3)*(a**3*x + b* *2)**(1/3)*b),x)*a - int(1/(2*x**(2/3)*(a**3*x + b**2)**(1/3)*a*x**2 - x** (2/3)*(a**3*x + b**2)**(1/3)*b),x)*b