3.25.78 \(\int \frac {-b+a x^2}{(-b+2 a x^2) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\)

Optimal. Leaf size=204 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6-2 a b^3\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]-\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-a x\right )}{2 a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b^2 x^2}+a x}\right )}{2 a}+\frac {\log \left (a x \sqrt [3]{a^3 x^3+b^2 x^2}+\left (a^3 x^3+b^2 x^2\right )^{2/3}+a^2 x^2\right )}{4 a} \]

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Rubi [B]  time = 1.05, antiderivative size = 870, normalized size of antiderivative = 4.26, number of steps used = 8, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2056, 6725, 59, 912, 91} \begin {gather*} -\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 a \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{x a^3+b^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log (x)}{4 a \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt {b}-\sqrt {2} \sqrt {a} x\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt {2} \sqrt {a} x+\sqrt {b}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {3 x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{a \sqrt [3]{x}}-1\right )}{4 a \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[3]*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a*x^(1/3))])/(a*
(b^2*x^2 + a^3*x^3)^(1/3)) - (Sqrt[3]*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/(
Sqrt[3]*a^(1/6)*(a^(5/2) - Sqrt[2]*b^(3/2))^(1/3)*x^(1/3))])/(4*a^(1/6)*(a^(5/2) - Sqrt[2]*b^(3/2))^(1/3)*(b^2
*x^2 + a^3*x^3)^(1/3)) - (Sqrt[3]*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/(Sqrt
[3]*a^(1/6)*(a^(5/2) + Sqrt[2]*b^(3/2))^(1/3)*x^(1/3))])/(4*a^(1/6)*(a^(5/2) + Sqrt[2]*b^(3/2))^(1/3)*(b^2*x^2
+ a^3*x^3)^(1/3)) - (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[x])/(4*a*(b^2*x^2 + a^3*x^3)^(1/3)) + (x^(2/3)*(b^2 + a^
3*x)^(1/3)*Log[Sqrt[b] - Sqrt[2]*Sqrt[a]*x])/(8*a^(1/6)*(a^(5/2) + Sqrt[2]*b^(3/2))^(1/3)*(b^2*x^2 + a^3*x^3)^
(1/3)) + (x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[Sqrt[b] + Sqrt[2]*Sqrt[a]*x])/(8*a^(1/6)*(a^(5/2) - Sqrt[2]*b^(3/2))
^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (3*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^(1/6)
*(a^(5/2) - Sqrt[2]*b^(3/2))^(1/3))])/(8*a^(1/6)*(a^(5/2) - Sqrt[2]*b^(3/2))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))
- (3*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^(1/6)*(a^(5/2) + Sqrt[2]*b^(3/2))^(1/3)
)])/(8*a^(1/6)*(a^(5/2) + Sqrt[2]*b^(3/2))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (3*x^(2/3)*(b^2 + a^3*x)^(1/3)*L
og[-1 + (b^2 + a^3*x)^(1/3)/(a*x^(1/3))])/(4*a*(b^2*x^2 + a^3*x^3)^(1/3))

Rule 59

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, -Simp[(Sqrt
[3]*q*ArcTan[(2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/3)) + 1/Sqrt[3]])/d, x] + (-Simp[(3*q*Log[(q*(a + b*x
)^(1/3))/(c + d*x)^(1/3) - 1])/(2*d), x] - Simp[(q*Log[c + d*x])/(2*d), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[
b*c - a*d, 0] && PosQ[d/b]

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 912

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \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 {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {-b+a x^2}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (-b+2 a x^2\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (\frac {1}{2 x^{2/3} \sqrt [3]{b^2+a^3 x}}-\frac {b}{2 x^{2/3} \sqrt [3]{b^2+a^3 x} \left (-b+2 a x^2\right )}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (b x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x} \left (-b+2 a x^2\right )} \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (b x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-\sqrt {2} \sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+\sqrt {2} \sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{2 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-\sqrt {2} \sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{4 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+\sqrt {2} \sqrt {a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{4 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}-\sqrt {2} \sqrt {a} x\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt {b}+\sqrt {2} \sqrt {a} x\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}}}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}-\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}}}\right )}{8 \sqrt [6]{a} \sqrt [3]{a^{5/2}+\sqrt {2} b^{3/2}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 145, normalized size = 0.71 \begin {gather*} \frac {3 x \left (2 \sqrt [3]{\frac {a^3 x}{b^2}+1} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a^3 x-\sqrt {2} \sqrt {a} b^{3/2} x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x a^3+\sqrt {2} b^{3/2} x \sqrt {a}}{x a^3+b^2}\right )\right )}{4 \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*(2*(1 + (a^3*x)/b^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -((a^3*x)/b^2)] + Hypergeometric2F1[1/3, 1, 4
/3, (a^3*x - Sqrt[2]*Sqrt[a]*b^(3/2)*x)/(b^2 + a^3*x)] + Hypergeometric2F1[1/3, 1, 4/3, (a^3*x + Sqrt[2]*Sqrt[
a]*b^(3/2)*x)/(b^2 + a^3*x)]))/(4*(x^2*(b^2 + a^3*x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 208, normalized size = 1.02 \begin {gather*} \frac {\sqrt {3} \tan ^{-1}\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^2 x-a \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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-b + a*x^2)/((-b + 2*a*x^2)*(b^2*x^2 + a^3*x^3)^(1/3)),x]

[Out]

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

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fricas [B]  time = 0.65, size = 2150, normalized size = 10.54

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*sqrt(3)*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)*arctan(1/3*(2*sqrt(3)*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))^(1/3)*sqrt(((a^3*x^3 + b^2*x^2)^(1/3)*(a^3*x - 2*sqrt(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^2 - 2*sqrt(1/2)*(a^6 - 2*a*b^3)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6))*x^2)*(-(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) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2) + sqr
t(3)*x - 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3)*(-(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))/x) + 4*sqrt(3)*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)*arctan(1/3*(2*sqrt(3)*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))^(1/3)*sqrt(((a^3*x^3 + b^2*x^2)^(1/3)*(a^3*x + 2*sqrt(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^2 + 2*sqrt(1/2)*(a^6 - 2*a*b^3)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6))*x
^2)*((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) + (a^3*x^3
+ b^2*x^2)^(2/3))/x^2) + sqrt(3)*x - 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3)*((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))/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*sqrt(1/2)*(a^6 - 2*a*b^3)*sqrt(b^3/(a^1
1 - 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) + 2*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(-((a^3*x + 2*sqrt(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) - 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(((a^3*x^3 + b^2*x^2)^(1/3)*(a^3*x - 2*sqrt(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^2 - 2*sqrt(1/2)*(a^6 - 2*a*b^3)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6))*x^2)*(-(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) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2)
- 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(((a^3*
x^3 + b^2*x^2)^(1/3)*(a^3*x + 2*sqrt(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^2 + 2*sqrt(1/2)*
(a^6 - 2*a*b^3)*sqrt(b^3/(a^11 - 4*a^6*b^3 + 4*a*b^6))*x^2)*((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) + (a^3*x^3 + b^2*x^2)^(2/3))/x^2) - 4*sqrt(3)*arctan(1/3*(sqrt(3)
*a*x + 2*sqrt(3)*(a^3*x^3 + b^2*x^2)^(1/3))/(a*x)) - 4*log(-(a*x - (a^3*x^3 + b^2*x^2)^(1/3))/x) + 2*log((a^2*
x^2 + (a^3*x^3 + b^2*x^2)^(1/3)*a*x + (a^3*x^3 + b^2*x^2)^(2/3))/x^2))/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b}{\left (2 a \,x^{2}-b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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