3.24.68 \(\int \frac {-b+a^3 x^2}{(b+a^3 x^2) \sqrt [3]{-b x^2+a^3 x^3}} \, dx\) [2368]

Optimal. Leaf size=189 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b x^2+a^3 x^3}}\right )}{a}-\frac {\log \left (-a x+\sqrt [3]{-b x^2+a^3 x^3}\right )}{a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{-b x^2+a^3 x^3}+\left (-b x^2+a^3 x^3\right )^{2/3}\right )}{2 a}+\text {RootSum}\left [a^6+a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\& ,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(821\) vs. \(2(189)=378\).
time = 0.75, antiderivative size = 821, normalized size of antiderivative = 4.34, 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 = {2081, 6857, 61, 926, 93} \begin {gather*} -\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log (x)}{2 a \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {b}-\sqrt {-a^3} x\right )}{2 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {-a^3} x+\sqrt {b}\right )}{2 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a^3+\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{a \sqrt [3]{x}}-1\right )}{2 a \sqrt [3]{a^3 x^3-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[3]*x^(2/3)*(-b + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[3]*a*x^(1/3))])/(a*(-(b*
x^2) + a^3*x^3)^(1/3))) + (Sqrt[3]*x^(2/3)*(-b + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[
3]*(a^3 - Sqrt[-a^3]*Sqrt[b])^(1/3)*x^(1/3))])/((a^3 - Sqrt[-a^3]*Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) +
(Sqrt[3]*x^(2/3)*(-b + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[3]*(a^3 + Sqrt[-a^3]*Sqrt
[b])^(1/3)*x^(1/3))])/((a^3 + Sqrt[-a^3]*Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1
/3)*Log[x])/(2*a*(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1/3)*Log[Sqrt[b] - Sqrt[-a^3]*x])/(2*(a^
3 - Sqrt[-a^3]*Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1/3)*Log[Sqrt[b] + Sqrt[-a^
3]*x])/(2*(a^3 + Sqrt[-a^3]*Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) + (3*x^(2/3)*(-b + a^3*x)^(1/3)*Log[-x^
(1/3) + (-b + a^3*x)^(1/3)/(a^3 - Sqrt[-a^3]*Sqrt[b])^(1/3)])/(2*(a^3 - Sqrt[-a^3]*Sqrt[b])^(1/3)*(-(b*x^2) +
a^3*x^3)^(1/3)) + (3*x^(2/3)*(-b + a^3*x)^(1/3)*Log[-x^(1/3) + (-b + a^3*x)^(1/3)/(a^3 + Sqrt[-a^3]*Sqrt[b])^(
1/3)])/(2*(a^3 + Sqrt[-a^3]*Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) - (3*x^(2/3)*(-b + a^3*x)^(1/3)*Log[-1
+ (-b + a^3*x)^(1/3)/(a*x^(1/3))])/(2*a*(-(b*x^2) + a^3*x^3)^(1/3))

Rule 61

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

Rule 93

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 926

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 2081

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 6857

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

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Mathematica [A]
time = 0.47, size = 226, normalized size = 1.20 \begin {gather*} \frac {x^{2/3} \sqrt [3]{-b+a^3 x} \left (2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{-b+a^3 x}}\right )-2 \log \left (a \left (a \sqrt [3]{x}-\sqrt [3]{-b+a^3 x}\right )\right )+\log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{-b+a^3 x}+\left (-b+a^3 x\right )^{2/3}\right )+2 a \text {RootSum}\left [a^6+a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{2 a \sqrt [3]{x^2 \left (-b+a^3 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {a^{3} x^{2}-b}{\left (a^{3} x^{2}+b \right ) \left (a^{3} x^{3}-b \,x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.44, size = 1727, normalized size = 9.14 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(4*sqrt(3)*a*(((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) + 1)/(a^3 + b))^(1/3)*arctan(1/3*(2*sqrt(3)*x
*(((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) + 1)/(a^3 + b))^(1/3)*sqrt(((a^3*x^3 - b*x^2)^(1/3)*(a^3*x - (
a^6 + a^3*b)*x*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)))*(((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) + 1)/(a^3 +
b))^(2/3) + (a^3*x^2 - (a^6 + a^3*b)*x^2*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)))*(((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*
b + a^3*b^2)) + 1)/(a^3 + b))^(1/3) + (a^3*x^3 - b*x^2)^(2/3))/x^2) - sqrt(3)*x - 2*sqrt(3)*(a^3*x^3 - b*x^2)^
(1/3)*(((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) + 1)/(a^3 + b))^(1/3))/x) + 4*sqrt(3)*a*(-((a^3 + b)*sqrt
(-b/(a^9 + 2*a^6*b + a^3*b^2)) - 1)/(a^3 + b))^(1/3)*arctan(1/3*(2*sqrt(3)*x*(-((a^3 + b)*sqrt(-b/(a^9 + 2*a^6
*b + a^3*b^2)) - 1)/(a^3 + b))^(1/3)*sqrt(((a^3*x^3 - b*x^2)^(1/3)*(a^3*x + (a^6 + a^3*b)*x*sqrt(-b/(a^9 + 2*a
^6*b + a^3*b^2)))*(-((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) - 1)/(a^3 + b))^(2/3) + (a^3*x^2 + (a^6 + a^
3*b)*x^2*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)))*(-((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) - 1)/(a^3 + b))^(
1/3) + (a^3*x^3 - b*x^2)^(2/3))/x^2) - sqrt(3)*x - 2*sqrt(3)*(a^3*x^3 - b*x^2)^(1/3)*(-((a^3 + b)*sqrt(-b/(a^9
+ 2*a^6*b + a^3*b^2)) - 1)/(a^3 + b))^(1/3))/x) - 2*a*(((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) + 1)/(a^
3 + b))^(1/3)*log(-2*((a^3*x - (a^6 + a^3*b)*x*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)))*(((a^3 + b)*sqrt(-b/(a^9 +
2*a^6*b + a^3*b^2)) + 1)/(a^3 + b))^(2/3) - (a^3*x^3 - b*x^2)^(1/3))/x) - 2*a*(-((a^3 + b)*sqrt(-b/(a^9 + 2*a^
6*b + a^3*b^2)) - 1)/(a^3 + b))^(1/3)*log(-2*((a^3*x + (a^6 + a^3*b)*x*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)))*(-(
(a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) - 1)/(a^3 + b))^(2/3) - (a^3*x^3 - b*x^2)^(1/3))/x) + a*(((a^3 +
b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) + 1)/(a^3 + b))^(1/3)*log(16*((a^3*x^3 - b*x^2)^(1/3)*(a^3*x - (a^6 + a^
3*b)*x*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)))*(((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) + 1)/(a^3 + b))^(2/3
) + (a^3*x^2 - (a^6 + a^3*b)*x^2*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)))*(((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*
b^2)) + 1)/(a^3 + b))^(1/3) + (a^3*x^3 - b*x^2)^(2/3))/x^2) + a*(-((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)
) - 1)/(a^3 + b))^(1/3)*log(16*((a^3*x^3 - b*x^2)^(1/3)*(a^3*x + (a^6 + a^3*b)*x*sqrt(-b/(a^9 + 2*a^6*b + a^3*
b^2)))*(-((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) - 1)/(a^3 + b))^(2/3) + (a^3*x^2 + (a^6 + a^3*b)*x^2*sq
rt(-b/(a^9 + 2*a^6*b + a^3*b^2)))*(-((a^3 + b)*sqrt(-b/(a^9 + 2*a^6*b + a^3*b^2)) - 1)/(a^3 + b))^(1/3) + (a^3
*x^3 - b*x^2)^(2/3))/x^2) + 2*sqrt(3)*arctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*(a^3*x^3 - b*x^2)^(1/3))/(a*x)) + 2*
log(-(a*x - (a^3*x^3 - b*x^2)^(1/3))/x) - log((a^2*x^2 + (a^3*x^3 - b*x^2)^(1/3)*a*x + (a^3*x^3 - b*x^2)^(2/3)
)/x^2))/a

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{3} x^{2} - b}{\sqrt [3]{x^{2} \left (a^{3} x - b\right )} \left (a^{3} x^{2} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {b-a^3\,x^2}{\left (a^3\,x^2+b\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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