3.27.4 \(\int \frac {-b+a x^2}{(-b+2 a x^3) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\) [2604]

Optimal. Leaf size=226 \[ -\frac {1}{3} \text {RootSum}\left [a^9+2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\& ,\frac {-a^6 \log (x)+a b^3 \log (x)+a^6 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-a b^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+2 a^3 \log (x) \text {$\#$1}^3-2 a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3-\log (x) \text {$\#$1}^6+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^6}{a^6 \text {$\#$1}-2 a^3 \text {$\#$1}^4+\text {$\#$1}^7}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1265\) vs. \(2(226)=452\).
time = 1.18, antiderivative size = 1265, normalized size of antiderivative = 5.60, number of steps used = 6, number of rules used = 3, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.073, Rules used = {2081, 6857, 93} \begin {gather*} -\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {2 \sqrt [3]{x a^3+b^2}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{-2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\frac {\sqrt [3]{x a^3+b^2}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}-\sqrt [3]{x}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(((-2)^(1/3)*a^(1/3) + 2*b^(1/3))*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^(1/9)*(a^(8/3) - (-2)^(1/3)*b^(5/3))^(1/3)*x^(1/3))])/(Sqrt[3]*a^(1/9)*b^(1/3)*(a^(8/3) - (-2)^(1/3
)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + ((2^(1/3)*a^(1/3) - 2*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTa
n[1/Sqrt[3] + (2*(b^2 + a^3*x)^(1/3))/(Sqrt[3]*a^(1/9)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3)*x^(1/3))])/(2*Sqrt[3]
*a^(1/9)*b^(1/3)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (((-1)^(2/3)*2^(1/3)*a^(1/3) -
2*b^(1/3))*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^(1/9)*(a^(8/3) +
(-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(1/9)*b^(1/3)*(a^(8/3) + (-1)^(2/3)*2^(1/3)*b^(5/3)
)^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (((-2)^(1/3)*a^(1/3) + 2*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3)
+ (-2)^(1/3)*a^(1/3)*x])/(12*a^(1/9)*b^(1/3)*(a^(8/3) - (-2)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))
- ((2^(1/3)*a^(1/3) - 2*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3) - 2^(1/3)*a^(1/3)*x])/(12*a^(1/9)*b^(
1/3)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (((-1)^(2/3)*2^(1/3)*a^(1/3) - 2*b^(1/3))*
x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3) - (-1)^(2/3)*2^(1/3)*a^(1/3)*x])/(12*a^(1/9)*b^(1/3)*(a^(8/3) + (-1)^(
2/3)*2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - (((-2)^(1/3)*a^(1/3) + 2*b^(1/3))*x^(2/3)*(b^2 + a^3*
x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^(1/9)*(a^(8/3) - (-2)^(1/3)*b^(5/3))^(1/3))])/(4*a^(1/9)*b^(1/3
)*(a^(8/3) - (-2)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + ((2^(1/3)*a^(1/3) - 2*b^(1/3))*x^(2/3)*(b^
2 + a^3*x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^(1/9)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3))])/(4*a^(1/9)*b
^(1/3)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (((-1)^(2/3)*2^(1/3)*a^(1/3) - 2*b^(1/3)
)*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[-x^(1/3) + (b^2 + a^3*x)^(1/3)/(a^(1/9)*(a^(8/3) + (-1)^(2/3)*2^(1/3)*b^(5/3
))^(1/3))])/(4*a^(1/9)*b^(1/3)*(a^(8/3) + (-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))

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 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 x^2}{\left (-b+2 a x^3\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^3\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 {-\frac {\sqrt [3]{-1} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}}{3 b x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {\frac {\sqrt [3]{a} b}{2^{2/3}}-b^{4/3}}{3 b x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {\frac {(-1)^{2/3} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}}{3 b x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) 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 [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{12 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}\right )}{4 \sqrt [9]{a} \sqrt [3]{b} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}

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Mathematica [A]
time = 10.08, size = 230, normalized size = 1.02 \begin {gather*} \frac {x \left (\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}-\sqrt [3]{-2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (-\sqrt [3]{2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a^3 x+\sqrt [3]{2} \sqrt [3]{a} b^{5/3} x}{b^2+a^3 x}\right )\right )}{2 \sqrt [3]{b} \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(((-2)^(1/3)*a^(1/3) + 2*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/3)*(a^(8/3) - (-2)^(1/3)*b^(5/3))*x)
/(b^2 + a^3*x)] + (-((-1)^(2/3)*2^(1/3)*a^(1/3)) + 2*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/3)*(a^(8/3)
+ (-1)^(2/3)*2^(1/3)*b^(5/3))*x)/(b^2 + a^3*x)] + (-(2^(1/3)*a^(1/3)) + 2*b^(1/3))*Hypergeometric2F1[1/3, 1,
4/3, (a^3*x + 2^(1/3)*a^(1/3)*b^(5/3)*x)/(b^2 + a^3*x)]))/(2*b^(1/3)*(x^2*(b^2 + a^3*x))^(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^2-b)/(2*a*x^3-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*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 108.94, size = 149068, normalized size = 659.59 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*(1/6)^(1/3)*(1/9)^(1/3)*((6*a^6 - 24*a^5*b + 36*a^3*b^2 + 72*b^4 + (a^8*b + 2*b^6)*(2*(1/2)^(2/3)*(-I
*sqrt(3) + 1)*((a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^2/(a^8*b + 2*b^6)^2 - (a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b
^2 + 48*b^3)/(a^8*b^2 + 2*b^7))/(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^8*b + 2*b^6)^3 - 3*(a^6 - 4*a^5*b
+ 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^8*b^2 + 2*b^7)*(a^8*b + 2*b^6)) +
(a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^9*b^3 + 2*a*b^8) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48*a*b^6 + 6
4*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)/((a^8 + 2*b^5)^2*a*b^2))^(1/3) + (1
/2)^(1/3)*(I*sqrt(3) + 1)*(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^8*b + 2*b^6)^3 - 3*(a^6 - 4*a^5*b + 6*a
^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^8*b^2 + 2*b^7)*(a^8*b + 2*b^6)) + (a^3 -
12*a^2*b + 48*a*b^2 - 64*b^3)/(a^9*b^3 + 2*a*b^8) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48*a*b^6 + 64*b^7
+ 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)/((a^8 + 2*b^5)^2*a*b^2))^(1/3) - 2*(a^6 -
4*a^5*b + 6*a^3*b^2 + 12*b^4)/(a^8*b + 2*b^6)) + 3*sqrt(1/3)*(a^8*b + 2*b^6)*sqrt(-(4*a^12 + 544*a^11*b + 134
4*a^8*b^3 - 720*a^6*b^4 - 832*a^3*b^6 + 64*(8*a^2 + 3*a)*b^7 - 192*b^8 + 32*(36*a^5 + a^4)*b^5 + (a^16*b^2 + 4
*a^8*b^7 + 4*b^12)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^2/(a^8*b + 2*b^6)^2 -
(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/(a^8*b^2 + 2*b^7))/(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3
/(a^8*b + 2*b^6)^3 - 3*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/(
(a^8*b^2 + 2*b^7)*(a^8*b + 2*b^6)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^9*b^3 + 2*a*b^8) + 2*(54*a^10 + 2
16*a^9*b + 216*a^7*b^2 - 48*a*b^6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3
)/((a^8 + 2*b^5)^2*a*b^2))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^8*
b + 2*b^6)^3 - 3*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^8*b
^2 + 2*b^7)*(a^8*b + 2*b^6)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^9*b^3 + 2*a*b^8) + 2*(54*a^10 + 216*a^9
*b + 216*a^7*b^2 - 48*a*b^6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)/((a^
8 + 2*b^5)^2*a*b^2))^(1/3) - 2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)/(a^8*b + 2*b^6))^2 + 16*(4*a^10 - 3*a^9)*b
^2 + 4*(a^14*b - 4*a^13*b^2 + 6*a^11*b^3 + 12*a^8*b^5 + 2*a^6*b^6 - 8*a^5*b^7 + 12*a^3*b^8 + 24*b^10)*(2*(1/2)
^(2/3)*(-I*sqrt(3) + 1)*((a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^2/(a^8*b + 2*b^6)^2 - (a^4 + 28*a^3*b + 16*a^2*b
^2 + 6*a*b^2 + 48*b^3)/(a^8*b^2 + 2*b^7))/(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^8*b + 2*b^6)^3 - 3*(a^6
- 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^8*b^2 + 2*b^7)*(a^8*b +
2*b^6)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^9*b^3 + 2*a*b^8) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48
*a*b^6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)/((a^8 + 2*b^5)^2*a*b^2))^
(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^8*b + 2*b^6)^3 - 3*(a^6 - 4*a
^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^8*b^2 + 2*b^7)*(a^8*b + 2*b^6)
) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^9*b^3 + 2*a*b^8) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48*a*b^6
+ 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)/((a^8 + 2*b^5)^2*a*b^2))^(1/3)
- 2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)/(a^8*b + 2*b^6)))/(a^16*b^2 + 4*a^8*b^7 + 4*b^12)))/(a^8*b + 2*b^6))^
(1/3)*arctan(-1/96*(32*sqrt(3)*(2916*a^23 - 11664*a^22*b + 589824*a*b^16 - 262144*b^17 - 294912*(8*a^3 - a^2)*
b^15 + 983040*(3*a^4 - a^3)*b^14 - 46080*(192*a^6 - 64*a^5 + a^4)*b^13 + 2304*(2112*a^7 - 976*a^6 + 137*a^5)*b
^12 - 192*(92160*a^9 - 38016*a^8 + 912*a^7 + 487*a^6)*b^11 + 144*(15360*a^10 - 19968*a^9 + 2672*a^8 + 49*a^7)*
b^10 - 36*(552960*a^12 - 221184*a^11 + 24384*a^10 + 2464*a^9 - 5*a^8)*b^9 - (995328*a^13 + 2350080*a^12 - 4924
80*a^11 - 5904*a^10 - a^9)*b^8 - 72*(165888*a^15 - 79488*a^14 + 7296*a^13 + 804*a^12 - a^11)*b^7 - 864*(648*a^
15 - 306*a^14 - a^13)*b^6 - 864*(3456*a^18 - 3456*a^17 + 468*a^16 + 29*a^15)*b^5 + 108*(6912*a^19 - 3456*a^18
+ 792*a^17 - a^16)*b^4 + 3888*(96*a^20 - 36*a^19 - a^18)*b^3 - 23328*(4*a^21 - a^20)*b^2)*x + 3*(1/6)^(1/3)*(1
/9)^(1/3)*(sqrt(3)*(12*a^15*b^2 - a^13*b^3 + 16*a^12*b^4 - 8*a^10*b^5 + 8*a^9*b^6 + 24*a^7*b^7 - 2*a^5*b^8 + 3
2*a^4*b^9 - 16*a^2*b^10 + 16*a*b^11)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^2/(
a^8*b + 2*b^6)^2 - (a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/(a^8*b^2 + 2*b^7))/(2*(a^6 - 4*a^5*b + 6*a
^3*b^2 + 12*b^4)^3/(a^8*b + 2*b^6)^3 - 3*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6
*a*b^2 + 48*b^3)/((a^8*b^2 + 2*b^7)*(a^8*b + 2*b^6)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^9*b^3 + 2*a*b^8
) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48*a...

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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^{3} - b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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