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

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

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Rubi [B]  time = 2.08, 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 = {2056, 6725, 91} \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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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 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 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^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 = 0.64, size = 230, normalized size = 1.02 \begin {gather*} \frac {x \left (\left (2 \sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {x a^3+\sqrt [3]{2} b^{5/3} x \sqrt [3]{a}}{x a^3+b^2}\right )+\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}{x a^3+b^2}\right )+\left (2 \sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}\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}{x a^3+b^2}\right )\right )}{2 \sqrt [3]{b} \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^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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IntegrateAlgebraic [A]  time = 2.95, size = 226, normalized size = 1.00 \begin {gather*} -\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 ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*RootSum[a^9 + 2*a*b^5 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-(a^6*Log[x]) + a*b^3*Log[x] + a^6*Log[(b^2*x
^2 + a^3*x^3)^(1/3) - x*#1] - a*b^3*Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1] + 2*a^3*Log[x]*#1^3 - 2*a^3*Log[(b^2
*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^3 - Log[x]*#1^6 + Log[(b^2*x^2 + a^3*x^3)^(1/3) - x*#1]*#1^6)/(a^6*#1 - 2*a^3
*#1^4 + #1^7) & ]

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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]

Timed out

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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^{3} - 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^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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maple [F]  time = 0.07, 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*} \int \frac {a x^{2} - b}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{3} - 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^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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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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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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