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

Optimal. Leaf size=299 \[ \frac {1}{2} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6+a b^3\& ,\frac {\text {$\#$1}^3 a^2 \log (x)-\text {$\#$1}^3 a^2 \log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-\text {$\#$1} x\right )+b^3 \log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-\text {$\#$1} x\right )+a^5 \log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-\text {$\#$1} x\right )+a^5 (-\log (x))-b^3 \log (x)}{\text {$\#$1}^2 a^3-\text {$\#$1}^5}\& \right ]-\log \left (\sqrt [3]{a^3 x^3-b^2 x^2}-a x\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3-b^2 x^2}+a x}\right )+\frac {1}{2} \log \left (a x \sqrt [3]{a^3 x^3-b^2 x^2}+\left (a^3 x^3-b^2 x^2\right )^{2/3}+a^2 x^2\right ) \]

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Rubi [B]  time = 0.79, antiderivative size = 866, normalized size of antiderivative = 2.90, number of steps used = 7, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2056, 906, 59, 6725, 91} \begin {gather*} -\frac {\sqrt {3} \sqrt [3]{a^3 x^3-b^2 x^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{x} a}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {\sqrt {3} \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{2 a x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {\sqrt {3} \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{2 a x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt {b}-\sqrt {-a} x\right )}{4 a x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt {-a} x+\sqrt {b}\right )}{4 a x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^3 x^3-b^2 x^2} \log \left (a^3 x-b^2\right )}{2 x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {3 \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}-1\right )}{2 x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {3 \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 a x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {3 \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 a x^{2/3} \sqrt [3]{a^3 x-b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[3]*(-(b^2*x^2) + a^3*x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*a*x^(1/3))/(Sqrt[3]*(-b^2 + a^3*x)^(1/3))])/(x^(
2/3)*(-b^2 + a^3*x)^(1/3))) + (Sqrt[3]*(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*ArcTan[1/Sq
rt[3] + (2*(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))/(Sqrt[3]*(-b^2 + a^3*x)^(1/3))])/(2*a*x^(2/3)*(-b^2 + a^3*x
)^(1/3)) + (Sqrt[3]*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(a^3 + S
qrt[-a]*b^(3/2))^(1/3)*x^(1/3))/(Sqrt[3]*(-b^2 + a^3*x)^(1/3))])/(2*a*x^(2/3)*(-b^2 + a^3*x)^(1/3)) - ((a^3 -
Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*Log[Sqrt[b] - Sqrt[-a]*x])/(4*a*x^(2/3)*(-b^2 + a^3*x)^(1
/3)) - ((a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*Log[Sqrt[b] + Sqrt[-a]*x])/(4*a*x^(2/3)*(-
b^2 + a^3*x)^(1/3)) - ((-(b^2*x^2) + a^3*x^3)^(1/3)*Log[-b^2 + a^3*x])/(2*x^(2/3)*(-b^2 + a^3*x)^(1/3)) - (3*(
-(b^2*x^2) + a^3*x^3)^(1/3)*Log[-1 + (a*x^(1/3))/(-b^2 + a^3*x)^(1/3)])/(2*x^(2/3)*(-b^2 + a^3*x)^(1/3)) + (3*
(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*Log[(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3) - (-b^2
+ a^3*x)^(1/3)])/(4*a*x^(2/3)*(-b^2 + a^3*x)^(1/3)) + (3*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3
)^(1/3)*Log[(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3) - (-b^2 + a^3*x)^(1/3)])/(4*a*x^(2/3)*(-b^2 + a^3*x)^(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 906

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

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

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Mathematica [A]  time = 0.10, size = 188, normalized size = 0.63 \begin {gather*} \frac {3 x^2 \left (2 (-a)^{5/2} \left (1-\frac {a^3 x}{b^2}\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {a^3 x}{b^2}\right )-\left ((-a)^{5/2}+b^{3/2}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\left (a^3-\sqrt {-a} b^{3/2}\right ) x}{a^3 x-b^2}\right )+\left (a (-a)^{3/2}+b^{3/2}\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\left (a^3+\sqrt {-a} b^{3/2}\right ) x}{a^3 x-b^2}\right )\right )}{4 \sqrt {-a} \left (x^2 \left (a^3 x-b^2\right )\right )^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x^2*(2*(-a)^(5/2)*(1 - (a^3*x)/b^2)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, (a^3*x)/b^2] - ((-a)^(5/2) + b^(
3/2))*Hypergeometric2F1[2/3, 1, 5/3, ((a^3 - Sqrt[-a]*b^(3/2))*x)/(-b^2 + a^3*x)] + ((-a)^(3/2)*a + b^(3/2))*H
ypergeometric2F1[2/3, 1, 5/3, ((a^3 + Sqrt[-a]*b^(3/2))*x)/(-b^2 + a^3*x)]))/(4*Sqrt[-a]*(x^2*(-b^2 + a^3*x))^
(2/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 298, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )-\log \left (-a x+\sqrt [3]{-b^2 x^2+a^3 x^3}\right )+\frac {1}{2} \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 )+\frac {1}{2} \text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {a^5 \log (x)+b^3 \log (x)-a^5 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-b^3 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-a^2 \log (x) \text {$\#$1}^3+a^2 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a^3 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.63, size = 925, normalized size = 3.09

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*(sqrt(-b^3/a^5) + 1)^(1/3)*arctan(1/3*(sqrt(3)*(a^5 + b^3)*x + 2*(sqrt(3)*a^4*x*sqrt(-b^3/a^5) - sqrt
(3)*a^4*x)*(sqrt(-b^3/a^5) + 1)^(2/3)*sqrt((a^2*x^2*(sqrt(-b^3/a^5) + 1)^(2/3) + (a^3*x^3 - b^2*x^2)^(1/3)*a*x
*(sqrt(-b^3/a^5) + 1)^(1/3) + (a^3*x^3 - b^2*x^2)^(2/3))/x^2) - 2*(a^3*x^3 - b^2*x^2)^(1/3)*(sqrt(3)*a^4*sqrt(
-b^3/a^5) - sqrt(3)*a^4)*(sqrt(-b^3/a^5) + 1)^(2/3))/((a^5 + b^3)*x)) + sqrt(3)*(-sqrt(-b^3/a^5) + 1)^(1/3)*ar
ctan(-1/3*(sqrt(3)*(a^5 + b^3)*x - 2*(sqrt(3)*a^4*x*sqrt(-b^3/a^5) + sqrt(3)*a^4*x)*(-sqrt(-b^3/a^5) + 1)^(2/3
)*sqrt((a^2*x^2*(-sqrt(-b^3/a^5) + 1)^(2/3) + (a^3*x^3 - b^2*x^2)^(1/3)*a*x*(-sqrt(-b^3/a^5) + 1)^(1/3) + (a^3
*x^3 - b^2*x^2)^(2/3))/x^2) + 2*(a^3*x^3 - b^2*x^2)^(1/3)*(sqrt(3)*a^4*sqrt(-b^3/a^5) + sqrt(3)*a^4)*(-sqrt(-b
^3/a^5) + 1)^(2/3))/((a^5 + b^3)*x)) + sqrt(3)*arctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*(a^3*x^3 - b^2*x^2)^(1/3))/
(a*x)) + 1/2*(sqrt(-b^3/a^5) + 1)^(1/3)*log(-(a*x*(sqrt(-b^3/a^5) + 1)^(1/3) - (a^3*x^3 - b^2*x^2)^(1/3))/x) +
1/2*(-sqrt(-b^3/a^5) + 1)^(1/3)*log(-(a*x*(-sqrt(-b^3/a^5) + 1)^(1/3) - (a^3*x^3 - b^2*x^2)^(1/3))/x) - 1/4*(
sqrt(-b^3/a^5) + 1)^(1/3)*log(4*(a^2*x^2*(sqrt(-b^3/a^5) + 1)^(2/3) + (a^3*x^3 - b^2*x^2)^(1/3)*a*x*(sqrt(-b^3
/a^5) + 1)^(1/3) + (a^3*x^3 - b^2*x^2)^(2/3))/x^2) - 1/4*(-sqrt(-b^3/a^5) + 1)^(1/3)*log(4*(a^2*x^2*(-sqrt(-b^
3/a^5) + 1)^(2/3) + (a^3*x^3 - b^2*x^2)^(1/3)*a*x*(-sqrt(-b^3/a^5) + 1)^(1/3) + (a^3*x^3 - b^2*x^2)^(2/3))/x^2
) - log(-(a*x - (a^3*x^3 - b^2*x^2)^(1/3))/x) + 1/2*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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} + b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} + b}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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