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

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 = 0.96, antiderivative size = 874, normalized size of antiderivative = 4.28, number of steps used = 8, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, 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 [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} 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 [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} 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 [3]{a^3+\sqrt {2} \sqrt {-a} 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 [3]{a^3-\sqrt {2} \sqrt {-a} 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 [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} 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 [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} 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^3 - Sqrt[2]*Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))])/(4*(a^3 - Sqrt[2]*Sqrt[-a]*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
^3 + Sqrt[2]*Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))])/(4*(a^3 + Sqrt[2]*Sqrt[-a]*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)*L
og[Sqrt[b] - Sqrt[2]*Sqrt[-a]*x])/(8*(a^3 + Sqrt[2]*Sqrt[-a]*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^3 - Sqrt[2]*Sqrt[-a]*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^3 - Sqrt[2]*Sqrt[-a]*
b^(3/2))^(1/3)])/(8*(a^3 - Sqrt[2]*Sqrt[-a]*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^3 + Sqrt[2]*Sqrt[-a]*b^(3/2))^(1/3)])/(8*(a^3 + Sqrt[2]*Sqrt[-a
]*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[-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 [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} 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 [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}} \sqrt [3]{x}}\right )}{4 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} 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 [3]{a^3+\sqrt {2} \sqrt {-a} 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 [3]{a^3-\sqrt {2} \sqrt {-a} 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 [3]{a^3-\sqrt {2} \sqrt {-a} b^{3/2}}}\right )}{8 \sqrt [3]{a^3-\sqrt {2} \sqrt {-a} 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 [3]{a^3+\sqrt {2} \sqrt {-a} b^{3/2}}}\right )}{8 \sqrt [3]{a^3+\sqrt {2} \sqrt {-a} 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 = 0.09, 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 {\left (a^3-\sqrt {2} \sqrt {-a} b^{3/2}\right ) x}{x a^3+b^2}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (a^3+\sqrt {2} \sqrt {-a} b^{3/2}\right ) x}{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 - Sqrt[2]*Sqrt[-a]*b^(3/2))*x)/(b^2 + a^3*x)] + Hypergeometric2F1[1/3, 1, 4/3, ((a^3 + 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 = 0.55, size = 2180, normalized size = 10.69

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)
+ 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) + 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^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) + 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)*s
qrt(-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 {a\,x^2+b}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a\,x^2+b\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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