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

Optimal. Leaf size=310 \[ \frac {1}{6} \text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a^3-3 \text {$\#$1}^3 a^6+a^9-2 a b^5\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+a^3 \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )+2 b^2 \log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-\text {$\#$1} x\right )+a^3 (-\log (x))-2 b^2 \log (x)}{\text {$\#$1} a^3-\text {$\#$1}^4}\& \right ]-\frac {\log \left (\sqrt [3]{a^3 x^3+b^2 x^2}-a x\right )}{2 a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+b^2 x^2}+a x}\right )}{2 a}+\frac {\log \left (a x \sqrt [3]{a^3 x^3+b^2 x^2}+\left (a^3 x^3+b^2 x^2\right )^{2/3}+a^2 x^2\right )}{4 a} \]

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Rubi [B]  time = 2.07, antiderivative size = 2012, normalized size of antiderivative = 6.49, number of steps used = 13, number of rules used = 6, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1593, 2056, 6725, 105, 59, 91}

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*((a^(1/3) + (-2)^(2/3)*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))/(S
qrt[3]*a*x^(1/3))])/(Sqrt[3]*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) + 2^(2/3)*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*x^(1/3))])/(2*Sqrt[3]*a^(4/3)*(b^2*x^2 + a^
3*x^3)^(1/3)) + ((-1)^(1/3)*((-1)^(2/3)*a^(1/3) + 2^(2/3)*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*x^(1/3))])/(2*Sqrt[3]*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + ((a^(1/3) +
(-2)^(2/3)*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))])/(2*Sqrt[3]*a^(4/9)*(a^(8/3) + (-2)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2
+ a^3*x^3)^(1/3)) + ((a^(1/3) + 2^(2/3)*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))])/(2*Sqrt[3]*a^(4/9)*(a^(8/3) - 2^(1/3)
*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((-1)^(1/3)*((-1)^(2/3)*a^(1/3) + 2^(2/3)*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^(4/9)*(a^(8/3) - (-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))
- ((a^(1/3) + (-2)^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[x])/(12*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3))
- ((a^(1/3) + 2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[x])/(12*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((
a^(1/3) - (-1)^(1/3)*2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[x])/(12*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3
)) - ((a^(1/3) + (-2)^(2/3)*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^(
4/9)*(a^(8/3) + (-2)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) + 2^(2/3)*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^(4/9)*(a^(8/3) - 2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 +
a^3*x^3)^(1/3)) - ((a^(1/3) - (-1)^(1/3)*2^(2/3)*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^(4/9)*(a^(8/3) - (-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + ((
a^(1/3) + (-2)^(2/3)*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^(4/9)*(a^(8/3) + (-2)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + (
(a^(1/3) + 2^(2/3)*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^(4/9)*(a^(8/3) - 2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + ((a^(1/3)
- (-1)^(1/3)*2^(2/3)*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^(4/9)*(a^(8/3) - (-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^
3*x^3)^(1/3)) - ((a^(1/3) + (-2)^(2/3)*b^(1/3))*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^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) + 2^(2/3)*b^(1/3))*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^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) - (-1)^(1/3)*2^(2/3)*
b^(1/3))*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^(4/3)*(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 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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

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Mathematica [A]  time = 0.66, size = 268, normalized size = 0.86 \begin {gather*} \frac {3 \sqrt [3]{a} x \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 )-x \left (\left (\sqrt [3]{a}+2^{2/3} \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}{x a^3+b^2}\right )+\left (\sqrt [3]{a}+(-2)^{2/3} \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 (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \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}{x a^3+b^2}\right )\right )}{2 \sqrt [3]{a} \sqrt [3]{x^2 \left (a^3 x+b^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.03, size = 313, normalized size = 1.01 \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}{6} \text {RootSum}\left [a^9-2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {a^3 \log (x)+2 b^2 \log (x)-a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-2 b^2 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^3+\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a^3 \text {$\#$1}+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

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

[Out]

Timed out

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{3}+b x}{\left (2 a \,x^{3}+b \right ) \left (a^{3} x^{3}+b^{2} x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a*x^3+b*x)/(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^{3} + b x}{{\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^3+b*x)/(2*a*x^3+b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="maxima")

[Out]

integrate((a*x^3 + b*x)/((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 {a\,x^3+b\,x}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a\,x^3+b\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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