\(\int \frac {b x+a x^3}{(b+2 a x^3) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\) [2891]

Optimal result

Integrand size = 39, antiderivative size = 316 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\frac {x}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2 x^2+a^3 x^3}}{\sqrt {3} a}}{x}\right )}{2 a}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a}+\frac {1}{6} \text {RootSum}\left [a^9-2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+2 b^2 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{a^3 \text {$\#$1}-\text {$\#$1}^4}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1777\) vs. \(2(316)=632\).

Time = 1.56 (sec) , antiderivative size = 1777, normalized size of antiderivative = 5.62, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {1607, 2081, 6857, 129, 494, 245, 384} \[ \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=\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \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]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \arctan \left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \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]{a}+(-2)^{2/3} \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 a^{4/9} \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]{a}+2^{2/3} \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 a^{4/9} \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]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 a^{4/9} \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]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \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]{a}+2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \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]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}} \]

[In]

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

[Out]

((a^(1/3) + (-2)^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqr
t[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 + (2*a*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[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 + (2*a*x^(1/3))/(b^2 + a
^3*x)^(1/3))/Sqrt[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 + (2*a^(1/9)*(a^(8/3) + (-2)^(1/3)*b^(5/3))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))
/Sqrt[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 + (2*a^(1/9)*(a^(8/3) - 2^(1/3)*b^(5/3))^(1/3)*x^(1/3))/(
b^2 + a^3*x)^(1/3))/Sqrt[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 + (2*a^(1/9)*(a^(8/
3) - (-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[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[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])/(1
2*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[a*x^(1/3) - (b^2 + a^3*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[a*x^(1/3) - (b^2 + a^3*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[a*x^(1/3) - (b^2 + a^3*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[a^(1/9)*(a^(8/3
) + (-2)^(1/3)*b^(5/3))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(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[a^(1/9)*(a^(8/3) - 2
^(1/3)*b^(5/3))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(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[a^(1/9)*(a^(8/3) - (
-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(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))

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 494

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[a*(e^n/b), Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 1607

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 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*} \text {integral}& = \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 (\left (\sqrt [3]{-2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 ) \text {Subst}\left (\int \frac {x^3}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \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 ) \text {Subst}\left (\int \frac {1}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x^3\right ) \sqrt [3]{b^2+a^3 x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.85 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\frac {3 \sqrt [3]{a} x \sqrt [3]{1+\frac {a^3 x}{b^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {a^3 x}{b^2}\right )-x \left (\left (\sqrt [3]{a}+(-2)^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}+\sqrt [3]{-2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}-\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {\sqrt [3]{a} \left (a^{8/3}-(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}+2^{2/3} \sqrt [3]{b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},1,\frac {4}{3},\frac {a^3 x-\sqrt [3]{2} \sqrt [3]{a} b^{5/3} x}{b^2+a^3 x}\right )\right )}{2 \sqrt [3]{a} \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \]

[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))

Maple [N/A] (verified)

Time = 0.40 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.67

method	result	size
pseudoelliptic	\(\frac {-6 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a^{3} \textit {\_Z}^{6}+3 a^{6} \textit {\_Z}^{3}-a^{9}+2 a \,b^{5}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right ) \left (\textit {\_R}^{3}-a^{3}-2 b^{2}\right )}{\textit {\_R} \left (\textit {\_R}^{3}-a^{3}\right )}\right ) a -6 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+3 \ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x +b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{12 a}\)	\(212\)

[In]

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

[Out]

1/12*(-6*3^(1/2)*arctan(1/3*(a*x+2*(x^2*(a^3*x+b^2))^(1/3))*3^(1/2)/a/x)+2*sum(ln((-_R*x+(x^2*(a^3*x+b^2))^(1/
3))/x)*(_R^3-a^3-2*b^2)/_R/(_R^3-a^3),_R=RootOf(_Z^9-3*_Z^6*a^3+3*_Z^3*a^6-a^9+2*a*b^5))*a-6*ln((-a*x+(x^2*(a^
3*x+b^2))^(1/3))/x)+3*ln((a^2*x^2+a*(x^2*(a^3*x+b^2))^(1/3)*x+(x^2*(a^3*x+b^2))^(2/3))/x^2))/a

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 5.36 (sec) , antiderivative size = 66610, normalized size of antiderivative = 210.79 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Too large to display} \]

[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]

Too large to include

Sympy [N/A]

Not integrable

Time = 21.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.10 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {x \left (a x^{2} + b\right )}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{3} + b\right )}\, dx \]

[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)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.12 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int { \frac {a x^{3} + b x}{{\left (a^{3} x^{3} + b^{2} x^{2}\right )}^{\frac {1}{3}} {\left (2 \, a x^{3} + b\right )}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\text {Timed out} \]

[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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.12 \[ \int \frac {b x+a x^3}{\left (b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx=\int \frac {a\,x^3+b\,x}{{\left (a^3\,x^3+b^2\,x^2\right )}^{1/3}\,\left (2\,a\,x^3+b\right )} \,d x \]

[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)