3.24.87 \(\int \frac {\sqrt [3]{x+x^3} (b+a x^6)}{d+c x^6} \, dx\) [2387]

Optimal. Leaf size=191 \[ \frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \text {ArcTan}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 \sqrt {3} c}-\frac {a \log \left (-x+\sqrt [3]{x+x^3}\right )}{6 c}+\frac {a \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )}{12 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\& ,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\& \right ]}{6 c d} \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1609\) vs. \(2(191)=382\).
time = 4.95, antiderivative size = 1609, normalized size of antiderivative = 8.42, number of steps used = 61, number of rules used = 12, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2081, 6857, 285, 335, 281, 337, 973, 477, 476, 495, 503, 524} \begin {gather*} \frac {a \sqrt [3]{x^3+x} x}{2 c}-\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^3+x} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^3+x} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(b c-a d) \sqrt [3]{x^3+x} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {a \sqrt [3]{x^3+x} \text {ArcTan}\left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} c \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{-1} \sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x^2+1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {\sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{d}-\sqrt [3]{-c} x^2\right )}{12 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{-1} \sqrt [3]{-c} x^2+\sqrt [3]{d}\right )}{12 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {\sqrt [3]{-1} \sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^2\right )}{12 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(-1)^{2/3} (b c-a d) \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{-1} (b c-a d) \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(b c-a d) \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{2/3} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {a \sqrt [3]{x^3+x} \log \left (x^{2/3}-\sqrt [3]{x^2+1}\right )}{4 c \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{\sqrt [3]{-c}+\sqrt [3]{d}} x^{2/3}-\sqrt [9]{d} \sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {(-1)^{2/3} \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{-c}-\sqrt [3]{d}} x^{2/3}+\sqrt [9]{d} \sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {\sqrt [3]{-1} \sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} (b c-a d) \sqrt [3]{x^3+x} \log \left (\sqrt [3]{-(-1)^{2/3} \sqrt [3]{-c}-\sqrt [3]{d}} x^{2/3}+\sqrt [9]{d} \sqrt [3]{x^2+1}\right )}{4 (-c)^{4/3} d^{7/9} \sqrt [3]{x^2+1} \sqrt [3]{x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((x + x^3)^(1/3)*(b + a*x^6))/(d + c*x^6),x]

[Out]

(a*x*(x + x^3)^(1/3))/(2*c) - (a*(x + x^3)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(1 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]
*c*x^(1/3)*(1 + x^2)^(1/3)) - ((b*c - a*d)*(x + x^3)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(1 + x^2)^(1/3))/Sqrt[3]])/
(2*Sqrt[3]*(-c)^(4/3)*d^(2/3)*x^(1/3)*(1 + x^2)^(1/3)) + ((-1)^(1/3)*(b*c - a*d)*(x + x^3)^(1/3)*ArcTan[(1 + (
2*x^(2/3))/(1 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(-c)^(4/3)*d^(2/3)*x^(1/3)*(1 + x^2)^(1/3)) - ((-1)^(2/3)*(b*
c - a*d)*(x + x^3)^(1/3)*ArcTan[(1 + (2*x^(2/3))/(1 + x^2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(-c)^(4/3)*d^(2/3)*x^(1
/3)*(1 + x^2)^(1/3)) - ((-1)^(2/3)*((-1)^(1/3)*(-c)^(1/3) - d^(1/3))^(1/3)*(b*c - a*d)*(x + x^3)^(1/3)*ArcTan[
(1 - (2*((-1)^(1/3)*(-c)^(1/3) - d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt[3]*(-c)^
(4/3)*d^(7/9)*x^(1/3)*(1 + x^2)^(1/3)) + ((-1)^(1/3)*(-((-1)^(2/3)*(-c)^(1/3)) - d^(1/3))^(1/3)*(b*c - a*d)*(x
+ x^3)^(1/3)*ArcTan[(1 - (2*(-((-1)^(2/3)*(-c)^(1/3)) - d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(1 + x^2)^(1/3)))/Sq
rt[3]])/(2*Sqrt[3]*(-c)^(4/3)*d^(7/9)*x^(1/3)*(1 + x^2)^(1/3)) + (((-c)^(1/3) + d^(1/3))^(1/3)*(b*c - a*d)*(x
+ x^3)^(1/3)*ArcTan[(1 + (2*((-c)^(1/3) + d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(1 + x^2)^(1/3)))/Sqrt[3]])/(2*Sqrt
[3]*(-c)^(4/3)*d^(7/9)*x^(1/3)*(1 + x^2)^(1/3)) - (((-c)^(1/3) + d^(1/3))^(1/3)*(b*c - a*d)*(x + x^3)^(1/3)*Lo
g[d^(1/3) - (-c)^(1/3)*x^2])/(12*(-c)^(4/3)*d^(7/9)*x^(1/3)*(1 + x^2)^(1/3)) + ((-1)^(2/3)*((-1)^(1/3)*(-c)^(1
/3) - d^(1/3))^(1/3)*(b*c - a*d)*(x + x^3)^(1/3)*Log[d^(1/3) + (-1)^(1/3)*(-c)^(1/3)*x^2])/(12*(-c)^(4/3)*d^(7
/9)*x^(1/3)*(1 + x^2)^(1/3)) - ((-1)^(1/3)*(-((-1)^(2/3)*(-c)^(1/3)) - d^(1/3))^(1/3)*(b*c - a*d)*(x + x^3)^(1
/3)*Log[d^(1/3) - (-1)^(2/3)*(-c)^(1/3)*x^2])/(12*(-c)^(4/3)*d^(7/9)*x^(1/3)*(1 + x^2)^(1/3)) - (a*(x + x^3)^(
1/3)*Log[x^(2/3) - (1 + x^2)^(1/3)])/(4*c*x^(1/3)*(1 + x^2)^(1/3)) - ((b*c - a*d)*(x + x^3)^(1/3)*Log[x^(2/3)
- (1 + x^2)^(1/3)])/(4*(-c)^(4/3)*d^(2/3)*x^(1/3)*(1 + x^2)^(1/3)) + ((-1)^(1/3)*(b*c - a*d)*(x + x^3)^(1/3)*L
og[x^(2/3) - (1 + x^2)^(1/3)])/(4*(-c)^(4/3)*d^(2/3)*x^(1/3)*(1 + x^2)^(1/3)) - ((-1)^(2/3)*(b*c - a*d)*(x + x
^3)^(1/3)*Log[x^(2/3) - (1 + x^2)^(1/3)])/(4*(-c)^(4/3)*d^(2/3)*x^(1/3)*(1 + x^2)^(1/3)) + (((-c)^(1/3) + d^(1
/3))^(1/3)*(b*c - a*d)*(x + x^3)^(1/3)*Log[((-c)^(1/3) + d^(1/3))^(1/3)*x^(2/3) - d^(1/9)*(1 + x^2)^(1/3)])/(4
*(-c)^(4/3)*d^(7/9)*x^(1/3)*(1 + x^2)^(1/3)) - ((-1)^(2/3)*((-1)^(1/3)*(-c)^(1/3) - d^(1/3))^(1/3)*(b*c - a*d)
*(x + x^3)^(1/3)*Log[((-1)^(1/3)*(-c)^(1/3) - d^(1/3))^(1/3)*x^(2/3) + d^(1/9)*(1 + x^2)^(1/3)])/(4*(-c)^(4/3)
*d^(7/9)*x^(1/3)*(1 + x^2)^(1/3)) + ((-1)^(1/3)*(-((-1)^(2/3)*(-c)^(1/3)) - d^(1/3))^(1/3)*(b*c - a*d)*(x + x^
3)^(1/3)*Log[(-((-1)^(2/3)*(-c)^(1/3)) - d^(1/3))^(1/3)*x^(2/3) + d^(1/9)*(1 + x^2)^(1/3)])/(4*(-c)^(4/3)*d^(7
/9)*x^(1/3)*(1 + x^2)^(1/3))

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 285

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + n
*p + 1))), x] + Dist[a*n*(p/(m + n*p + 1)), Int[(c*x)^m*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x]
&& IGtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

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

Rule 476

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 477

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/e^n))^p*(c + d*(x^(k*n)/e^n))^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 495

Int[((x_)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[x*(a + b*x^n)^(p
- 1), x], x] - Dist[(b*c - a*d)/d, Int[x*((a + b*x^n)^(p - 1)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[b*c - a*d, 0] && IGtQ[n, 0] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, 1, 1, n, p, -1, x]

Rule 503

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

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 973

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d*((g*x)^n/x^n), In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[e*((g*x)^n/x^n), Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*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*} \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx &=\frac {\sqrt [3]{x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2} \left (b+a x^6\right )}{d+c x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {\sqrt [3]{x+x^3} \int \left (\frac {a \sqrt [3]{x} \sqrt [3]{1+x^2}}{c}+\frac {(b c-a d) \sqrt [3]{x} \sqrt [3]{1+x^2}}{c \left (d+c x^6\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {\left (a \sqrt [3]{x+x^3}\right ) \int \sqrt [3]{x} \sqrt [3]{1+x^2} \, dx}{c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{d+c x^6} \, dx}{c \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x}}{\left (1+x^2\right )^{2/3}} \, dx}{3 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \left (\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{2 \sqrt {d} \left (\sqrt {d}-\sqrt {-c} x^3\right )}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{2 \sqrt {d} \left (\sqrt {d}+\sqrt {-c} x^3\right )}\right ) \, dx}{c \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt {d}-\sqrt {-c} x^3} \, dx}{2 c \sqrt {d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt {d}+\sqrt {-c} x^3} \, dx}{2 c \sqrt {d} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{2 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \left (-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (-\sqrt [6]{d}-\sqrt [6]{-c} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (-\sqrt [6]{d}+\sqrt [3]{-1} \sqrt [6]{-c} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (-\sqrt [6]{d}-(-1)^{2/3} \sqrt [6]{-c} x\right )}\right ) \, dx}{2 c \sqrt {d} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \left (\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (\sqrt [6]{d}-\sqrt [6]{-c} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (\sqrt [6]{d}+\sqrt [3]{-1} \sqrt [6]{-c} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{3 \sqrt [3]{d} \left (\sqrt [6]{d}-(-1)^{2/3} \sqrt [6]{-c} x\right )}\right ) \, dx}{2 c \sqrt {d} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{-\sqrt [6]{d}-\sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [6]{d}-\sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{-\sqrt [6]{d}+\sqrt [3]{-1} \sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [6]{d}+\sqrt [3]{-1} \sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{-\sqrt [6]{d}-(-1)^{2/3} \sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [6]{d}-(-1)^{2/3} \sqrt [6]{-c} x} \, dx}{6 c d^{5/6} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [3]{d}-\sqrt [3]{-c} x^2} \, dx}{6 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{-c} x^2} \, dx}{6 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \int \frac {\sqrt [3]{x} \sqrt [3]{1+x^2}}{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^2} \, dx}{6 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \sqrt [3]{x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{4 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6}}{\sqrt [3]{d}-\sqrt [3]{-c} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6}}{\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{-c} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x^3 \sqrt [3]{1+x^6}}{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \sqrt [3]{x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {a \sqrt [3]{x+x^3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {\left (a \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{2 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3}}{\sqrt [3]{d}-\sqrt [3]{-c} x^3} \, dx,x,x^{2/3}\right )}{4 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3}}{\sqrt [3]{d}+\sqrt [3]{-1} \sqrt [3]{-c} x^3} \, dx,x,x^{2/3}\right )}{4 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}+2 \frac {\left ((b c-a d) \sqrt [3]{x+x^3}\right ) \text {Subst}\left (\int \frac {x \sqrt [3]{1+x^3}}{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{-c} x^3} \, dx,x,x^{2/3}\right )}{4 c d^{2/3} \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ &=\frac {a x \sqrt [3]{x+x^3}}{2 c}+\frac {(b c-a d) x \sqrt [3]{x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};-x^2,\frac {\sqrt [3]{-c} x^2}{\sqrt [3]{d}}\right )}{4 c d \sqrt [3]{1+x^2}}+\frac {(b c-a d) x \sqrt [3]{x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};-x^2,-\frac {\sqrt [3]{-1} \sqrt [3]{-c} x^2}{\sqrt [3]{d}}\right )}{4 c d \sqrt [3]{1+x^2}}+\frac {(b c-a d) x \sqrt [3]{x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};-x^2,\frac {(-1)^{2/3} \sqrt [3]{-c} x^2}{\sqrt [3]{d}}\right )}{4 c d \sqrt [3]{1+x^2}}-\frac {a \sqrt [3]{x+x^3} \tan ^{-1}\left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{1+x^2}}}{\sqrt {3}}\right )}{2 \sqrt {3} c \sqrt [3]{x} \sqrt [3]{1+x^2}}-\frac {a \sqrt [3]{x+x^3} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{6 c \sqrt [3]{x} \sqrt [3]{1+x^2}}+\frac {a \sqrt [3]{x+x^3} \log \left (1+\frac {x^{4/3}}{\left (1+x^2\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{1+x^2}}\right )}{12 c \sqrt [3]{x} \sqrt [3]{1+x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 226, normalized size = 1.18 \begin {gather*} \frac {\sqrt [3]{x+x^3} \left (a d \left (6 x^{4/3} \sqrt [3]{1+x^2}-2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{1+x^2}}\right )-2 \log \left (c \left (-x^{2/3}+\sqrt [3]{1+x^2}\right )\right )+\log \left (x^{4/3}+x^{2/3} \sqrt [3]{1+x^2}+\left (1+x^2\right )^{2/3}\right )\right )+(-2 b c+2 a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-2 \log \left (\sqrt [3]{x}\right ) \text {$\#$1}+\log \left (\sqrt [3]{1+x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]\right )}{12 c d \sqrt [3]{x} \sqrt [3]{1+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((x + x^3)^(1/3)*(b + a*x^6))/(d + c*x^6),x]

[Out]

((x + x^3)^(1/3)*(a*d*(6*x^(4/3)*(1 + x^2)^(1/3) - 2*Sqrt[3]*ArcTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(1 + x^2)^(
1/3))] - 2*Log[c*(-x^(2/3) + (1 + x^2)^(1/3))] + Log[x^(4/3) + x^(2/3)*(1 + x^2)^(1/3) + (1 + x^2)^(2/3)]) + (
-2*b*c + 2*a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , (-2*Log[x^(1/3)]*#1 + Log[(1 + x^2)^(1/3) - x
^(2/3)*#1]*#1)/(-1 + #1^3) & ]))/(12*c*d*x^(1/3)*(1 + x^2)^(1/3))

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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}+x \right )^{\frac {1}{3}} \left (a \,x^{6}+b \right )}{c \,x^{6}+d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+x)^(1/3)*(a*x^6+b)/(c*x^6+d),x)

[Out]

int((x^3+x)^(1/3)*(a*x^6+b)/(c*x^6+d),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x)^(1/3)*(a*x^6+b)/(c*x^6+d),x, algorithm="maxima")

[Out]

integrate((a*x^6 + b)*(x^3 + x)^(1/3)/(c*x^6 + d), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x)^(1/3)*(a*x^6+b)/(c*x^6+d),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (re
sidue poly has multiple non-linear factors)

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Sympy [F(-1)] Timed out
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((x**3+x)**(1/3)*(a*x**6+b)/(c*x**6+d),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+x)^(1/3)*(a*x^6+b)/(c*x^6+d),x, algorithm="giac")

[Out]

integrate((a*x^6 + b)*(x^3 + x)^(1/3)/(c*x^6 + d), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a\,x^6+b\right )\,{\left (x^3+x\right )}^{1/3}}{c\,x^6+d} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b + a*x^6)*(x + x^3)^(1/3))/(d + c*x^6),x)

[Out]

int(((b + a*x^6)*(x + x^3)^(1/3))/(d + c*x^6), x)

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