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

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

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Rubi [B]  time = 3.06, antiderivative size = 412, normalized size of antiderivative = 2.16, number of steps used = 55, number of rules used = 16, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2056, 6725, 279, 329, 275, 331, 292, 31, 634, 618, 204, 628, 959, 466, 465, 510} \begin {gather*} \frac {\sqrt [3]{x^3+x} x (b c-a d) 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]{x^2+1}}+\frac {\sqrt [3]{x^3+x} x (b c-a d) 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]{x^2+1}}+\frac {\sqrt [3]{x^3+x} x (b c-a d) 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]{x^2+1}}+\frac {a \sqrt [3]{x^3+x} x}{2 c}-\frac {a \sqrt [3]{x^3+x} \log \left (1-\frac {x^{2/3}}{\sqrt [3]{x^2+1}}\right )}{6 c \sqrt [3]{x^2+1} \sqrt [3]{x}}+\frac {a \sqrt [3]{x^3+x} \log \left (\frac {x^{4/3}}{\left (x^2+1\right )^{2/3}}+\frac {x^{2/3}}{\sqrt [3]{x^2+1}}+1\right )}{12 c \sqrt [3]{x^2+1} \sqrt [3]{x}}-\frac {a \sqrt [3]{x^3+x} \tan ^{-1}\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}} \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) + ((b*c - a*d)*x*(x + x^3)^(1/3)*AppellF1[2/3, -1/3, 1, 5/3, -x^2, ((-c)^(1/3)*x^2
)/d^(1/3)])/(4*c*d*(1 + x^2)^(1/3)) + ((b*c - a*d)*x*(x + x^3)^(1/3)*AppellF1[2/3, -1/3, 1, 5/3, -x^2, -(((-1)
^(1/3)*(-c)^(1/3)*x^2)/d^(1/3))])/(4*c*d*(1 + x^2)^(1/3)) + ((b*c - a*d)*x*(x + x^3)^(1/3)*AppellF1[2/3, -1/3,
1, 5/3, -x^2, ((-1)^(2/3)*(-c)^(1/3)*x^2)/d^(1/3)])/(4*c*d*(1 + x^2)^(1/3)) - (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)) - (a*(x + x^3)^(1/3)*Log[1 - x^(2
/3)/(1 + x^2)^(1/3)])/(6*c*x^(1/3)*(1 + x^2)^(1/3)) + (a*(x + x^3)^(1/3)*Log[1 + x^(4/3)/(1 + x^2)^(2/3) + x^(
2/3)/(1 + x^2)^(1/3)])/(12*c*x^(1/3)*(1 + x^2)^(1/3))

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 275

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 279

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 292

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

Rule 329

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 331

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 465

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 466

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 510

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)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), 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 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 959

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 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]{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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [F]  time = 7.19, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x+x^3} \left (b+a x^6\right )}{d+c x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.93, size = 194, normalized size = 1.02 \begin {gather*} \frac {a x \sqrt [3]{x+x^3}}{2 c}-\frac {a \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 \sqrt {3} c}-\frac {a \log \left (-c x+c \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

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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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{c x^{6} + d}\,{d x} \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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maple [F]  time = 0.01, 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*} \int \frac {{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}{c x^{6} + d}\,{d x} \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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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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sympy [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((x**3+x)**(1/3)*(a*x**6+b)/(c*x**6+d),x)

[Out]

Timed out

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