3.10.37 \(\int \frac {\sqrt [3]{-x+x^3}}{b+a x^6} \, dx\)

Optimal. Leaf size=71 \[ \frac {\text {RootSum}\left [\text {$\#$1}^9 (-b)+3 \text {$\#$1}^6 b-3 \text {$\#$1}^3 b+a+b\& ,\frac {\text {$\#$1} \log \left (\sqrt [3]{x^3-x}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^3-1}\& \right ]}{6 b} \]

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Rubi [B]  time = 3.66, antiderivative size = 192, normalized size of antiderivative = 2.70, number of steps used = 55, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2056, 6725, 959, 466, 465, 511, 510} \begin {gather*} \frac {x \sqrt [3]{x^3-x} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {\sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {x \sqrt [3]{x^3-x} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,-\frac {\sqrt [3]{-1} \sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {x \sqrt [3]{x^3-x} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {(-1)^{2/3} \sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-x + x^3)^(1/3)/(b + a*x^6),x]

[Out]

(x*(-x + x^3)^(1/3)*AppellF1[2/3, -1/3, 1, 5/3, x^2, ((-a)^(1/3)*x^2)/b^(1/3)])/(4*b*(1 - x^2)^(1/3)) + (x*(-x
+ x^3)^(1/3)*AppellF1[2/3, -1/3, 1, 5/3, x^2, -(((-1)^(1/3)*(-a)^(1/3)*x^2)/b^(1/3))])/(4*b*(1 - x^2)^(1/3))
+ (x*(-x + x^3)^(1/3)*AppellF1[2/3, -1/3, 1, 5/3, x^2, ((-1)^(2/3)*(-a)^(1/3)*x^2)/b^(1/3)])/(4*b*(1 - x^2)^(1
/3))

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 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{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])

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}}{b+a x^6} \, dx &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{b+a x^6} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^3\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^3\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt {b}-\sqrt {-a} x^3} \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt {b}+\sqrt {-a} x^3} \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}-\sqrt [6]{-a} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x\right )}-\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (-\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x\right )}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}-\sqrt [6]{-a} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x\right )}+\frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{3 \sqrt [3]{b} \left (\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x\right )}\right ) \, dx}{2 \sqrt {b} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}-\sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}-\sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}+\sqrt [3]{-1} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{-\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [6]{b}-(-1)^{2/3} \sqrt [6]{-a} x} \, dx}{6 b^{5/6} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}-\sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{x} \sqrt [3]{-1+x^2}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^2} \, dx}{6 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}-\sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{-1+x^6}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^6} \, dx,x,\sqrt [3]{x}\right )}{2 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}-\sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{-1+x^3}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}-\sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}+2 \frac {\sqrt [3]{-x+x^3} \operatorname {Subst}\left (\int \frac {x \sqrt [3]{1-x^3}}{\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{-a} x^3} \, dx,x,x^{2/3}\right )}{4 b^{2/3} \sqrt [3]{x} \sqrt [3]{1-x^2}}\\ &=\frac {x \sqrt [3]{-x+x^3} F_1\left (\frac {2}{3};-\frac {1}{3},1;\frac {5}{3};x^2,\frac {\sqrt [3]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {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]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}+\frac {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]{-a} x^2}{\sqrt [3]{b}}\right )}{4 b \sqrt [3]{1-x^2}}\\ \end {align*}

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Mathematica [F]  time = 2.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-x+x^3}}{b+a x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.36, size = 71, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [a+b-3 b \text {$\#$1}^3+3 b \text {$\#$1}^6-b \text {$\#$1}^9\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^3}\&\right ]}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-x + x^3)^(1/3)/(b + a*x^6),x]

[Out]

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

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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),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x\right )}^{\frac {1}{3}}}{a x^{6} + b}\,{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),x, algorithm="giac")

[Out]

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

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-x \right )^{\frac {1}{3}}}{a \,x^{6}+b}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x\right )}^{\frac {1}{3}}}{a x^{6} + b}\,{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),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3 - x)^(1/3)/(b + a*x^6),x)

[Out]

int((x^3 - x)^(1/3)/(b + a*x^6), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}}{a x^{6} + b}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-x)**(1/3)/(a*x**6+b),x)

[Out]

Integral((x*(x - 1)*(x + 1))**(1/3)/(a*x**6 + b), x)

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