3.9.4 \(\int \frac {1}{\sqrt [3]{x+x^3} (b+a x^6)} \, dx\)

Optimal. Leaf size=61 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^9 b-3 \text {$\#$1}^6 b+3 \text {$\#$1}^3 b+a-b\& ,\frac {\log \left (\sqrt [3]{x^3+x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{6 b} \]

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Rubi [F]  time = 0.92, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/6*(x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-b^(1/9) - a^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/
3)])/(b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-b^(1/9) + (-1)^(1/9)*a^
(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst
][Defer[Int][1/((-b^(1/9) - (-1)^(2/9)*a^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*b^(8/9)*(x + x^3)^(1/3
)) - (x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-b^(1/9) + (-1)^(1/3)*a^(1/9)*x)*(1 + x^3)^(1/3)), x
], x, x^(2/3)])/(6*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-b^(1/9) -
(-1)^(4/9)*a^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3
)*Defer[Subst][Defer[Int][1/((-b^(1/9) + (-1)^(5/9)*a^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*b^(8/9)*(
x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-b^(1/9) - (-1)^(2/3)*a^(1/9)*x)*(1 + x
^3)^(1/3)), x], x, x^(2/3)])/(6*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/
((-b^(1/9) + (-1)^(7/9)*a^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])/(6*b^(8/9)*(x + x^3)^(1/3)) - (x^(1/3)*(
1 + x^2)^(1/3)*Defer[Subst][Defer[Int][1/((-b^(1/9) - (-1)^(8/9)*a^(1/9)*x)*(1 + x^3)^(1/3)), x], x, x^(2/3)])
/(6*b^(8/9)*(x + x^3)^(1/3))

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{x+x^3} \left (b+a x^6\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \int \frac {1}{\sqrt [3]{x} \sqrt [3]{1+x^2} \left (b+a x^6\right )} \, dx}{\sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^3} \left (b+a x^9\right )} \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=\frac {\left (3 \sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-\sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+\sqrt [9]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{2/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+\sqrt [3]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{4/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+(-1)^{5/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{2/3} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}+(-1)^{7/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}-\frac {1}{9 b^{8/9} \left (-\sqrt [9]{b}-(-1)^{8/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}}\right ) \, dx,x,x^{2/3}\right )}{2 \sqrt [3]{x+x^3}}\\ &=-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-\sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+\sqrt [9]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{2/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+\sqrt [3]{-1} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{4/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+(-1)^{5/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{2/3} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}+(-1)^{7/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}-\frac {\left (\sqrt [3]{x} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-\sqrt [9]{b}-(-1)^{8/9} \sqrt [9]{a} x\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{6 b^{8/9} \sqrt [3]{x+x^3}}\\ \end {align*}

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Mathematica [A]  time = 2.93, size = 74, normalized size = 1.21 \begin {gather*} -\frac {\sqrt [3]{\frac {1}{x^2}+1} x \text {RootSum}\left [\text {$\#$1}^9 b-3 \text {$\#$1}^6 b+3 \text {$\#$1}^3 b+a-b\&,\frac {\log \left (\sqrt [3]{\frac {1}{x^2}+1}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b \sqrt [3]{x^3+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.78, size = 61, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [a-b+3 b \text {$\#$1}^3-3 b \text {$\#$1}^6+b \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*RootSum[a - b + 3*b*#1^3 - 3*b*#1^6 + b*#1^9 & , (-Log[x] + Log[(x + x^3)^(1/3) - x*#1])/#1 & ]/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(1/(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 {1}{{\left (a x^{6} + b\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (x^{3}+x \right )^{\frac {1}{3}} \left (a \,x^{6}+b \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(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*} -\frac {3 \, {\left (9 \, x^{7} + 3 \, x^{5} - x^{3} + 5 \, x\right )}}{80 \, {\left (a x^{\frac {19}{3}} + b x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}} + \int \frac {9 \, {\left (9 \, b x^{6} + 3 \, b x^{4} - b x^{2} + 5 \, b\right )}}{40 \, {\left (a^{2} x^{\frac {37}{3}} + 2 \, a b x^{\frac {19}{3}} + b^{2} x^{\frac {1}{3}}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/80*(9*x^7 + 3*x^5 - x^3 + 5*x)/((a*x^(19/3) + b*x^(1/3))*(x^2 + 1)^(1/3)) + integrate(9/40*(9*b*x^6 + 3*b*x
^4 - b*x^2 + 5*b)/((a^2*x^(37/3) + 2*a*b*x^(19/3) + b^2*x^(1/3))*(x^2 + 1)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\left (a\,x^6+b\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (a x^{6} + b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((x*(x**2 + 1))**(1/3)*(a*x**6 + b)), x)

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