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

Optimal. Leaf size=77 \[ -\frac {\text {RootSum}\left [\text {$\#$1}^9 (-d)+3 \text {$\#$1}^6 a d-3 \text {$\#$1}^3 a^2 d+a^3 d+b^3 c\& ,\frac {\log \left (\sqrt [3]{a x^3-b x}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{6 d} \]

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Rubi [F]  time = 1.28, 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]{-b x+a x^3} \left (d+c x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [B]  time = 0.49, size = 571, normalized size = 7.42 \begin {gather*} -\frac {x \sqrt [3]{\frac {b}{x^2}-a} \left (a^2 \text {RootSum}\left [\text {$\#$1}^3 d+3 \text {$\#$1}^2 a d+3 \text {$\#$1} a^2 d+a^3 d+b^3 c\&,\frac {-\frac {\log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {b}{x^2}-a}+\left (\frac {b}{x^2}-a\right )^{2/3}\right )}{\sqrt [3]{\text {$\#$1}}}+\frac {2 \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {b}{x^2}-a}\right )}{\sqrt [3]{\text {$\#$1}}}+\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b}{x^2}-a}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\sqrt [3]{\text {$\#$1}}}}{\text {$\#$1}^2+2 \text {$\#$1} a+a^2}\&\right ]+2 a \text {RootSum}\left [\text {$\#$1}^3 d+3 \text {$\#$1}^2 a d+3 \text {$\#$1} a^2 d+a^3 d+b^3 c\&,\frac {2 \text {$\#$1}^{2/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{2/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {b}{x^2}-a}+\left (\frac {b}{x^2}-a\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{2/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b}{x^2}-a}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\text {$\#$1}^2+2 \text {$\#$1} a+a^2}\&\right ]+\text {RootSum}\left [\text {$\#$1}^3 d+3 \text {$\#$1}^2 a d+3 \text {$\#$1} a^2 d+a^3 d+b^3 c\&,\frac {2 \text {$\#$1}^{5/3} \log \left (\sqrt [3]{\text {$\#$1}}-\sqrt [3]{\frac {b}{x^2}-a}\right )-\text {$\#$1}^{5/3} \log \left (\text {$\#$1}^{2/3}+\sqrt [3]{\text {$\#$1}} \sqrt [3]{\frac {b}{x^2}-a}+\left (\frac {b}{x^2}-a\right )^{2/3}\right )+2 \sqrt {3} \text {$\#$1}^{5/3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{\frac {b}{x^2}-a}}{\sqrt [3]{\text {$\#$1}}}+1}{\sqrt {3}}\right )}{\text {$\#$1}^2+2 \text {$\#$1} a+a^2}\&\right ]\right )}{12 d \sqrt [3]{a x^3-b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/12*((-a + b/x^2)^(1/3)*x*(a^2*RootSum[b^3*c + a^3*d + 3*a^2*d*#1 + 3*a*d*#1^2 + d*#1^3 & , ((2*Sqrt[3]*ArcT
an[(1 + (2*(-a + b/x^2)^(1/3))/#1^(1/3))/Sqrt[3]])/#1^(1/3) + (2*Log[-(-a + b/x^2)^(1/3) + #1^(1/3)])/#1^(1/3)
- Log[(-a + b/x^2)^(2/3) + (-a + b/x^2)^(1/3)*#1^(1/3) + #1^(2/3)]/#1^(1/3))/(a^2 + 2*a*#1 + #1^2) & ] + 2*a*
RootSum[b^3*c + a^3*d + 3*a^2*d*#1 + 3*a*d*#1^2 + d*#1^3 & , (2*Sqrt[3]*ArcTan[(1 + (2*(-a + b/x^2)^(1/3))/#1^
(1/3))/Sqrt[3]]*#1^(2/3) + 2*Log[-(-a + b/x^2)^(1/3) + #1^(1/3)]*#1^(2/3) - Log[(-a + b/x^2)^(2/3) + (-a + b/x
^2)^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(2/3))/(a^2 + 2*a*#1 + #1^2) & ] + RootSum[b^3*c + a^3*d + 3*a^2*d*#1 + 3*a*
d*#1^2 + d*#1^3 & , (2*Sqrt[3]*ArcTan[(1 + (2*(-a + b/x^2)^(1/3))/#1^(1/3))/Sqrt[3]]*#1^(5/3) + 2*Log[-(-a + b
/x^2)^(1/3) + #1^(1/3)]*#1^(5/3) - Log[(-a + b/x^2)^(2/3) + (-a + b/x^2)^(1/3)*#1^(1/3) + #1^(2/3)]*#1^(5/3))/
(a^2 + 2*a*#1 + #1^2) & ]))/(d*(-(b*x) + a*x^3)^(1/3))

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IntegrateAlgebraic [A]  time = 0.00, size = 77, normalized size = 1.00 \begin {gather*} -\frac {\text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*RootSum[b^3*c + a^3*d - 3*a^2*d*#1^3 + 3*a*d*#1^6 - d*#1^9 & , (-Log[x] + Log[(-(b*x) + a*x^3)^(1/3) - x*
#1])/#1 & ]/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(1/(a*x^3-b*x)^(1/3)/(c*x^6+d),x, algorithm="fricas")

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/((a*x^3 - b*x)^(1/3)*(d + c*x^6)), 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 (a x^{2} - b\right )} \left (c x^{6} + d\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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