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

Optimal. Leaf size=65 \[ \frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3 a^3+a^6+4 a b\& ,\frac {\log \left (\sqrt [3]{a^3 x^3+a x^6-b}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]

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Rubi [F]  time = 0.79, 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 {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(x*(1 + (2*Sqrt[a]*x^3)/(a^(5/2) - Sqrt[a^5 + 4*b]))^(1/3)*(1 + (2*Sqrt[a]*x^3)/(a^(5/2) + Sqrt[a^5 + 4*b]))^(
1/3)*AppellF1[1/3, 1/3, 1/3, 4/3, (-2*a*x^3)/(a^3 - Sqrt[a^6 + 4*a*b]), (-2*a*x^3)/(a^3 + Sqrt[a^6 + 4*a*b])])
/(-b + a^3*x^3 + a*x^6)^(1/3) - Sqrt[b]*Defer[Int][1/((Sqrt[b] - Sqrt[-a]*x^3)*(-b + a^3*x^3 + a*x^6)^(1/3)),
x] - Sqrt[b]*Defer[Int][1/((Sqrt[b] + Sqrt[-a]*x^3)*(-b + a^3*x^3 + a*x^6)^(1/3)), x]

Rubi steps

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

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Mathematica [F]  time = 0.64, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.75, size = 65, normalized size = 1.00 \begin {gather*} \frac {1}{6} \text {RootSum}\left [a^6+4 a b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b+a^3 x^3+a x^6}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[a^6 + 4*a*b - 2*a^3*#1^3 + #1^6 & , (-Log[x] + Log[(-b + a^3*x^3 + a*x^6)^(1/3) - x*#1])/#1 & ]/6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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