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

Optimal. Leaf size=140 \[ \frac {\log \left (\sqrt [3]{a^3 x^3+a x^6-b}-a x\right )}{3 a}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} a x}{2 \sqrt [3]{a^3 x^3+a x^6-b}+a x}\right )}{\sqrt {3} a}-\frac {\log \left (a x \sqrt [3]{a^3 x^3+a x^6-b}+\left (a^3 x^3+a x^6-b\right )^{2/3}+a^2 x^2\right )}{6 a} \]

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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\\ &=(2 b) \int \frac {1}{\left (-b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx+\int \frac {1}{\sqrt [3]{-b+a^3 x^3+a x^6}} \, dx\\ &=(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+\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 = 0.60, 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 = 2.78, size = 144, normalized size = 1.03 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b+a^3 x^3+a x^6}}\right )}{\sqrt {3} a}+\frac {\log \left (a^2 x-a \sqrt [3]{-b+a^3 x^3+a x^6}\right )}{3 a}-\frac {\log \left (a^2 x^2+a x \sqrt [3]{-b+a^3 x^3+a x^6}+\left (-b+a^3 x^3+a x^6\right )^{2/3}\right )}{6 a} \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]

-(ArcTan[(Sqrt[3]*a*x)/(a*x + 2*(-b + a^3*x^3 + a*x^6)^(1/3))]/(Sqrt[3]*a)) + Log[a^2*x - a*(-b + a^3*x^3 + a*
x^6)^(1/3)]/(3*a) - Log[a^2*x^2 + a*x*(-b + a^3*x^3 + a*x^6)^(1/3) + (-b + a^3*x^3 + a*x^6)^(2/3)]/(6*a)

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

[Out]

Timed out

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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 = 0.03, 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 = -0.01 \begin {gather*} \int -\frac {a\,x^6+b}{\left (b-a\,x^6\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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