3.11.75 \(\int \frac {1}{(-b+a^3 x^3) \sqrt [3]{-b x^2+a^3 x^3}} \, dx\)

Optimal. Leaf size=81 \[ \frac {\text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a^3-3 \text {$\#$1}^3 a^6+a^9-a^3 b^2\& ,\frac {\log \left (\sqrt [3]{a^3 x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{3 b} \]

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Rubi [B]  time = 0.99, antiderivative size = 946, normalized size of antiderivative = 11.68, number of steps used = 6, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {2056, 6725, 91} \begin {gather*} \frac {x^{2/3} \sqrt [3]{a^3 x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt [3]{b}-a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt [3]{-1} a x+\sqrt [3]{b}\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt [3]{b}-(-1)^{2/3} a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 91

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)*((e_.) + (f_.)*(x_))), x_Symbol] :> With[{q = Rt[
(d*e - c*f)/(b*e - a*f), 3]}, -Simp[(Sqrt[3]*q*ArcTan[1/Sqrt[3] + (2*q*(a + b*x)^(1/3))/(Sqrt[3]*(c + d*x)^(1/
3))])/(d*e - c*f), x] + (Simp[(q*Log[e + f*x])/(2*(d*e - c*f)), x] - Simp[(3*q*Log[q*(a + b*x)^(1/3) - (c + d*
x)^(1/3)])/(2*(d*e - c*f)), x])] /; FreeQ[{a, b, c, d, e, f}, x]

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

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Mathematica [A]  time = 0.07, size = 132, normalized size = 1.63 \begin {gather*} -\frac {x \left (\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2-b^{2/3}\right ) x}{a^3 x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2+\sqrt [3]{-1} b^{2/3}\right ) x}{a^3 x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a \left (a^2-(-1)^{2/3} b^{2/3}\right ) x}{a^3 x-b}\right )\right )}{b \sqrt [3]{x^2 \left (a^3 x-b\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*(Hypergeometric2F1[1/3, 1, 4/3, (a*(a^2 - b^(2/3))*x)/(-b + a^3*x)] + Hypergeometric2F1[1/3, 1, 4/3, (a*(
a^2 + (-1)^(1/3)*b^(2/3))*x)/(-b + a^3*x)] + Hypergeometric2F1[1/3, 1, 4/3, (a*(a^2 - (-1)^(2/3)*b^(2/3))*x)/(
-b + a^3*x)]))/(b*(x^2*(-b + a^3*x))^(1/3)))

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IntegrateAlgebraic [A]  time = 0.00, size = 81, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [a^9-a^3 b^2-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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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(1/(a^3*x^3-b)/(a^3*x^3-b*x^2)^(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 {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^3*x^3-b)/(a^3*x^3-b*x^2)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((a^3*x^3 - b*x^2)^(1/3)*(a^3*x^3 - b)), x)

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maple [F]  time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a^{3} x^{3}-b \right ) \left (a^{3} x^{3}-b \,x^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a^3*x^3-b)/(a^3*x^3-b*x^2)^(1/3),x)

[Out]

int(1/(a^3*x^3-b)/(a^3*x^3-b*x^2)^(1/3),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} - b\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a^3*x^3-b)/(a^3*x^3-b*x^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((a^3*x^3 - b*x^2)^(1/3)*(a^3*x^3 - b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {1}{\left (b-a^3\,x^3\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/((b - a^3*x^3)*(a^3*x^3 - b*x^2)^(1/3)),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a**3*x**3-b)/(a**3*x**3-b*x**2)**(1/3),x)

[Out]

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

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