3.11.75 \(\int \frac {1}{(-b+a^3 x^3) \sqrt [3]{-b x^2+a^3 x^3}} \, dx\) [1075]

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

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(946\) vs. \(2(81)=162\).
time = 0.59, 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 = {2081, 6857, 93} \begin {gather*} \frac {x^{2/3} \sqrt [3]{a^3 x-b} \text {ArcTan}\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} \text {ArcTan}\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} \text {ArcTan}\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 93

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 2081

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 6857

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 = 0.00, size = 119, normalized size = 1.47 \begin {gather*} \frac {x^{2/3} \sqrt [3]{-b+a^3 x} \text {RootSum}\left [a^9-a^3 b^2-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b \sqrt [3]{x^2 \left (-b+a^3 x\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^(2/3)*(-b + a^3*x)^(1/3)*RootSum[a^9 - a^3*b^2 - 3*a^6*#1^3 + 3*a^3*#1^6 - #1^9 & , (-Log[x^(1/3)] + Log[(-
b + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(3*b*(x^2*(-b + a^3*x))^(1/3))

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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*} \text {Failed to integrate} \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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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 1.48, size = 39968, normalized size = 493.43 \begin {gather*} \text {Too large to display} \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]

-2*sqrt(3)*(1/6)^(1/3)*(1/18)^(1/3)*((6*a^3 + (a^6*b^3 - b^5)*(2*(1/2)^(2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b
^6 - b^8))*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 -
a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) + (1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 -
3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3)
+ 1)) + 3*sqrt(1/3)*(a^6*b^3 - b^5)*sqrt(-(4*a^6 + (a^12*b^6 - 2*a^6*b^8 + b^10)*(2*(1/2)^(2/3)*(a^6/(a^6*b^3
- b^5)^2 - 1/(a^6*b^6 - b^8))*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b
^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) + (1/2)^(1/3)*(2*a^9/
(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^
7))^(1/3)*(I*sqrt(3) + 1))^2 + 4*(a^9*b^3 - a^3*b^5)*(2*(1/2)^(2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b^6 - b^8)
)*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11)
+ 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) + (1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a
^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3) + 1)) -
16*b^2)/(a^12*b^6 - 2*a^6*b^8 + b^10)))/(a^6*b^3 - b^5))^(1/3)*arctan(1/6*(3*sqrt(3)*sqrt(2)*(1/6)^(1/3)*(1/18
)^(1/3)*b*x*((6*a^3 + (a^6*b^3 - b^5)*(2*(1/2)^(2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b^6 - b^8))*(-I*sqrt(3) +
1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^
2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) + (1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(
a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3) + 1)) + 3*sqrt(1/3)*(a^
6*b^3 - b^5)*sqrt(-(4*a^6 + (a^12*b^6 - 2*a^6*b^8 + b^10)*(2*(1/2)^(2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b^6 -
b^8))*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*
b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) + (1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^
3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3) + 1
))^2 + 4*(a^9*b^3 - a^3*b^5)*(2*(1/2)^(2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b^6 - b^8))*(-I*sqrt(3) + 1)/(2*a^
9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*
b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) + (1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 -
b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3) + 1)) - 16*b^2)/(a^12*b^6 - 2*a^
6*b^8 + b^10)))/(a^6*b^3 - b^5))^(1/3)*sqrt(-(9*(1/6)^(2/3)*(1/18)^(2/3)*(6*(a^3*x^3 - b*x^2)^(1/3)*(2*(1/2)^(
2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b^6 - b^8))*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 -
b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) + (
1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a
^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3) + 1))*a^6*b^5*x + (a^9*b^8 - a^3*b^10)*(a^3*x^3 - b*x^2)^(1/3)*(2*(1/2)
^(2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b^6 - b^8))*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6
- b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) +
(1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/(
(a^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3) + 1))^2*x - 3*sqrt(1/3)*(a^9*b^8 - a^3*b^10)*(a^3*x^3 - b*x^2)^(1/3)*
(2*(1/2)^(2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b^6 - b^8))*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/(
(a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3
- b^5) + (1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^1
1) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3) + 1))*x*sqrt(-(4*a^6 + (a^12*b^6 - 2*a^6*b^8 + b^10)*(2*(1/2)
^(2/3)*(a^6/(a^6*b^3 - b^5)^2 - 1/(a^6*b^6 - b^8))*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6
- b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) +
(1/2)^(1/3)*(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/(
(a^6 - b^2)^2*a^3*b^7))^(1/3)*(I*sqrt(3) + 1))^2 + 4*(a^9*b^3 - a^3*b^5)*(2*(1/2)^(2/3)*(a^6/(a^6*b^3 - b^5)^2
- 1/(a^6*b^6 - b^8))*(-I*sqrt(3) + 1)/(2*a^9/(a^6*b^3 - b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/
(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3) - 2*a^3/(a^6*b^3 - b^5) + (1/2)^(1/3)*(2*a^9/(a^6*b^3
- b^5)^3 - 3*a^3/((a^6*b^6 - b^8)*(a^6*b^3 - b^5)) + 1/(a^9*b^9 - a^3*b^11) + 1/((a^6 - b^2)^2*a^3*b^7))^(1/3)
*(I*sqrt(3) + 1)) - 16*b^2)/(a^12*b^6 - 2*a^6*b...

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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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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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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