3.6.71 \(\int \frac {\sqrt [3]{a+b x^3}}{x (a d-b d x^3)} \, dx\) [571]

Optimal. Leaf size=214 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}-\frac {\log (x)}{2 a^{2/3} d}+\frac {\log \left (a-b x^3\right )}{3\ 2^{2/3} a^{2/3} d}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} d}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{2/3} d} \]

[Out]

-1/2*ln(x)/a^(2/3)/d+1/6*ln(-b*x^3+a)*2^(1/3)/a^(2/3)/d+1/2*ln(a^(1/3)-(b*x^3+a)^(1/3))/a^(2/3)/d-1/2*ln(2^(1/
3)*a^(1/3)-(b*x^3+a)^(1/3))*2^(1/3)/a^(2/3)/d-1/3*arctan(1/3*(a^(1/3)+2*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/a^(2
/3)/d*3^(1/2)+1/3*2^(1/3)*arctan(1/3*(a^(1/3)+2^(2/3)*(b*x^3+a)^(1/3))/a^(1/3)*3^(1/2))/a^(2/3)/d*3^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {457, 85, 59, 631, 210, 31} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}+\frac {\sqrt [3]{2} \text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} d}+\frac {\log \left (a-b x^3\right )}{3\ 2^{2/3} a^{2/3} d}+\frac {\log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2 a^{2/3} d}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} a^{2/3} d}-\frac {\log (x)}{2 a^{2/3} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 59

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/b, 3]}, Simp[-L
og[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Dist[3/(2*b*q), Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x
)^(1/3)], x] - Dist[3/(2*b*q^2), Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x]
&& PosQ[(b*c - a*d)/b]

Rule 85

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[(b*e - a*f)/(b*c
- a*d), Int[(e + f*x)^(p - 1)/(a + b*x), x], x] - Dist[(d*e - c*f)/(b*c - a*d), Int[(e + f*x)^(p - 1)/(c + d*x
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rubi steps

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

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Mathematica [A]
time = 0.27, size = 235, normalized size = 1.10 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )+2 \sqrt [3]{2} \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )+\log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-\sqrt [3]{2} \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{6 a^{2/3} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x \left (-b d \,x^{3}+a d \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1046 vs. \(2 (161) = 322\).
time = 4.63, size = 1046, normalized size = 4.89 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {1}{3}} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{12} \, \left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (\sqrt {3} a^{3} d^{5} \sqrt {\frac {1}{a^{4} d^{6}}} - 3 \, \sqrt {3} a d^{2}\right )} \sqrt {-4 \, \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{3} d^{4} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \sqrt {\frac {1}{a^{4} d^{6}}} + 4 \, \left (\frac {1}{2}\right )^{\frac {2}{3}} a^{2} d^{2} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {2}{3}} + 4 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {2}{3}} - \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (\sqrt {3} a^{3} d^{5} \sqrt {\frac {1}{a^{4} d^{6}}} - 3 \, \sqrt {3} a d^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {2}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {2}{3} \, \sqrt {3} \left (\frac {1}{2}\right )^{\frac {1}{3}} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (\sqrt {3} a^{3} d^{5} \sqrt {\frac {1}{a^{4} d^{6}}} + 3 \, \sqrt {3} a d^{2}\right )} \sqrt {\left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{3} d^{4} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \sqrt {\frac {1}{a^{4} d^{6}}} + \left (\frac {1}{2}\right )^{\frac {2}{3}} a^{2} d^{2} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {2}{3}} - \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {2}{3}} {\left (\sqrt {3} a^{3} d^{5} \sqrt {\frac {1}{a^{4} d^{6}}} + 3 \, \sqrt {3} a d^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{3}} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left (-4 \, \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{3} d^{4} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \sqrt {\frac {1}{a^{4} d^{6}}} + 4 \, \left (\frac {1}{2}\right )^{\frac {2}{3}} a^{2} d^{2} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {2}{3}} + 4 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) - \frac {1}{6} \, \left (\frac {1}{2}\right )^{\frac {1}{3}} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left (4 \, \left (\frac {1}{2}\right )^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{3} d^{4} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \sqrt {\frac {1}{a^{4} d^{6}}} + 4 \, \left (\frac {1}{2}\right )^{\frac {2}{3}} a^{2} d^{2} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {2}{3}} + 4 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) + \frac {1}{3} \, \left (\frac {1}{2}\right )^{\frac {1}{3}} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left (\left (\frac {1}{2}\right )^{\frac {1}{3}} a^{3} d^{4} \left (-\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} + 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \sqrt {\frac {1}{a^{4} d^{6}}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + \frac {1}{3} \, \left (\frac {1}{2}\right )^{\frac {1}{3}} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \log \left (-\left (\frac {1}{2}\right )^{\frac {1}{3}} a^{3} d^{4} \left (\frac {3 \, a^{2} d^{3} \sqrt {\frac {1}{a^{4} d^{6}}} - 1}{a^{2} d^{3}}\right )^{\frac {1}{3}} \sqrt {\frac {1}{a^{4} d^{6}}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*sqrt(3)*(1/2)^(1/3)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(1/3)*arctan(1/12*(1/2)^(2/3)*(sqrt(3)
*a^3*d^5*sqrt(1/(a^4*d^6)) - 3*sqrt(3)*a*d^2)*sqrt(-4*(1/2)^(1/3)*(b*x^3 + a)^(1/3)*a^3*d^4*(-(3*a^2*d^3*sqrt(
1/(a^4*d^6)) + 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + 4*(1/2)^(2/3)*a^2*d^2*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) +
1)/(a^2*d^3))^(2/3) + 4*(b*x^3 + a)^(2/3))*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(2/3) - 1/6*(1/2)^(2
/3)*(sqrt(3)*a^3*d^5*sqrt(1/(a^4*d^6)) - 3*sqrt(3)*a*d^2)*(b*x^3 + a)^(1/3)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1
)/(a^2*d^3))^(2/3) + 1/3*sqrt(3)) + 2/3*sqrt(3)*(1/2)^(1/3)*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3
)*arctan(1/6*(1/2)^(2/3)*(sqrt(3)*a^3*d^5*sqrt(1/(a^4*d^6)) + 3*sqrt(3)*a*d^2)*sqrt((1/2)^(1/3)*(b*x^3 + a)^(1
/3)*a^3*d^4*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + (1/2)^(2/3)*a^2*d^2*((3*a^
2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(2/3) + (b*x^3 + a)^(2/3))*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3
))^(2/3) - 1/6*(1/2)^(2/3)*(sqrt(3)*a^3*d^5*sqrt(1/(a^4*d^6)) + 3*sqrt(3)*a*d^2)*(b*x^3 + a)^(1/3)*((3*a^2*d^3
*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(2/3) - 1/3*sqrt(3)) - 1/6*(1/2)^(1/3)*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/
(a^2*d^3))^(1/3)*log(-4*(1/2)^(1/3)*(b*x^3 + a)^(1/3)*a^3*d^4*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(
1/3)*sqrt(1/(a^4*d^6)) + 4*(1/2)^(2/3)*a^2*d^2*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(2/3) + 4*(b*x^3
+ a)^(2/3)) - 1/6*(1/2)^(1/3)*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*log(4*(1/2)^(1/3)*(b*x^3 +
a)^(1/3)*a^3*d^4*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + 4*(1/2)^(2/3)*a^2*d^2
*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(2/3) + 4*(b*x^3 + a)^(2/3)) + 1/3*(1/2)^(1/3)*(-(3*a^2*d^3*sqr
t(1/(a^4*d^6)) + 1)/(a^2*d^3))^(1/3)*log((1/2)^(1/3)*a^3*d^4*(-(3*a^2*d^3*sqrt(1/(a^4*d^6)) + 1)/(a^2*d^3))^(1
/3)*sqrt(1/(a^4*d^6)) + (b*x^3 + a)^(1/3)) + 1/3*(1/2)^(1/3)*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/
3)*log(-(1/2)^(1/3)*a^3*d^4*((3*a^2*d^3*sqrt(1/(a^4*d^6)) - 1)/(a^2*d^3))^(1/3)*sqrt(1/(a^4*d^6)) + (b*x^3 + a
)^(1/3))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\sqrt [3]{a + b x^{3}}}{- a x + b x^{4}}\, dx}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral((a + b*x**3)**(1/3)/(-a*x + b*x**4), x)/d

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Algebraic extensions not allowed in a ro
otofAlgebraic extensions not allowed in a rootofAlgebraic extensions not allowed in a rootofAlgebraic extensio
ns not allowed in a r

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Mupad [B]
time = 5.10, size = 345, normalized size = 1.61 \begin {gather*} \ln \left ({\left (b\,x^3+a\right )}^{1/3}-a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}\right )\,{\left (\frac {1}{27\,a^2\,d^3}\right )}^{1/3}+\ln \left ({\left (b\,x^3+a\right )}^{1/3}+2^{1/3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}\right )\,{\left (-\frac {2}{27\,a^2\,d^3}\right )}^{1/3}-\ln \left (2^{1/3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}-2\,{\left (b\,x^3+a\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2}{27\,a^2\,d^3}\right )}^{1/3}+\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}-2^{1/3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}+2^{1/3}\,\sqrt {3}\,a\,d\,{\left (-\frac {1}{a^2\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {2}{27\,a^2\,d^3}\right )}^{1/3}+\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}+a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}-\sqrt {3}\,a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{27\,a^2\,d^3}\right )}^{1/3}-\ln \left (2\,{\left (b\,x^3+a\right )}^{1/3}+a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}+\sqrt {3}\,a\,d\,{\left (\frac {1}{a^2\,d^3}\right )}^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {1}{27\,a^2\,d^3}\right )}^{1/3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((a + b*x^3)^(1/3) - a*d*(1/(a^2*d^3))^(1/3))*(1/(27*a^2*d^3))^(1/3) + log((a + b*x^3)^(1/3) + 2^(1/3)*a*d*
(-1/(a^2*d^3))^(1/3))*(-2/(27*a^2*d^3))^(1/3) - log(2^(1/3)*a*d*(-1/(a^2*d^3))^(1/3) - 2*(a + b*x^3)^(1/3) + 2
^(1/3)*3^(1/2)*a*d*(-1/(a^2*d^3))^(1/3)*1i)*((3^(1/2)*1i)/2 + 1/2)*(-2/(27*a^2*d^3))^(1/3) + log(2*(a + b*x^3)
^(1/3) - 2^(1/3)*a*d*(-1/(a^2*d^3))^(1/3) + 2^(1/3)*3^(1/2)*a*d*(-1/(a^2*d^3))^(1/3)*1i)*((3^(1/2)*1i)/2 - 1/2
)*(-2/(27*a^2*d^3))^(1/3) + log(2*(a + b*x^3)^(1/3) + a*d*(1/(a^2*d^3))^(1/3) - 3^(1/2)*a*d*(1/(a^2*d^3))^(1/3
)*1i)*((3^(1/2)*1i)/2 - 1/2)*(1/(27*a^2*d^3))^(1/3) - log(2*(a + b*x^3)^(1/3) + a*d*(1/(a^2*d^3))^(1/3) + 3^(1
/2)*a*d*(1/(a^2*d^3))^(1/3)*1i)*((3^(1/2)*1i)/2 + 1/2)*(1/(27*a^2*d^3))^(1/3)

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