3.9.66 \(\int \frac {\sqrt {-b+a x^3}}{x (2 b+a x^3)} \, dx\)

Optimal. Leaf size=66 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {3} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \]

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Rubi [A]  time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {446, 83, 63, 205, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {3} \sqrt {b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 83

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 203

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

Rule 205

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

Rule 446

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]]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 62, normalized size = 0.94 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {b}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {a x^3-b}}{\sqrt {3} \sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.06, size = 66, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {-b+a x^3}}{\sqrt {b}}\right )}{3 \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-b+a x^3}}{\sqrt {3} \sqrt {b}}\right )}{\sqrt {3} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.54, size = 151, normalized size = 2.29 \begin {gather*} \left [-\frac {\sqrt {3} \sqrt {-b} \log \left (\frac {a x^{3} - 2 \, \sqrt {3} \sqrt {a x^{3} - b} \sqrt {-b} - 4 \, b}{a x^{3} + 2 \, b}\right ) + \sqrt {-b} \log \left (\frac {a x^{3} + 2 \, \sqrt {a x^{3} - b} \sqrt {-b} - 2 \, b}{x^{3}}\right )}{6 \, b}, \frac {\sqrt {3} \sqrt {b} \arctan \left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \, \sqrt {b}}\right ) - \sqrt {b} \arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/6*(sqrt(3)*sqrt(-b)*log((a*x^3 - 2*sqrt(3)*sqrt(a*x^3 - b)*sqrt(-b) - 4*b)/(a*x^3 + 2*b)) + sqrt(-b)*log((
a*x^3 + 2*sqrt(a*x^3 - b)*sqrt(-b) - 2*b)/x^3))/b, 1/3*(sqrt(3)*sqrt(b)*arctan(1/3*sqrt(3)*sqrt(a*x^3 - b)/sqr
t(b)) - sqrt(b)*arctan(sqrt(a*x^3 - b)/sqrt(b)))/b]

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giac [A]  time = 0.26, size = 50, normalized size = 0.76 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \, \sqrt {b}}\right )}{3 \, \sqrt {b}} - \frac {\arctan \left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}}\right )}{3 \, \sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*sqrt(a*x^3 - b)/sqrt(b))/sqrt(b) - 1/3*arctan(sqrt(a*x^3 - b)/sqrt(b))/sqrt(b)

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maple [C]  time = 0.46, size = 458, normalized size = 6.94

method	result	size
default	\(-\frac {a \left (\frac {2 \sqrt {a \,x^{3}-b}}{3 a}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\frac {\left (a^{2} b \right )^{\frac {1}{3}} \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+\left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{2 \left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{-3 \left (a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+\left (a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (-i \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (a^{2} b \right )^{\frac {2}{3}}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {-\frac {i \left (x +\frac {\left (a^{2} b \right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{3}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \sqrt {a \,x^{3}-b}}\right )}{3 a^{3}}\right )}{2 b}+\frac {\frac {2 \sqrt {a \,x^{3}-b}}{3}+\frac {2 b \arctanh \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}}{2 b}\)	\(458\)
elliptic	\(\frac {\sqrt {2}\, \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {3}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha \right )}{12 a b \sqrt {a \,x^{3}-b}}-\frac {\sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {3}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right )\right )}{12 \sqrt {a \,x^{3}-b}}+\frac {i \sqrt {2}\, \left (a^{2} b \right )^{\frac {1}{3}} \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}\right )}{6 b \sqrt {a \,x^{3}-b}}-\frac {i \sqrt {2}\, \left (a^{2} b \right )^{\frac {2}{3}} \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha \right )}{12 a b \sqrt {a \,x^{3}-b}}-\frac {i \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \sqrt {\frac {-a x +\left (a^{2} b \right )^{\frac {1}{3}}}{\left (a^{2} b \right )^{\frac {1}{3}} \left (3+i \sqrt {3}\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}-2 a x -\left (a^{2} b \right )^{\frac {1}{3}}\right )}{\left (a^{2} b \right )^{\frac {1}{3}}}}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+2 b \right )}{\sum }\EllipticPi \left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}}+2 a x +\left (a^{2} b \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (a^{2} b \right )^{\frac {1}{3}}}}}{6}, \frac {-2 i \sqrt {3}\, \left (a^{2} b \right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \sqrt {3}\, \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, a b -3 \left (a^{2} b \right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha +3 a b}{6 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{3+i \sqrt {3}}}\right )\right )}{12 \sqrt {a \,x^{3}-b}}+\frac {\arctanh \left (\frac {\sqrt {a \,x^{3}-b}}{\sqrt {-b}}\right )}{3 \sqrt {-b}}\)	\(1430\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^3-b)^(1/2)/x/(a*x^3+2*b),x,method=_RETURNVERBOSE)

[Out]

-1/2*a/b*(2/3*(a*x^3-b)^(1/2)/a-1/3*I/a^3*2^(1/2)*sum((a^2*b)^(1/3)*(-1/2*I*a*(2*x+1/a*(I*3^(1/2)*(a^2*b)^(1/3
)+(a^2*b)^(1/3)))/(a^2*b)^(1/3))^(1/2)*(a*(x-1/a*(a^2*b)^(1/3))/(-3*(a^2*b)^(1/3)-I*3^(1/2)*(a^2*b)^(1/3)))^(1
/2)*(1/2*I*a*(2*x+1/a*(-I*3^(1/2)*(a^2*b)^(1/3)+(a^2*b)^(1/3)))/(a^2*b)^(1/3))^(1/2)/(a*x^3-b)^(1/2)*(-I*(a^2*
b)^(1/3)*_alpha*3^(1/2)*a+I*3^(1/2)*(a^2*b)^(2/3)+2*_alpha^2*a^2-(a^2*b)^(1/3)*_alpha*a-(a^2*b)^(2/3))*Ellipti
cPi(1/3*3^(1/2)*(-I*(x+1/2/a*(a^2*b)^(1/3)+1/2*I*3^(1/2)/a*(a^2*b)^(1/3))*3^(1/2)*a/(a^2*b)^(1/3))^(1/2),1/6/a
*(-2*I*3^(1/2)*(a^2*b)^(1/3)*_alpha^2*a+I*3^(1/2)*(a^2*b)^(2/3)*_alpha+I*3^(1/2)*a*b-3*(a^2*b)^(2/3)*_alpha+3*
a*b)/b,(-I*3^(1/2)/a*(a^2*b)^(1/3)/(-3/2/a*(a^2*b)^(1/3)-1/2*I*3^(1/2)/a*(a^2*b)^(1/3)))^(1/2)),_alpha=RootOf(
_Z^3*a+2*b)))+1/2/b*(2/3*(a*x^3-b)^(1/2)+2/3*b*arctanh((a*x^3-b)^(1/2)/(-b)^(1/2))/(-b)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.50, size = 96, normalized size = 1.45 \begin {gather*} \frac {\ln \left (\frac {2\,b-a\,x^3+\sqrt {b}\,\sqrt {a\,x^3-b}\,2{}\mathrm {i}}{x^3}\right )\,1{}\mathrm {i}}{6\,\sqrt {b}}+\frac {\sqrt {3}\,\ln \left (\frac {\sqrt {3}\,b\,4{}\mathrm {i}+6\,\sqrt {b}\,\sqrt {a\,x^3-b}-\sqrt {3}\,a\,x^3\,1{}\mathrm {i}}{2\,a\,x^3+4\,b}\right )\,1{}\mathrm {i}}{6\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((2*b + b^(1/2)*(a*x^3 - b)^(1/2)*2i - a*x^3)/x^3)*1i)/(6*b^(1/2)) + (3^(1/2)*log((3^(1/2)*b*4i + 6*b^(1/2
)*(a*x^3 - b)^(1/2) - 3^(1/2)*a*x^3*1i)/(4*b + 2*a*x^3))*1i)/(6*b^(1/2))

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sympy [A]  time = 6.49, size = 63, normalized size = 0.95 \begin {gather*} \frac {2 \left (- \frac {a \operatorname {atan}{\left (\frac {\sqrt {a x^{3} - b}}{\sqrt {b}} \right )}}{6 \sqrt {b}} + \frac {\sqrt {3} a \operatorname {atan}{\left (\frac {\sqrt {3} \sqrt {a x^{3} - b}}{3 \sqrt {b}} \right )}}{6 \sqrt {b}}\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-a*atan(sqrt(a*x**3 - b)/sqrt(b))/(6*sqrt(b)) + sqrt(3)*a*atan(sqrt(3)*sqrt(a*x**3 - b)/(3*sqrt(b)))/(6*sqr
t(b)))/a

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