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

Optimal. Leaf size=63 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {2} \sqrt {b}}\right )}{3 \sqrt {b}} \]

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Rubi [A]  time = 0.09, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {573, 156, 63, 208, 207} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {2} \sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/Sqrt[b] - (4*Sqrt[2]*ArcTanh[Sqrt[b + a*x^3]/(Sqrt[2]*Sqrt[b])])/(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 156

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

Rule 207

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

Rule 208

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

Rule 573

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

Rubi steps

\begin {align*} \int \frac {3 b+a x^3}{x \left (-b+a x^3\right ) \sqrt {b+a x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {3 b+a x}{x (-b+a x) \sqrt {b+a x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} (4 a) \operatorname {Subst}\left (\int \frac {1}{(-b+a x) \sqrt {b+a x}} \, dx,x,x^3\right )-\operatorname {Subst}\left (\int \frac {1}{x \sqrt {b+a x}} \, dx,x,x^3\right )\\ &=\frac {8}{3} \operatorname {Subst}\left (\int \frac {1}{-2 b+x^2} \, dx,x,\sqrt {b+a x^3}\right )-\frac {2 \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x^3}\right )}{a}\\ &=\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {2} \sqrt {b}}\right )}{3 \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 60, normalized size = 0.95 \begin {gather*} \frac {2 \left (3 \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {b}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a x^3+b}}{\sqrt {2} \sqrt {b}}\right )\right )}{3 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.07, size = 63, normalized size = 1.00 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {b+a x^3}}{\sqrt {2} \sqrt {b}}\right )}{3 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*ArcTanh[Sqrt[b + a*x^3]/Sqrt[b]])/Sqrt[b] - (4*Sqrt[2]*ArcTanh[Sqrt[b + a*x^3]/(Sqrt[2]*Sqrt[b])])/(3*Sqrt[
b])

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fricas [A]  time = 0.49, size = 154, normalized size = 2.44 \begin {gather*} \left [\frac {2 \, \sqrt {2} \sqrt {b} \log \left (\frac {a x^{3} - 2 \, \sqrt {2} \sqrt {a x^{3} + b} \sqrt {b} + 3 \, b}{a x^{3} - b}\right ) + 3 \, \sqrt {b} \log \left (\frac {a x^{3} + 2 \, \sqrt {a x^{3} + b} \sqrt {b} + 2 \, b}{x^{3}}\right )}{3 \, b}, \frac {2 \, {\left (2 \, \sqrt {2} b \sqrt {-\frac {1}{b}} \arctan \left (\frac {\sqrt {2} b \sqrt {-\frac {1}{b}}}{\sqrt {a x^{3} + b}}\right ) - 3 \, \sqrt {-b} \arctan \left (\frac {\sqrt {a x^{3} + b} \sqrt {-b}}{b}\right )\right )}}{3 \, b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.13, size = 54, normalized size = 0.86 \begin {gather*} \frac {4 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{3} + b}}{2 \, \sqrt {-b}}\right )}{3 \, \sqrt {-b}} - \frac {2 \, \arctan \left (\frac {\sqrt {a x^{3} + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*sqrt(2)*arctan(1/2*sqrt(2)*sqrt(a*x^3 + b)/sqrt(-b))/sqrt(-b) - 2*arctan(sqrt(a*x^3 + b)/sqrt(-b))/sqrt(-b
)

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maple [C]  time = 0.29, size = 433, normalized size = 6.87

method	result	size
default	\(\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}-i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{\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 {-\frac {i a \left (2 x +\frac {\left (-a^{2} b \right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a^{2} b \right )^{\frac {1}{3}}}{a}\right )}{2 \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 \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {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}{4 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^{2} b}+\frac {2 \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{\sqrt {b}}\)	\(433\)
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}-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 \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {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}{4 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha \right )}{3 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}-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 \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {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}{4 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\right )\right )}{3 \sqrt {a \,x^{3}+b}}+\frac {2 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}-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 \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {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}{4 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha ^{2}\right )}{3 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}-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 \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {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}{4 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\right ) \underline {\hspace {1.25 ex}}\alpha \right )}{3 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}-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 \left (-a^{2} b \right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a +i \left (-a^{2} b \right )^{\frac {2}{3}} \sqrt {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}{4 a b}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\right )\right )}{3 \sqrt {a \,x^{3}+b}}+\frac {2 \arctanh \left (\frac {\sqrt {a \,x^{3}+b}}{\sqrt {b}}\right )}{\sqrt {b}}\)	\(1502\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*I/a^2/b*2^(1/2)*sum((-a^2*b)^(1/3)*(1/2*I*a*(2*x+1/a*((-a^2*b)^(1/3)-I*3^(1/2)*(-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*(
(-a^2*b)^(1/3)+I*3^(1/2)*(-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))*EllipticPi(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),-1/4/a*(2*I*(-a^2*b)^
(1/3)*3^(1/2)*_alpha^2*a-I*(-a^2*b)^(2/3)*3^(1/2)*_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-b))+
2*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 {a x^{3} + 3 \, b}{\sqrt {a x^{3} + b} {\left (a x^{3} - b\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.32, size = 89, normalized size = 1.41 \begin {gather*} \frac {\ln \left (\frac {\left (\sqrt {a\,x^3+b}-\sqrt {b}\right )\,{\left (\sqrt {a\,x^3+b}+\sqrt {b}\right )}^3}{x^6}\right )}{\sqrt {b}}+\frac {2\,\sqrt {2}\,\ln \left (\frac {3\,\sqrt {2}\,b-4\,\sqrt {b}\,\sqrt {a\,x^3+b}+\sqrt {2}\,a\,x^3}{b-a\,x^3}\right )}{3\,\sqrt {b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 37.51, size = 66, normalized size = 1.05 \begin {gather*} - \frac {2 \operatorname {atan}{\left (\frac {\sqrt {a x^{3} + b}}{\sqrt {- b}} \right )}}{\sqrt {- b}} + \frac {4 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {a x^{3} + b}}{2 \sqrt {- b}} \right )}}{3 \sqrt {- b}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*atan(sqrt(a*x**3 + b)/sqrt(-b))/sqrt(-b) + 4*sqrt(2)*atan(sqrt(2)*sqrt(a*x**3 + b)/(2*sqrt(-b)))/(3*sqrt(-b
))

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