3.12.62 \(\int \frac {\sqrt {-x+x^4}}{-b+a x^3} \, dx\)

Optimal. Leaf size=86 \[ \frac {2 \sqrt {a-b} \tan ^{-1}\left (\frac {x \sqrt {x^4-x} \sqrt {a-b}}{\sqrt {b} (x-1) \left (x^2+x+1\right )}\right )}{3 a \sqrt {b}}+\frac {2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {x^4-x}}\right )}{3 a} \]

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Rubi [A]  time = 0.15, antiderivative size = 125, normalized size of antiderivative = 1.45, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2056, 466, 465, 402, 217, 206, 377, 205} \begin {gather*} \frac {2 \sqrt {x^4-x} \sqrt {a-b} \tan ^{-1}\left (\frac {x^{3/2} \sqrt {a-b}}{\sqrt {b} \sqrt {x^3-1}}\right )}{3 a \sqrt {b} \sqrt {x} \sqrt {x^3-1}}+\frac {2 \sqrt {x^4-x} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {x^3-1}}\right )}{3 a \sqrt {x} \sqrt {x^3-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-x + x^4]/(-b + a*x^3),x]

[Out]

(2*Sqrt[a - b]*Sqrt[-x + x^4]*ArcTan[(Sqrt[a - b]*x^(3/2))/(Sqrt[b]*Sqrt[-1 + x^3])])/(3*a*Sqrt[b]*Sqrt[x]*Sqr
t[-1 + x^3]) + (2*Sqrt[-x + x^4]*ArcTanh[x^(3/2)/Sqrt[-1 + x^3]])/(3*a*Sqrt[x]*Sqrt[-1 + x^3])

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 206

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 377

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

Rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Dist[b/d, Int[(a + b*x^2)^(p - 1), x], x]
- Dist[(b*c - a*d)/d, Int[(a + b*x^2)^(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d,
0] && GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4])

Rule 465

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

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {-x+x^4}}{-b+a x^3} \, dx &=\frac {\sqrt {-x+x^4} \int \frac {\sqrt {x} \sqrt {-1+x^3}}{-b+a x^3} \, dx}{\sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {-1+x^6}}{-b+a x^6} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^2}}{-b+a x^2} \, dx,x,x^{3/2}\right )}{3 \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2}} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (2 (a-b) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^2} \left (-b+a x^2\right )} \, dx,x,x^{3/2}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {\left (2 \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}-\frac {\left (2 (a-b) \sqrt {-x+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-b-(a-b) x^2} \, dx,x,\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}\\ &=\frac {2 \sqrt {a-b} \sqrt {-x+x^4} \tan ^{-1}\left (\frac {\sqrt {a-b} x^{3/2}}{\sqrt {b} \sqrt {-1+x^3}}\right )}{3 a \sqrt {b} \sqrt {x} \sqrt {-1+x^3}}+\frac {2 \sqrt {-x+x^4} \tanh ^{-1}\left (\frac {x^{3/2}}{\sqrt {-1+x^3}}\right )}{3 a \sqrt {x} \sqrt {-1+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.03, size = 52, normalized size = 0.60 \begin {gather*} -\frac {2 x \sqrt {x \left (x^3-1\right )} F_1\left (\frac {1}{2};-\frac {1}{2},1;\frac {3}{2};x^3,\frac {a x^3}{b}\right )}{3 b \sqrt {1-x^3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[-x + x^4]/(-b + a*x^3),x]

[Out]

(-2*x*Sqrt[x*(-1 + x^3)]*AppellF1[1/2, -1/2, 1, 3/2, x^3, (a*x^3)/b])/(3*b*Sqrt[1 - x^3])

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IntegrateAlgebraic [A]  time = 0.53, size = 86, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a-b} x \sqrt {-x+x^4}}{\sqrt {b} (-1+x) \left (1+x+x^2\right )}\right )}{3 a \sqrt {b}}+\frac {2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {-x+x^4}}\right )}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[-x + x^4]/(-b + a*x^3),x]

[Out]

(2*Sqrt[a - b]*ArcTan[(Sqrt[a - b]*x*Sqrt[-x + x^4])/(Sqrt[b]*(-1 + x)*(1 + x + x^2))])/(3*a*Sqrt[b]) + (2*Arc
Tanh[x^2/Sqrt[-x + x^4]])/(3*a)

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fricas [A]  time = 1.38, size = 219, normalized size = 2.55 \begin {gather*} \left [\frac {\sqrt {-\frac {a - b}{b}} \log \left (-\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} x^{6} + 2 \, {\left (3 \, a b - 4 \, b^{2}\right )} x^{3} + b^{2} + 4 \, {\left ({\left (a b - 2 \, b^{2}\right )} x^{4} + b^{2} x\right )} \sqrt {x^{4} - x} \sqrt {-\frac {a - b}{b}}}{a^{2} x^{6} - 2 \, a b x^{3} + b^{2}}\right ) + 2 \, \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right )}{6 \, a}, \frac {\sqrt {\frac {a - b}{b}} \arctan \left (-\frac {2 \, \sqrt {x^{4} - x} b x \sqrt {\frac {a - b}{b}}}{{\left (a - 2 \, b\right )} x^{3} + b}\right ) + \log \left (-2 \, x^{3} - 2 \, \sqrt {x^{4} - x} x + 1\right )}{3 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x)^(1/2)/(a*x^3-b),x, algorithm="fricas")

[Out]

[1/6*(sqrt(-(a - b)/b)*log(-((a^2 - 8*a*b + 8*b^2)*x^6 + 2*(3*a*b - 4*b^2)*x^3 + b^2 + 4*((a*b - 2*b^2)*x^4 +
b^2*x)*sqrt(x^4 - x)*sqrt(-(a - b)/b))/(a^2*x^6 - 2*a*b*x^3 + b^2)) + 2*log(-2*x^3 - 2*sqrt(x^4 - x)*x + 1))/a
, 1/3*(sqrt((a - b)/b)*arctan(-2*sqrt(x^4 - x)*b*x*sqrt((a - b)/b)/((a - 2*b)*x^3 + b)) + log(-2*x^3 - 2*sqrt(
x^4 - x)*x + 1))/a]

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giac [A]  time = 0.30, size = 80, normalized size = 0.93 \begin {gather*} -\frac {2 \, {\left (a - b\right )} \arctan \left (\frac {b \sqrt {-\frac {1}{x^{3}} + 1}}{\sqrt {a b - b^{2}}}\right )}{3 \, \sqrt {a b - b^{2}} a} + \frac {\log \left (\sqrt {-\frac {1}{x^{3}} + 1} + 1\right )}{3 \, a} - \frac {\log \left ({\left | \sqrt {-\frac {1}{x^{3}} + 1} - 1 \right |}\right )}{3 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x)^(1/2)/(a*x^3-b),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.21, size = 636, normalized size = 7.40

method	result	size
default	\(\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{a \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}-\frac {2 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \sqrt {3}\, b +3 b}{6 b}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3 a}\)	\(636\)
elliptic	\(\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (-1+x \right )^{2} \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {-\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{a \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {x \left (-1+x \right ) \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}-\frac {2 \sqrt {4}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\left (-1+x \right )^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (1-i \sqrt {3}\right ) \sqrt {\frac {x \left (-3+i \sqrt {3}\right )}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{\left (-1+x \right ) \left (-1-i \sqrt {3}\right )}}\, \sqrt {\frac {1+2 x -i \sqrt {3}}{\left (-1+x \right ) \left (-1+i \sqrt {3}\right )}}\, \left (\EllipticF \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} a \EllipticPi \left (\sqrt {\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-1+x \right )}}, \frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, a -3 \underline {\hspace {1.25 ex}}\alpha ^{2} a -i \sqrt {3}\, b +3 b}{6 b}, \sqrt {\frac {\left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\right )}{b}\right )}{\underline {\hspace {1.25 ex}}\alpha \left (-3+i \sqrt {3}\right ) \sqrt {x \left (-1+x \right ) \left (i \sqrt {3}+2 x +1\right ) \left (1+2 x -i \sqrt {3}\right )}}\right )}{3 a}\)	\(636\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-x)^(1/2)/(a*x^3-b),x,method=_RETURNVERBOSE)

[Out]

2/a*(1/2-1/2*I*3^(1/2))*((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2)*(-1+x)^2*((x+1/2+1/2*I*3^(1
/2))/(-1/2-1/2*I*3^(1/2))/(-1+x))^(1/2)*((x+1/2-1/2*I*3^(1/2))/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2)/(-3/2+1/2*I*
3^(1/2))/(x*(-1+x)*(x+1/2+1/2*I*3^(1/2))*(x+1/2-1/2*I*3^(1/2)))^(1/2)*(EllipticF(((-3/2+1/2*I*3^(1/2))*x/(-1/2
+1/2*I*3^(1/2))/(-1+x))^(1/2),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2))
)^(1/2))-EllipticPi(((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2),(-1/2+1/2*I*3^(1/2))/(-3/2+1/2*
I*3^(1/2)),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2)))-2/3/a*4^(
1/2)*sum(1/_alpha*(-1+x)^2*(_alpha^2+_alpha+1)*(1-I*3^(1/2))*(x/(-1+x)*(-3+I*3^(1/2))/(-1+I*3^(1/2)))^(1/2)*(1
/(-1+x)*(I*3^(1/2)+2*x+1)/(-1-I*3^(1/2)))^(1/2)*(1/(-1+x)*(1+2*x-I*3^(1/2))/(-1+I*3^(1/2)))^(1/2)/(-3+I*3^(1/2
))/(x*(-1+x)*(I*3^(1/2)+2*x+1)*(1+2*x-I*3^(1/2)))^(1/2)*(EllipticF(((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2)
)/(-1+x))^(1/2),((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1/2))-_alph
a^2*a/b*EllipticPi(((-3/2+1/2*I*3^(1/2))*x/(-1/2+1/2*I*3^(1/2))/(-1+x))^(1/2),1/6*(I*_alpha^2*3^(1/2)*a-3*_alp
ha^2*a-I*3^(1/2)*b+3*b)/b,((3/2+1/2*I*3^(1/2))*(1/2-1/2*I*3^(1/2))/(1/2+1/2*I*3^(1/2))/(3/2-1/2*I*3^(1/2)))^(1
/2))),_alpha=RootOf(_Z^3*a-b))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-x)^(1/2)/(a*x^3-b),x, algorithm="maxima")

[Out]

integrate(sqrt(x^4 - x)/(a*x^3 - b), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {\sqrt {x^4-x}}{b-a\,x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^4 - x)^(1/2)/(b - a*x^3),x)

[Out]

-int((x^4 - x)^(1/2)/(b - a*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-x)**(1/2)/(a*x**3-b),x)

[Out]

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

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