3.20.78 \(\int \frac {\sqrt {-b x+a x^3}}{-b^2+a^2 x^4} \, dx\) [1978]

Optimal. Leaf size=140 \[ -\frac {\text {ArcTan}\left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}{-b-2 \sqrt {a} \sqrt {b} x+a x^2}\right )}{4 a^{3/4} b^{3/4}}+\frac {\tanh ^{-1}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right )}{4 a^{3/4} b^{3/4}} \]

[Out]

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

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Rubi [A]
time = 0.29, antiderivative size = 181, normalized size of antiderivative = 1.29, number of steps used = 14, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2081, 1268, 477, 504, 1225, 230, 227, 1713, 214, 211} \begin {gather*} \frac {\sqrt {a x^3-b x} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{2 \sqrt {2} (-a)^{3/4} b^{3/4} \sqrt {x} \sqrt {a x^2-b}}-\frac {\sqrt {a x^3-b x} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{2 \sqrt {2} (-a)^{3/4} b^{3/4} \sqrt {x} \sqrt {a x^2-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 211

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

Rule 214

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

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 230

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + b*(x^4/a)]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + b*(x^4/
a)], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rule 477

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 504

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s
= Denominator[Rt[-a/b, 2]]}, Dist[s/(2*b), Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Dist[s/(2*b), Int[1/
((r - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 1225

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[1/(2*d), Int[1/Sqrt[a + c*x^4], x],
x] + Dist[1/(2*d), Int[(d - e*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d
^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]

Rule 1268

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[(f*x)^m*(d +
e*x^2)^(q + p)*(a/d + (c/e)*x^2)^p, x] /; FreeQ[{a, c, d, e, f, q, m, q}, x] && EqQ[c*d^2 + a*e^2, 0] && Integ
erQ[p]

Rule 1713

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Dist[A, Subst[Int[1/
(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ
[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]

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]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.33, size = 125, normalized size = 0.89 \begin {gather*} -\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {-b x+a x^3} \left (\text {ArcTan}\left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )+i \text {ArcTan}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b+a x^2}}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )\right )}{a^{3/4} b^{3/4} \sqrt {x} \sqrt {-b+a x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.08, size = 601, normalized size = 4.29

method	result	size
elliptic	\(\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}+\frac {\sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\)	\(284\)
default	\(\frac {\sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \left (-\frac {2 \sqrt {a b}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {a b}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{2 b a \sqrt {a \,x^{3}-b x}}-\frac {-\frac {2 b \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticE \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}+\frac {b \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}}-\frac {b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a^{2} \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}}{2 b}\)	\(601\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/b/a*(a*b)^(1/2)*((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2)*(-2*(x-1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2)*(-x
*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)*(-2/a*(a*b)^(1/2)*EllipticE(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),
1/2*2^(1/2))+1/a*(a*b)^(1/2)*EllipticF(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),1/2*2^(1/2)))-1/2/b*(-2/a*b*(
x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)*EllipticE(((x
+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),1/2*2^(1/2))+1/a*b*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/2)+2)^(1/
2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)*EllipticF(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),1/2*2^(1/2))
-b/a^2*(a*b)^(1/2)*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^(1/2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)
^(1/2)/(-1/a*(-a*b)^(1/2)-1/a*(a*b)^(1/2))*EllipticPi(((x+1/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),-1/a*(a*b)^(1/
2)/(-1/a*(-a*b)^(1/2)-1/a*(a*b)^(1/2)),1/2*2^(1/2))-b/a^2*(a*b)^(1/2)*(x*a/(a*b)^(1/2)+1)^(1/2)*(-2*x*a/(a*b)^
(1/2)+2)^(1/2)*(-x*a/(a*b)^(1/2))^(1/2)/(a*x^3-b*x)^(1/2)/(-1/a*(a*b)^(1/2)+1/a*(-a*b)^(1/2))*EllipticPi(((x+1
/a*(a*b)^(1/2))*a/(a*b)^(1/2))^(1/2),-1/a*(a*b)^(1/2)/(-1/a*(a*b)^(1/2)+1/a*(-a*b)^(1/2)),1/2*2^(1/2)))

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 363 vs. \(2 (101) = 202\).
time = 0.43, size = 363, normalized size = 2.59 \begin {gather*} -\frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \arctan \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {a x^{3} - b x} a b \left (-\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}}{a x^{2} - b}\right ) + \frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - 6 \, a b x^{2} + b^{2} + 4 \, {\left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (-\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} - a b^{2}\right )} \left (-\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} - b x} + 4 \, {\left (a^{3} b^{2} x^{3} - a^{2} b^{3} x\right )} \sqrt {-\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) - \frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {a^{2} x^{4} - 6 \, a b x^{2} + b^{2} - 4 \, {\left (4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{3} b^{3} x \left (-\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + \left (\frac {1}{4}\right )^{\frac {1}{4}} {\left (a^{2} b x^{2} - a b^{2}\right )} \left (-\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}}\right )} \sqrt {a x^{3} - b x} + 4 \, {\left (a^{3} b^{2} x^{3} - a^{2} b^{3} x\right )} \sqrt {-\frac {1}{a^{3} b^{3}}}}{a^{2} x^{4} + 2 \, a b x^{2} + b^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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