\(\int \frac {(-b+a x^2) \sqrt {-b x+a x^3}}{x^2 (b+a x^2)} \, dx\) [2119]

Optimal result

Integrand size = 36, antiderivative size = 154 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\frac {2 \sqrt {-b x+a x^3}}{x}+\sqrt [4]{a} \sqrt [4]{b} \arctan \left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt {-b x+a x^3}}{-b-2 \sqrt {a} \sqrt {b} x+a x^2}\right )-\sqrt [4]{a} \sqrt [4]{b} \text {arctanh}\left (\frac {-\frac {b^{3/4}}{2 \sqrt [4]{a}}+\sqrt [4]{a} \sqrt [4]{b} x+\frac {a^{3/4} x^2}{2 \sqrt [4]{b}}}{\sqrt {-b x+a x^3}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.27, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {2081, 477, 485, 12, 504, 1225, 230, 227, 1713, 214, 211} \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {a x^3-b x} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{\sqrt {x} \sqrt {a x^2-b}}-\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {a x^3-b x} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a x^2-b}}\right )}{\sqrt {x} \sqrt {a x^2-b}}+\frac {2 \sqrt {a x^3-b x}}{x} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 485

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[c*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)
*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

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 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*} \text {integral}& = \frac {\sqrt {-b x+a x^3} \int \frac {\left (-b+a x^2\right )^{3/2}}{x^{3/2} \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 {\left (-b+a x^4\right )^{3/2}}{x^2 \left (b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}+\frac {\left (2 \sqrt {-b x+a x^3}\right ) \text {Subst}\left (\int -\frac {4 a b^2 x^2}{\sqrt {-b+a x^4} \left (b+a x^4\right )} \, dx,x,\sqrt {x}\right )}{b \sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}-\frac {\left (8 a b \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 {2 \sqrt {-b x+a x^3}}{x}-\frac {\left (4 a b \sqrt {-b x+a x^3}\right ) \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 {\left (4 a b \sqrt {-b x+a x^3}\right ) \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 {2 \sqrt {-b x+a x^3}}{x}+\frac {\left (2 a \sqrt {b} \sqrt {-b x+a x^3}\right ) \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 )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}}-\frac {\left (2 a \sqrt {b} \sqrt {-b x+a x^3}\right ) \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 )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}+\frac {\left (2 a b \sqrt {-b x+a x^3}\right ) \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 )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}}-\frac {\left (2 a b \sqrt {-b x+a x^3}\right ) \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 )}{\sqrt {-a} \sqrt {x} \sqrt {-b+a x^2}} \\ & = \frac {2 \sqrt {-b x+a x^3}}{x}+\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {-b x+a x^3} \arctan \left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )}{\sqrt {x} \sqrt {-b+a x^2}}-\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {-b x+a x^3} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{-a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )}{\sqrt {x} \sqrt {-b+a x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.44 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.02 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\frac {(1+i) \sqrt {-b x+a x^3} \left ((1-i) \sqrt {-b+a x^2}+\sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \arctan \left (\frac {(1+i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {-b+a x^2}}\right )+i \sqrt [4]{a} \sqrt [4]{b} \sqrt {x} \arctan \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-b+a x^2}}{\sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )\right )}{x \sqrt {-b+a x^2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.53 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.17

method	result	size
risch	\(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}-\frac {\left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right )+2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )-2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right )\right )}{2}\)	\(180\)
default	\(\frac {-\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right ) \left (a b \right )^{\frac {1}{4}} x -2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right ) \left (a b \right )^{\frac {1}{4}} x +2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right ) \left (a b \right )^{\frac {1}{4}} x +4 \sqrt {x \left (a \,x^{2}-b \right )}}{2 x}\)	\(188\)
pseudoelliptic	\(\frac {-\ln \left (\frac {a \,x^{2}+2 x \sqrt {a b}+2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}{a \,x^{2}+2 x \sqrt {a b}-2 \left (a b \right )^{\frac {1}{4}} \sqrt {x \left (a \,x^{2}-b \right )}-b}\right ) \left (a b \right )^{\frac {1}{4}} x -2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}+\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right ) \left (a b \right )^{\frac {1}{4}} x +2 \arctan \left (\frac {x \left (a b \right )^{\frac {1}{4}}-\sqrt {x \left (a \,x^{2}-b \right )}}{\left (a b \right )^{\frac {1}{4}} x}\right ) \left (a b \right )^{\frac {1}{4}} x +4 \sqrt {x \left (a \,x^{2}-b \right )}}{2 x}\)	\(188\)
elliptic	\(\frac {2 a \,x^{2}-2 b}{\sqrt {x \left (a \,x^{2}-b \right )}}-\frac {2 b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {-a b}}{a}-\frac {\sqrt {a b}}{a}\right )}-\frac {2 b \sqrt {a b}\, \sqrt {\frac {x a}{\sqrt {a b}}+1}\, \sqrt {-\frac {2 x a}{\sqrt {a b}}+2}\, \sqrt {-\frac {x a}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{a}\right ) a}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{a \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a \,x^{3}-b x}\, \left (-\frac {\sqrt {a b}}{a}+\frac {\sqrt {-a b}}{a}\right )}\)	\(310\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 41.62 (sec) , antiderivative size = 1362, normalized size of antiderivative = 8.84 \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/4*(4^(1/4)*(-a*b)^(1/4)*x*log((4^(3/4)*((5*a^3 - a^2*b)*x^4 - 8*(a^3 - 2*a^2*b)*x^3 + 5*a*b^2 - b^3 - 6*(5*
a^2*b - a*b^2)*x^2 + 8*(a^2*b - 2*a*b^2)*x)*(-a*b)^(3/4) + 8*(5*a^2*b^2 - a*b^3 - (5*a^3*b - a^2*b^2)*x^2 + 4*
(a^3*b - 2*a^2*b^2)*x + 2*(a^2*b - 2*a*b^2 - (a^3 - 2*a^2*b)*x^2 - (5*a^2*b - a*b^2)*x)*sqrt(-a*b))*sqrt(a*x^3
- b*x) - 4*4^(1/4)*((a^4 - 2*a^3*b)*x^4 + a^2*b^2 - 2*a*b^3 + 2*(5*a^3*b - a^2*b^2)*x^3 - 6*(a^3*b - 2*a^2*b^
2)*x^2 - 2*(5*a^2*b^2 - a*b^3)*x)*(-a*b)^(1/4))/(a^2*x^4 + 2*a*b*x^2 + b^2)) - 4^(1/4)*(-a*b)^(1/4)*x*log(-(4^
(3/4)*((5*a^3 - a^2*b)*x^4 - 8*(a^3 - 2*a^2*b)*x^3 + 5*a*b^2 - b^3 - 6*(5*a^2*b - a*b^2)*x^2 + 8*(a^2*b - 2*a*
b^2)*x)*(-a*b)^(3/4) - 8*(5*a^2*b^2 - a*b^3 - (5*a^3*b - a^2*b^2)*x^2 + 4*(a^3*b - 2*a^2*b^2)*x + 2*(a^2*b - 2
*a*b^2 - (a^3 - 2*a^2*b)*x^2 - (5*a^2*b - a*b^2)*x)*sqrt(-a*b))*sqrt(a*x^3 - b*x) - 4*4^(1/4)*((a^4 - 2*a^3*b)
*x^4 + a^2*b^2 - 2*a*b^3 + 2*(5*a^3*b - a^2*b^2)*x^3 - 6*(a^3*b - 2*a^2*b^2)*x^2 - 2*(5*a^2*b^2 - a*b^3)*x)*(-
a*b)^(1/4))/(a^2*x^4 + 2*a*b*x^2 + b^2)) - I*4^(1/4)*(-a*b)^(1/4)*x*log((4^(3/4)*(I*(5*a^3 - a^2*b)*x^4 - 8*I*
(a^3 - 2*a^2*b)*x^3 + 5*I*a*b^2 - I*b^3 - 6*I*(5*a^2*b - a*b^2)*x^2 + 8*I*(a^2*b - 2*a*b^2)*x)*(-a*b)^(3/4) +
8*(5*a^2*b^2 - a*b^3 - (5*a^3*b - a^2*b^2)*x^2 + 4*(a^3*b - 2*a^2*b^2)*x - 2*(a^2*b - 2*a*b^2 - (a^3 - 2*a^2*b
)*x^2 - (5*a^2*b - a*b^2)*x)*sqrt(-a*b))*sqrt(a*x^3 - b*x) - 4*4^(1/4)*(-I*(a^4 - 2*a^3*b)*x^4 - I*a^2*b^2 + 2
*I*a*b^3 - 2*I*(5*a^3*b - a^2*b^2)*x^3 + 6*I*(a^3*b - 2*a^2*b^2)*x^2 + 2*I*(5*a^2*b^2 - a*b^3)*x)*(-a*b)^(1/4)
)/(a^2*x^4 + 2*a*b*x^2 + b^2)) + I*4^(1/4)*(-a*b)^(1/4)*x*log((4^(3/4)*(-I*(5*a^3 - a^2*b)*x^4 + 8*I*(a^3 - 2*
a^2*b)*x^3 - 5*I*a*b^2 + I*b^3 + 6*I*(5*a^2*b - a*b^2)*x^2 - 8*I*(a^2*b - 2*a*b^2)*x)*(-a*b)^(3/4) + 8*(5*a^2*
b^2 - a*b^3 - (5*a^3*b - a^2*b^2)*x^2 + 4*(a^3*b - 2*a^2*b^2)*x - 2*(a^2*b - 2*a*b^2 - (a^3 - 2*a^2*b)*x^2 - (
5*a^2*b - a*b^2)*x)*sqrt(-a*b))*sqrt(a*x^3 - b*x) - 4*4^(1/4)*(I*(a^4 - 2*a^3*b)*x^4 + I*a^2*b^2 - 2*I*a*b^3 +
2*I*(5*a^3*b - a^2*b^2)*x^3 - 6*I*(a^3*b - 2*a^2*b^2)*x^2 - 2*I*(5*a^2*b^2 - a*b^3)*x)*(-a*b)^(1/4))/(a^2*x^4
+ 2*a*b*x^2 + b^2)) - 8*sqrt(a*x^3 - b*x))/x

Sympy [F]

\[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\int \frac {\sqrt {x \left (a x^{2} - b\right )} \left (a x^{2} - b\right )}{x^{2} \left (a x^{2} + b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\int { \frac {\sqrt {a x^{3} - b x} {\left (a x^{2} - b\right )}}{{\left (a x^{2} + b\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\int { \frac {\sqrt {a x^{3} - b x} {\left (a x^{2} - b\right )}}{{\left (a x^{2} + b\right )} x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-b+a x^2\right ) \sqrt {-b x+a x^3}}{x^2 \left (b+a x^2\right )} \, dx=\text {Hanged} \]

[In]

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

[Out]

\text{Hanged}