3.24.9 \(\int \frac {b^4+a^4 x^4}{\sqrt {-b^2 x+a^2 x^3} (-b^4+a^4 x^4)} \, dx\)

Optimal. Leaf size=177 \[ \frac {\sqrt {a^2 x^3-b^2 x}}{b^2-a^2 x^2}-\frac {\tan ^{-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}{a^2 x^2-2 a b x-b^2}\right )}{4 \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\frac {a^{3/2} x^2}{2 \sqrt {b}}-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^3-b^2 x}}\right )}{4 \sqrt {a} \sqrt {b}} \]

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Rubi [A]  time = 1.63, antiderivative size = 234, normalized size of antiderivative = 1.32, number of steps used = 22, number of rules used = 13, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2056, 6715, 6725, 224, 221, 1404, 414, 523, 409, 1211, 1699, 203, 206} \begin {gather*} -\frac {x}{\sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2-b^2}}\right )}{2 \sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {a^2 x^3-b^2 x}}-\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {x}}{\sqrt {a^2 x^2-b^2}}\right )}{2 \sqrt {2} \sqrt [4]{-a^2} \sqrt {b} \sqrt {a^2 x^3-b^2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 221

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 224

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 409

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

Rule 414

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(b*x*(a + b*x^n)^(p + 1)*(
c + d*x^n)^(q + 1))/(a*n*(p + 1)*(b*c - a*d)), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1)*
(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomial
Q[a, b, c, d, n, p, q, x]

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 1211

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 1404

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c*x^n)/e)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rule 1699

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 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]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

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

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Mathematica [C]  time = 0.56, size = 73, normalized size = 0.41 \begin {gather*} \frac {x \left (-\sqrt {1-\frac {a^2 x^2}{b^2}} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )-1\right )}{\sqrt {a^2 x^3-b^2 x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(-1 - Sqrt[1 - (a^2*x^2)/b^2]*AppellF1[1/4, -1/2, 1, 5/4, (a^2*x^2)/b^2, -((a^2*x^2)/b^2)]))/Sqrt[-(b^2*x)
+ a^2*x^3]

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IntegrateAlgebraic [A]  time = 0.50, size = 177, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-b^2 x+a^2 x^3}}{b^2-a^2 x^2}-\frac {\tan ^{-1}\left (\frac {2 \sqrt {a} \sqrt {b} \sqrt {-b^2 x+a^2 x^3}}{-b^2-2 a b x+a^2 x^2}\right )}{4 \sqrt {a} \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {-\frac {b^{3/2}}{2 \sqrt {a}}+\sqrt {a} \sqrt {b} x+\frac {a^{3/2} x^2}{2 \sqrt {b}}}{\sqrt {-b^2 x+a^2 x^3}}\right )}{4 \sqrt {a} \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.81, size = 1141, normalized size = 6.45

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*sqrt(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4)*arctan(1/2*((4*sqrt(2)*(1/4)^(3/4)*a^2*b^2*x*
(1/(a^2*b^2))^(3/4) - sqrt(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x) - (2*a^2*
x^3 - 2*b^2*x - (4*sqrt(2)*(1/4)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4) + sqrt(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/
(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x))*sqrt((a^4*x^4 + 2*a^2*b^2*x^2 + b^4 + 8*(sqrt(2)*(1/4)^(1/4)*a^2*b^2*
x*(1/(a^2*b^2))^(1/4) + sqrt(2)*(1/4)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x)
+ 8*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)))/(a^2*x^3 - b^2*x)) + 4*sqr
t(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4)*arctan(1/2*((4*sqrt(2)*(1/4)^(3/4)*a^2*b^2*x*(1/(a^2*b^2)
)^(3/4) - sqrt(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4))*sqrt(a^2*x^3 - b^2*x) + (2*a^2*x^3 - 2*b^2*
x + (4*sqrt(2)*(1/4)^(3/4)*a^2*b^2*x*(1/(a^2*b^2))^(3/4) + sqrt(2)*(1/4)^(1/4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(
1/4))*sqrt(a^2*x^3 - b^2*x))*sqrt((a^4*x^4 + 2*a^2*b^2*x^2 + b^4 - 8*(sqrt(2)*(1/4)^(1/4)*a^2*b^2*x*(1/(a^2*b^
2))^(1/4) + sqrt(2)*(1/4)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 8*(a^4*b^
2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)))/(a^2*x^3 - b^2*x)) + sqrt(2)*(1/4)^(1/
4)*(a^2*x^2 - b^2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 + 2*a^2*b^2*x^2 + b^4 + 8*(sqrt(2)*(1/4)^(1/4)*a^2*b^2*x*(
1/(a^2*b^2))^(1/4) + sqrt(2)*(1/4)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) +
8*(a^4*b^2*x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) - sqrt(2)*(1/4)^(1/4)*(a^2*x^2
- b^2)*(1/(a^2*b^2))^(1/4)*log((a^4*x^4 + 2*a^2*b^2*x^2 + b^4 - 8*(sqrt(2)*(1/4)^(1/4)*a^2*b^2*x*(1/(a^2*b^2)
)^(1/4) + sqrt(2)*(1/4)^(3/4)*(a^4*b^2*x^2 - a^2*b^4)*(1/(a^2*b^2))^(3/4))*sqrt(a^2*x^3 - b^2*x) + 8*(a^4*b^2*
x^3 - a^2*b^4*x)*sqrt(1/(a^2*b^2)))/(a^4*x^4 + 2*a^2*b^2*x^2 + b^4)) + 16*sqrt(a^2*x^3 - b^2*x))/(a^2*x^2 - b^
2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{4} x^{4} + b^{4}}{{\left (a^{4} x^{4} - b^{4}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.22, size = 322, normalized size = 1.82

method	result	size
elliptic	\(-\frac {x}{\sqrt {\left (x^{2}-\frac {b^{2}}{a^{2}}\right ) a^{2} x}}+\frac {b \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{2 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {i b^{2} \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}-\frac {i b^{2} \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {b}{a}+\frac {i b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {b}{a}+\frac {i b}{a}\right )}\)	\(322\)
default	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}-\frac {b \left (-\frac {a^{2} x^{2}-a b x}{b^{2} a \sqrt {\left (x +\frac {b}{a}\right ) \left (a^{2} x^{2}-a b x \right )}}+\frac {\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{2 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (-\frac {2 b \EllipticE \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {b \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{2 b \sqrt {a^{2} x^{3}-b^{2} x}}\right )}{2}+\frac {b \left (-\frac {a^{2} x^{2}+a b x}{b^{2} a \sqrt {\left (x -\frac {b}{a}\right ) \left (a^{2} x^{2}+a b x \right )}}-\frac {\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{2 a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \left (-\frac {2 b \EllipticE \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {b \EllipticF \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{2 b \sqrt {a^{2} x^{3}-b^{2} x}}\right )}{2}-b^{2} \left (-\frac {i \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {i b}{a}-\frac {b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {i b}{a}-\frac {b}{a}\right )}+\frac {i \sqrt {1+\frac {a x}{b}}\, \sqrt {2-\frac {2 a x}{b}}\, \sqrt {-\frac {a x}{b}}\, \EllipticPi \left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, -\frac {b}{a \left (-\frac {b}{a}+\frac {i b}{a}\right )}, \frac {\sqrt {2}}{2}\right )}{2 a^{2} \sqrt {a^{2} x^{3}-b^{2} x}\, \left (-\frac {b}{a}+\frac {i b}{a}\right )}\right )\)	\(787\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{4} x^{4} + b^{4}}{{\left (a^{4} x^{4} - b^{4}\right )} \sqrt {a^{2} x^{3} - b^{2} x}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

\text{Hanged}

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a^{4} x^{4} + b^{4}}{\sqrt {x \left (a x - b\right ) \left (a x + b\right )} \left (a x - b\right ) \left (a x + b\right ) \left (a^{2} x^{2} + b^{2}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a**4*x**4 + b**4)/(sqrt(x*(a*x - b)*(a*x + b))*(a*x - b)*(a*x + b)*(a**2*x**2 + b**2)), x)

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