\(\int \frac {x^2}{(-b+a x^4) \sqrt {b+a x^4}} \, dx\) [2533]

Optimal result

Integrand size = 26, antiderivative size = 212 \[ \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {6-4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {6+4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2+\sqrt {b+a x^4}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4}} \]

[Out]

1/4*arctan(2^(1/2)*a^(1/4)*b^(1/4)*x/(b^(1/2)+a^(1/2)*x^2+(a*x^4+b)^(1/2)))*2^(1/2)/a^(3/4)/b^(3/4)+1/8*arctan
h((2-2^(1/2))*a^(1/4)*b^(1/4)*x/(b^(1/2)+a^(1/2)*x^2+(a*x^4+b)^(1/2)))*2^(1/2)/a^(3/4)/b^(3/4)-1/8*arctanh((2+
2^(1/2))*a^(1/4)*b^(1/4)*x/(b^(1/2)+a^(1/2)*x^2+(a*x^4+b)^(1/2)))*2^(1/2)/a^(3/4)/b^(3/4)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.46, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {504, 1225, 226, 1713, 211, 214} \[ \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4}} \]

[In]

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

[Out]

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

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 226

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))*EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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]

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.38 \[ \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx=\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )-\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )}{4 \sqrt {2} a^{3/4} b^{3/4}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.43

method	result	size
pseudoelliptic	\(-\frac {\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {-\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {a \,x^{4}+b}}{\sqrt {2}\, \left (a b \right )^{\frac {1}{4}} x -\sqrt {a \,x^{4}+b}}\right )+2 \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right )}{16 a b}\)	\(92\)
default	\(-\frac {\left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}}{16 a b}\)	\(95\)
elliptic	\(-\frac {\left (a b \right )^{\frac {1}{4}} \left (\ln \left (\frac {\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )\right ) \sqrt {2}}{16 a b}\)	\(95\)

[In]

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

[Out]

-1/16*2^(1/2)*(a*b)^(1/4)*(ln((-2^(1/2)*(a*b)^(1/4)*x-(a*x^4+b)^(1/2))/(2^(1/2)*(a*b)^(1/4)*x-(a*x^4+b)^(1/2))
)+2*arctan(1/2*(a*x^4+b)^(1/2)*2^(1/2)/x/(a*b)^(1/4)))/a/b

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.42 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.85 \[ \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx=-\frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{2} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x^{3} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (a b^{2} \sqrt {\frac {1}{a^{3} b^{3}}} + x^{2}\right )}}{a x^{4} - b}\right ) + \frac {1}{8} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{2} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x^{3} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} - \sqrt {a x^{4} + b} {\left (a b^{2} \sqrt {\frac {1}{a^{3} b^{3}}} + x^{2}\right )}}{a x^{4} - b}\right ) + \frac {1}{8} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{2} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} - 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x^{3} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} - \sqrt {a x^{4} + b} {\left (a b^{2} \sqrt {\frac {1}{a^{3} b^{3}}} - x^{2}\right )}}{a x^{4} - b}\right ) - \frac {1}{8} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} \log \left (\frac {-4 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a^{2} b^{3} x \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {3}{4}} + 2 i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x^{3} \left (\frac {1}{a^{3} b^{3}}\right )^{\frac {1}{4}} - \sqrt {a x^{4} + b} {\left (a b^{2} \sqrt {\frac {1}{a^{3} b^{3}}} - x^{2}\right )}}{a x^{4} - b}\right ) \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx=\int \frac {x^{2}}{\left (a x^{4} - b\right ) \sqrt {a x^{4} + b}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {a x^{4} + b} {\left (a x^{4} - b\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx=\int { \frac {x^{2}}{\sqrt {a x^{4} + b} {\left (a x^{4} - b\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (-b+a x^4\right ) \sqrt {b+a x^4}} \, dx=-\int \frac {x^2}{\sqrt {a\,x^4+b}\,\left (b-a\,x^4\right )} \,d x \]

[In]

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

[Out]

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