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

Optimal. Leaf size=210 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}+\sqrt {a} x^2+\sqrt {b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {6-4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}+\sqrt {a} x^2+\sqrt {b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {6+4 \sqrt {2}} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}+\sqrt {a} x^2+\sqrt {b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \]

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Rubi [A]  time = 0.04, antiderivative size = 97, normalized size of antiderivative = 0.46, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {404, 212, 206, 203} \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a x^4+b}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[b + a*x^4]/(-b + a*x^4),x]

[Out]

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

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 212

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

Rule 404

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

Rubi steps

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

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Mathematica [C]  time = 0.21, size = 152, normalized size = 0.72 \begin {gather*} -\frac {5 b x \sqrt {a x^4+b} F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )}{\left (b-a x^4\right ) \left (2 a x^4 \left (2 F_1\left (\frac {5}{4};-\frac {1}{2},2;\frac {9}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )+F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )\right )+5 b F_1\left (\frac {1}{4};-\frac {1}{2},1;\frac {5}{4};-\frac {a x^4}{b},\frac {a x^4}{b}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-5*b*x*Sqrt[b + a*x^4]*AppellF1[1/4, -1/2, 1, 5/4, -((a*x^4)/b), (a*x^4)/b])/((b - a*x^4)*(5*b*AppellF1[1/4,
-1/2, 1, 5/4, -((a*x^4)/b), (a*x^4)/b] + 2*a*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, -((a*x^4)/b), (a*x^4)/b] + App
ellF1[5/4, 1/2, 1, 9/4, -((a*x^4)/b), (a*x^4)/b])))

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IntegrateAlgebraic [A]  time = 0.34, size = 97, normalized size = 0.46 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b+a x^4}}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.72, size = 259, normalized size = 1.23 \begin {gather*} \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \arctan \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} \sqrt {a x^{4} + b} \left (\frac {1}{a b}\right )^{\frac {1}{4}} - \frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a x^{2} \left (\frac {1}{a b}\right )^{\frac {1}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b \left (\frac {1}{a b}\right )^{\frac {3}{4}}}{\sqrt {a}}}{x}\right ) - \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} + \sqrt {a x^{4} + b} {\left (x^{2} + b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (\frac {1}{a b}\right )^{\frac {1}{4}} \log \left (-\frac {4 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} a b x^{3} \left (\frac {1}{a b}\right )^{\frac {3}{4}} + 2 \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b x \left (\frac {1}{a b}\right )^{\frac {1}{4}} - \sqrt {a x^{4} + b} {\left (x^{2} + b \sqrt {\frac {1}{a b}}\right )}}{a x^{4} - b}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.18, size = 96, normalized size = 0.46

method	result	size
default	\(\frac {\left (\frac {\arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )}{2 \left (a b \right )^{\frac {1}{4}}}-\frac {\ln \left (\frac {\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )}{4 \left (a b \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\)	\(96\)
elliptic	\(\frac {\left (\frac {\arctan \left (\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x \left (a b \right )^{\frac {1}{4}}}\right )}{2 \left (a b \right )^{\frac {1}{4}}}-\frac {\ln \left (\frac {\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}+\left (a b \right )^{\frac {1}{4}}}{\frac {\sqrt {a \,x^{4}+b}\, \sqrt {2}}{2 x}-\left (a b \right )^{\frac {1}{4}}}\right )}{4 \left (a b \right )^{\frac {1}{4}}}\right ) \sqrt {2}}{2}\)	\(96\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(b + a*x^4)^(1/2)/(b - a*x^4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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