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

Optimal. Leaf size=103 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c} \]

[Out]

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

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Rubi [A]  time = 0.0557869, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {404, 212, 206, 203} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c}+\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt{a+b x^4}}\right )}{2 \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 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 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])

Rubi steps

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

Mathematica [C]  time = 0.146802, size = 155, normalized size = 1.5 \[ \frac{5 a x \sqrt{a+b x^4} F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )}{c \left (a-b x^4\right ) \left (2 b x^4 \left (2 F_1\left (\frac{5}{4};-\frac{1}{2},2;\frac{9}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )+F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )\right )+5 a F_1\left (\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{b x^4}{a},\frac{b x^4}{a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(5*a*x*Sqrt[a + b*x^4]*AppellF1[1/4, -1/2, 1, 5/4, -((b*x^4)/a), (b*x^4)/a])/(c*(a - b*x^4)*(5*a*AppellF1[1/4,
-1/2, 1, 5/4, -((b*x^4)/a), (b*x^4)/a] + 2*b*x^4*(2*AppellF1[5/4, -1/2, 2, 9/4, -((b*x^4)/a), (b*x^4)/a] + Ap
pellF1[5/4, 1/2, 1, 9/4, -((b*x^4)/a), (b*x^4)/a])))

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Maple [A]  time = 0.106, size = 103, normalized size = 1. \begin{align*} -{\frac{\sqrt{2}}{4\,c}\arctan \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}{\frac{1}{\sqrt [4]{ab}}}} \right ){\frac{1}{\sqrt [4]{ab}}}}+{\frac{\sqrt{2}}{8\,c}\ln \left ({ \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}}+\sqrt [4]{ab} \right ) \left ({\frac{\sqrt{2}}{2\,x}\sqrt{b{x}^{4}+a}}-\sqrt [4]{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/c*2^(1/2)/(a*b)^(1/4)*arctan(1/2*(b*x^4+a)^(1/2)*2^(1/2)/x/(a*b)^(1/4))+1/8/c*2^(1/2)/(a*b)^(1/4)*ln((1/2
*(b*x^4+a)^(1/2)*2^(1/2)/x+(a*b)^(1/4))/(1/2*(b*x^4+a)^(1/2)*2^(1/2)/x-(a*b)^(1/4)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\sqrt{b x^{4} + a}}{b c x^{4} - a c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 5.7186, size = 787, normalized size = 7.64 \begin{align*} -\left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (\frac{1}{4}\right )^{\frac{1}{4}} \sqrt{b x^{4} + a} c \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} - \frac{2 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + \left (\frac{1}{4}\right )^{\frac{1}{4}} b c x^{2} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}}}{\sqrt{b}}}{x}\right ) + \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a}{\left (a c^{2} \sqrt{\frac{1}{a b c^{4}}} + x^{2}\right )}}{b x^{4} - a}\right ) - \frac{1}{4} \, \left (\frac{1}{4}\right )^{\frac{1}{4}} \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} \log \left (-\frac{4 \, \left (\frac{1}{4}\right )^{\frac{3}{4}} a b c^{3} x^{3} \left (\frac{1}{a b c^{4}}\right )^{\frac{3}{4}} + 2 \, \left (\frac{1}{4}\right )^{\frac{1}{4}} a c x \left (\frac{1}{a b c^{4}}\right )^{\frac{1}{4}} - \sqrt{b x^{4} + a}{\left (a c^{2} \sqrt{\frac{1}{a b c^{4}}} + x^{2}\right )}}{b x^{4} - a}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\sqrt{a + b x^{4}}}{- a + b x^{4}}\, dx}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\sqrt{b x^{4} + a}}{b c x^{4} - a c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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