3.825 \(\int \frac{1}{x^8 \sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=132 \[ \frac{5 b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{42 a^{9/4} \sqrt{a+b x^4}}+\frac{5 b \sqrt{a+b x^4}}{21 a^2 x^3}-\frac{\sqrt{a+b x^4}}{7 a x^7} \]

[Out]

-Sqrt[a + b*x^4]/(7*a*x^7) + (5*b*Sqrt[a + b*x^4])/(21*a^2*x^3) + (5*b^(7/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a +
b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(42*a^(9/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.0360192, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {325, 220} \[ \frac{5 b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 a^{9/4} \sqrt{a+b x^4}}+\frac{5 b \sqrt{a+b x^4}}{21 a^2 x^3}-\frac{\sqrt{a+b x^4}}{7 a x^7} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^8*Sqrt[a + b*x^4]),x]

[Out]

-Sqrt[a + b*x^4]/(7*a*x^7) + (5*b*Sqrt[a + b*x^4])/(21*a^2*x^3) + (5*b^(7/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a +
b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(42*a^(9/4)*Sqrt[a + b*x^4])

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 220

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)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{x^8 \sqrt{a+b x^4}} \, dx &=-\frac{\sqrt{a+b x^4}}{7 a x^7}-\frac{(5 b) \int \frac{1}{x^4 \sqrt{a+b x^4}} \, dx}{7 a}\\ &=-\frac{\sqrt{a+b x^4}}{7 a x^7}+\frac{5 b \sqrt{a+b x^4}}{21 a^2 x^3}+\frac{\left (5 b^2\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{21 a^2}\\ &=-\frac{\sqrt{a+b x^4}}{7 a x^7}+\frac{5 b \sqrt{a+b x^4}}{21 a^2 x^3}+\frac{5 b^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 a^{9/4} \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0093265, size = 51, normalized size = 0.39 \[ -\frac{\sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (-\frac{7}{4},\frac{1}{2};-\frac{3}{4};-\frac{b x^4}{a}\right )}{7 x^7 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*Sqrt[a + b*x^4]),x]

[Out]

-(Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[-7/4, 1/2, -3/4, -((b*x^4)/a)])/(7*x^7*Sqrt[a + b*x^4])

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Maple [C]  time = 0.013, size = 113, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,a{x}^{7}}\sqrt{b{x}^{4}+a}}+{\frac{5\,b}{21\,{x}^{3}{a}^{2}}\sqrt{b{x}^{4}+a}}+{\frac{5\,{b}^{2}}{21\,{a}^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^8/(b*x^4+a)^(1/2),x)

[Out]

-1/7*(b*x^4+a)^(1/2)/a/x^7+5/21*b*(b*x^4+a)^(1/2)/x^3/a^2+5/21/a^2*b^2/(I/a^(1/2)*b^(1/2))^(1/2)*(1-I/a^(1/2)*
b^(1/2)*x^2)^(1/2)*(1+I/a^(1/2)*b^(1/2)*x^2)^(1/2)/(b*x^4+a)^(1/2)*EllipticF(x*(I/a^(1/2)*b^(1/2))^(1/2),I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{4} + a} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^8), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{4} + a}}{b x^{12} + a x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^4 + a)/(b*x^12 + a*x^8), x)

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Sympy [C]  time = 1.65801, size = 44, normalized size = 0.33 \begin{align*} \frac{\Gamma \left (- \frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ - \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} x^{7} \Gamma \left (- \frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**8/(b*x**4+a)**(1/2),x)

[Out]

gamma(-7/4)*hyper((-7/4, 1/2), (-3/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*x**7*gamma(-3/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b x^{4} + a} x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^8/(b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^4 + a)*x^8), x)