3.779 \(\int \frac{\sqrt{a+c x^4}}{x^8} \, dx\)

Optimal. Leaf size=129 \[ -\frac{c^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{21 a^{5/4} \sqrt{a+c x^4}}-\frac{2 c \sqrt{a+c x^4}}{21 a x^3}-\frac{\sqrt{a+c x^4}}{7 x^7} \]

[Out]

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

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Rubi [A]  time = 0.0346284, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {277, 325, 220} \[ -\frac{c^{7/4} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{21 a^{5/4} \sqrt{a+c x^4}}-\frac{2 c \sqrt{a+c x^4}}{21 a x^3}-\frac{\sqrt{a+c x^4}}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^8,x]

[Out]

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

Rule 277

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

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

Mathematica [C]  time = 0.0104414, size = 51, normalized size = 0.4 \[ -\frac{\sqrt{a+c x^4} \, _2F_1\left (-\frac{7}{4},-\frac{1}{2};-\frac{3}{4};-\frac{c x^4}{a}\right )}{7 x^7 \sqrt{\frac{c x^4}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^8,x]

[Out]

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

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Maple [C]  time = 0.014, size = 110, normalized size = 0.9 \begin{align*} -{\frac{1}{7\,{x}^{7}}\sqrt{c{x}^{4}+a}}-{\frac{2\,c}{21\,a{x}^{3}}\sqrt{c{x}^{4}+a}}-{\frac{2\,{c}^{2}}{21\,a}\sqrt{1-{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{c}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{c}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{c{x}^{4}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*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{c x^{4} + a}}{x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^4 + a)/x^8, x)

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Sympy [C]  time = 1.52122, size = 46, normalized size = 0.36 \begin{align*} \frac{\sqrt{a} \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{c x^{4} e^{i \pi }}{a}} \right )}}{4 x^{7} \Gamma \left (- \frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(c*x^4 + a)/x^8, x)