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

Optimal. Leaf size=130 \[ \frac{5 a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{42 b^{9/4} \sqrt{a+b x^4}}-\frac{5 a x \sqrt{a+b x^4}}{21 b^2}+\frac{x^5 \sqrt{a+b x^4}}{7 b} \]

[Out]

(-5*a*x*Sqrt[a + b*x^4])/(21*b^2) + (x^5*Sqrt[a + b*x^4])/(7*b) + (5*a^(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*b^(9/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.0411811, antiderivative size = 130, 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 = {321, 220} \[ \frac{5 a^{7/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{42 b^{9/4} \sqrt{a+b x^4}}-\frac{5 a x \sqrt{a+b x^4}}{21 b^2}+\frac{x^5 \sqrt{a+b x^4}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[x^8/Sqrt[a + b*x^4],x]

[Out]

(-5*a*x*Sqrt[a + b*x^4])/(21*b^2) + (x^5*Sqrt[a + b*x^4])/(7*b) + (5*a^(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*b^(9/4)*Sqrt[a + b*x^4])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && 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{x^8}{\sqrt{a+b x^4}} \, dx &=\frac{x^5 \sqrt{a+b x^4}}{7 b}-\frac{(5 a) \int \frac{x^4}{\sqrt{a+b x^4}} \, dx}{7 b}\\ &=-\frac{5 a x \sqrt{a+b x^4}}{21 b^2}+\frac{x^5 \sqrt{a+b x^4}}{7 b}+\frac{\left (5 a^2\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{21 b^2}\\ &=-\frac{5 a x \sqrt{a+b x^4}}{21 b^2}+\frac{x^5 \sqrt{a+b x^4}}{7 b}+\frac{5 a^{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 b^{9/4} \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0199886, size = 79, normalized size = 0.61 \[ \frac{5 a^2 x \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{b x^4}{a}\right )-5 a^2 x-2 a b x^5+3 b^2 x^9}{21 b^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/Sqrt[a + b*x^4],x]

[Out]

(-5*a^2*x - 2*a*b*x^5 + 3*b^2*x^9 + 5*a^2*x*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, -((b*x^4)/a)]
)/(21*b^2*Sqrt[a + b*x^4])

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Maple [C]  time = 0.035, size = 111, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{7\,b}\sqrt{b{x}^{4}+a}}-{\frac{5\,ax}{21\,{b}^{2}}\sqrt{b{x}^{4}+a}}+{\frac{5\,{a}^{2}}{21\,{b}^{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(x^8/(b*x^4+a)^(1/2),x)

[Out]

1/7*x^5*(b*x^4+a)^(1/2)/b-5/21*a*x*(b*x^4+a)^(1/2)/b^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{x^{8}}{\sqrt{b x^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^8/sqrt(b*x^4 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^8/sqrt(b*x^4 + a), x)

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Sympy [C]  time = 1.28463, size = 37, normalized size = 0.28 \begin{align*} \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{13}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(13/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^8/sqrt(b*x^4 + a), x)