3.826 \(\int \frac{x^{10}}{\sqrt{a+b x^4}} \, dx\)

Optimal. Leaf size=261 \[ \frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{30 b^{11/4} \sqrt{a+b x^4}}+\frac{7 a^2 x \sqrt{a+b x^4}}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^4}}-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b} \]

[Out]

(-7*a*x^3*Sqrt[a + b*x^4])/(45*b^2) + (x^7*Sqrt[a + b*x^4])/(9*b) + (7*a^2*x*Sqrt[a + b*x^4])/(15*b^(5/2)*(Sqr
t[a] + Sqrt[b]*x^2)) - (7*a^(9/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Elliptic
E[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(15*b^(11/4)*Sqrt[a + b*x^4]) + (7*a^(9/4)*(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])/(30*b^(11/4)*Sqrt[a + b
*x^4])

________________________________________________________________________________________

Rubi [A]  time = 0.0918937, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {321, 305, 220, 1196} \[ \frac{7 a^2 x \sqrt{a+b x^4}}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{11/4} \sqrt{a+b x^4}}-\frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^4}}-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[x^10/Sqrt[a + b*x^4],x]

[Out]

(-7*a*x^3*Sqrt[a + b*x^4])/(45*b^2) + (x^7*Sqrt[a + b*x^4])/(9*b) + (7*a^2*x*Sqrt[a + b*x^4])/(15*b^(5/2)*(Sqr
t[a] + Sqrt[b]*x^2)) - (7*a^(9/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*Elliptic
E[2*ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(15*b^(11/4)*Sqrt[a + b*x^4]) + (7*a^(9/4)*(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])/(30*b^(11/4)*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 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

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]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int \frac{x^{10}}{\sqrt{a+b x^4}} \, dx &=\frac{x^7 \sqrt{a+b x^4}}{9 b}-\frac{(7 a) \int \frac{x^6}{\sqrt{a+b x^4}} \, dx}{9 b}\\ &=-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b}+\frac{\left (7 a^2\right ) \int \frac{x^2}{\sqrt{a+b x^4}} \, dx}{15 b^2}\\ &=-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b}+\frac{\left (7 a^{5/2}\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{15 b^{5/2}}-\frac{\left (7 a^{5/2}\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{15 b^{5/2}}\\ &=-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b}+\frac{7 a^2 x \sqrt{a+b x^4}}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^4}}+\frac{7 a^{9/4} \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 )}{30 b^{11/4} \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.0252055, size = 80, normalized size = 0.31 \[ \frac{x^3 \left (7 a^2 \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-7 a^2-2 a b x^4+5 b^2 x^8\right )}{45 b^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10/Sqrt[a + b*x^4],x]

[Out]

(x^3*(-7*a^2 - 2*a*b*x^4 + 5*b^2*x^8 + 7*a^2*Sqrt[1 + (b*x^4)/a]*Hypergeometric2F1[1/2, 3/4, 7/4, -((b*x^4)/a)
]))/(45*b^2*Sqrt[a + b*x^4])

________________________________________________________________________________________

Maple [C]  time = 0.007, size = 133, normalized size = 0.5 \begin{align*}{\frac{{x}^{7}}{9\,b}\sqrt{b{x}^{4}+a}}-{\frac{7\,a{x}^{3}}{45\,{b}^{2}}\sqrt{b{x}^{4}+a}}+{{\frac{7\,i}{15}}{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){b}^{-{\frac{5}{2}}}{\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^10/(b*x^4+a)^(1/2),x)

[Out]

1/9*x^7*(b*x^4+a)^(1/2)/b-7/45*a*x^3*(b*x^4+a)^(1/2)/b^2+7/15*I*a^(5/2)/b^(5/2)/(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)-EllipticE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{10}}{\sqrt{b x^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^10/sqrt(b*x^4 + a), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{10}}{\sqrt{b x^{4} + a}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

integral(x^10/sqrt(b*x^4 + a), x)

________________________________________________________________________________________

Sympy [C]  time = 1.66232, size = 37, normalized size = 0.14 \begin{align*} \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{15}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**10/(b*x**4+a)**(1/2),x)

[Out]

x**11*gamma(11/4)*hyper((1/2, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a)/(4*sqrt(a)*gamma(15/4))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{10}}{\sqrt{b x^{4} + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

integrate(x^10/sqrt(b*x^4 + a), x)