3.868 \(\int \frac{x^{10}}{(a+b x^4)^{3/2}} \, dx\)

Optimal. Leaf size=258 \[ -\frac{21 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),\frac{1}{2}\right )}{20 b^{11/4} \sqrt{a+b x^4}}+\frac{21 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a+b x^4}}+\frac{7 x^3 \sqrt{a+b x^4}}{10 b^2}-\frac{21 a x \sqrt{a+b x^4}}{10 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{x^7}{2 b \sqrt{a+b x^4}} \]

[Out]

-x^7/(2*b*Sqrt[a + b*x^4]) + (7*x^3*Sqrt[a + b*x^4])/(10*b^2) - (21*a*x*Sqrt[a + b*x^4])/(10*b^(5/2)*(Sqrt[a]
+ Sqrt[b]*x^2)) + (21*a^(5/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*
ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(10*b^(11/4)*Sqrt[a + b*x^4]) - (21*a^(5/4)*(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])/(20*b^(11/4)*Sqrt[a + b*x^
4])

________________________________________________________________________________________

Rubi [A]  time = 0.0876296, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {288, 321, 305, 220, 1196} \[ -\frac{21 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 b^{11/4} \sqrt{a+b x^4}}+\frac{21 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a+b x^4}}+\frac{7 x^3 \sqrt{a+b x^4}}{10 b^2}-\frac{21 a x \sqrt{a+b x^4}}{10 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{x^7}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[x^10/(a + b*x^4)^(3/2),x]

[Out]

-x^7/(2*b*Sqrt[a + b*x^4]) + (7*x^3*Sqrt[a + b*x^4])/(10*b^2) - (21*a*x*Sqrt[a + b*x^4])/(10*b^(5/2)*(Sqrt[a]
+ Sqrt[b]*x^2)) + (21*a^(5/4)*(Sqrt[a] + Sqrt[b]*x^2)*Sqrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*
ArcTan[(b^(1/4)*x)/a^(1/4)], 1/2])/(10*b^(11/4)*Sqrt[a + b*x^4]) - (21*a^(5/4)*(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])/(20*b^(11/4)*Sqrt[a + b*x^
4])

Rule 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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}}{\left (a+b x^4\right )^{3/2}} \, dx &=-\frac{x^7}{2 b \sqrt{a+b x^4}}+\frac{7 \int \frac{x^6}{\sqrt{a+b x^4}} \, dx}{2 b}\\ &=-\frac{x^7}{2 b \sqrt{a+b x^4}}+\frac{7 x^3 \sqrt{a+b x^4}}{10 b^2}-\frac{(21 a) \int \frac{x^2}{\sqrt{a+b x^4}} \, dx}{10 b^2}\\ &=-\frac{x^7}{2 b \sqrt{a+b x^4}}+\frac{7 x^3 \sqrt{a+b x^4}}{10 b^2}-\frac{\left (21 a^{3/2}\right ) \int \frac{1}{\sqrt{a+b x^4}} \, dx}{10 b^{5/2}}+\frac{\left (21 a^{3/2}\right ) \int \frac{1-\frac{\sqrt{b} x^2}{\sqrt{a}}}{\sqrt{a+b x^4}} \, dx}{10 b^{5/2}}\\ &=-\frac{x^7}{2 b \sqrt{a+b x^4}}+\frac{7 x^3 \sqrt{a+b x^4}}{10 b^2}-\frac{21 a x \sqrt{a+b x^4}}{10 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{21 a^{5/4} \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 )}{10 b^{11/4} \sqrt{a+b x^4}}-\frac{21 a^{5/4} \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 )}{20 b^{11/4} \sqrt{a+b x^4}}\\ \end{align*}

Mathematica [C]  time = 0.020603, size = 66, normalized size = 0.26 \[ \frac{x^3 \left (7 a \sqrt{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{3}{4},\frac{3}{2};\frac{7}{4};-\frac{b x^4}{a}\right )-7 a+b x^4\right )}{5 b^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^10/(a + b*x^4)^(3/2),x]

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.011, size = 137, normalized size = 0.5 \begin{align*}{\frac{a{x}^{3}}{2\,{b}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{{x}^{3}}{5\,{b}^{2}}\sqrt{b{x}^{4}+a}}-{{\frac{21\,i}{10}}{a}^{{\frac{3}{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)^(3/2),x)

[Out]

1/2/b^2*x^3*a/((x^4+1/b*a)*b)^(1/2)+1/5*x^3*(b*x^4+a)^(1/2)/b^2-21/10*I/b^(5/2)*a^(3/2)/(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}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^4+a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^10/(b*x^4 + a)^(3/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b x^{4} + a} x^{10}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^4+a)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^4 + a)*x^10/(b^2*x^8 + 2*a*b*x^4 + a^2), x)

________________________________________________________________________________________

Sympy [C]  time = 1.79673, size = 37, normalized size = 0.14 \begin{align*} \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \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)**(3/2),x)

[Out]

x**11*gamma(11/4)*hyper((3/2, 11/4), (15/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*gamma(15/4))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{10}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^10/(b*x^4+a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^10/(b*x^4 + a)^(3/2), x)