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

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

[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])

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Rubi [A]  time = 0.247386, 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 \[ -\frac{21 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 b^{11/4} \sqrt{a+b x^4}}+\frac{21 a^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 b^{11/4} \sqrt{a+b x^4}}-\frac{21 a x \sqrt{a+b x^4}}{10 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{7 x^3 \sqrt{a+b x^4}}{10 b^2}-\frac{x^7}{2 b \sqrt{a+b x^4}} \]

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])

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Rubi in Sympy [A]  time = 29.2571, size = 235, normalized size = 0.91 \[ \frac{21 a^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{10 b^{\frac{11}{4}} \sqrt{a + b x^{4}}} - \frac{21 a^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{20 b^{\frac{11}{4}} \sqrt{a + b x^{4}}} - \frac{21 a x \sqrt{a + b x^{4}}}{10 b^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} - \frac{x^{7}}{2 b \sqrt{a + b x^{4}}} + \frac{7 x^{3} \sqrt{a + b x^{4}}}{10 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(b*x**4+a)**(3/2),x)

[Out]

21*a**(5/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)*x*
*2)*elliptic_e(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(10*b**(11/4)*sqrt(a + b*x**4))
- 21*a**(5/4)*sqrt((a + b*x**4)/(sqrt(a) + sqrt(b)*x**2)**2)*(sqrt(a) + sqrt(b)
*x**2)*elliptic_f(2*atan(b**(1/4)*x/a**(1/4)), 1/2)/(20*b**(11/4)*sqrt(a + b*x**
4)) - 21*a*x*sqrt(a + b*x**4)/(10*b**(5/2)*(sqrt(a) + sqrt(b)*x**2)) - x**7/(2*b
*sqrt(a + b*x**4)) + 7*x**3*sqrt(a + b*x**4)/(10*b**2)

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Mathematica [C]  time = 0.229341, size = 172, normalized size = 0.67 \[ \frac{21 a^{3/2} \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-21 a^{3/2} \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{b} x^3 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (7 a+2 b x^4\right )}{10 b^{5/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/(a + b*x^4)^(3/2),x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*Sqrt[b]*x^3*(7*a + 2*b*x^4) - 21*a^(3/2)*Sqrt[1 + (b*
x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1] + 21*a^(3/2)*Sqrt[
1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/(10*Sqrt[(
I*Sqrt[b])/Sqrt[a]]*b^(5/2)*Sqrt[a + b*x^4])

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Maple [C]  time = 0.018, size = 137, normalized size = 0.5 \[{\frac{a{x}^{3}}{2\,{b}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{{x}^{3}}{5\,{b}^{2}}\sqrt{b{x}^{4}+a}}-{{\frac{21\,i}{10}}{a}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

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*a*x^3/((x^4+a/b)*b)^(1/2)+1/5*x^3*(b*x^4+a)^(1/2)/b^2-21/10*I*a^(3/2)/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)-Ellipt
icE(x*(I/a^(1/2)*b^(1/2))^(1/2),I))

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

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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{10}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}, x\right ) \]

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(x^10/(b*x^4 + a)^(3/2), x)

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Sympy [A]  time = 5.10265, size = 37, normalized size = 0.14 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{15}{4}\right )} \]

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))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]

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)