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

Optimal. Leaf size=158 \[ -\frac{7 a^{11/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 b^{11/4} \sqrt{a-b x^4}}+\frac{7 a^{11/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 b^{11/4} \sqrt{a-b x^4}}-\frac{7 a x^3 \sqrt{a-b x^4}}{45 b^2}-\frac{x^7 \sqrt{a-b x^4}}{9 b} \]

[Out]

(-7*a*x^3*Sqrt[a - b*x^4])/(45*b^2) - (x^7*Sqrt[a - b*x^4])/(9*b) + (7*a^(11/4)*
Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(15*b^(11/4)*Sqr
t[a - b*x^4]) - (7*a^(11/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(
1/4)], -1])/(15*b^(11/4)*Sqrt[a - b*x^4])

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Rubi [A]  time = 0.282767, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{7 a^{11/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 b^{11/4} \sqrt{a-b x^4}}+\frac{7 a^{11/4} \sqrt{1-\frac{b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{15 b^{11/4} \sqrt{a-b x^4}}-\frac{7 a x^3 \sqrt{a-b x^4}}{45 b^2}-\frac{x^7 \sqrt{a-b x^4}}{9 b} \]

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^(11/4)*
Sqrt[1 - (b*x^4)/a]*EllipticE[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(15*b^(11/4)*Sqr
t[a - b*x^4]) - (7*a^(11/4)*Sqrt[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(
1/4)], -1])/(15*b^(11/4)*Sqrt[a - b*x^4])

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Rubi in Sympy [A]  time = 45.3341, size = 143, normalized size = 0.91 \[ \frac{7 a^{\frac{11}{4}} \sqrt{1 - \frac{b x^{4}}{a}} E\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{15 b^{\frac{11}{4}} \sqrt{a - b x^{4}}} - \frac{7 a^{\frac{11}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{15 b^{\frac{11}{4}} \sqrt{a - b x^{4}}} - \frac{7 a x^{3} \sqrt{a - b x^{4}}}{45 b^{2}} - \frac{x^{7} \sqrt{a - b x^{4}}}{9 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10/(-b*x**4+a)**(1/2),x)

[Out]

7*a**(11/4)*sqrt(1 - b*x**4/a)*elliptic_e(asin(b**(1/4)*x/a**(1/4)), -1)/(15*b**
(11/4)*sqrt(a - b*x**4)) - 7*a**(11/4)*sqrt(1 - b*x**4/a)*elliptic_f(asin(b**(1/
4)*x/a**(1/4)), -1)/(15*b**(11/4)*sqrt(a - b*x**4)) - 7*a*x**3*sqrt(a - b*x**4)/
(45*b**2) - x**7*sqrt(a - b*x**4)/(9*b)

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Mathematica [C]  time = 1.37931, size = 134, normalized size = 0.85 \[ \frac{\left (b x^4-a\right ) \left (7 a x^3+5 b x^7\right )+\frac{21 i a^2 \sqrt{1-\frac{b x^4}{a}} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\left (-\frac{\sqrt{b}}{\sqrt{a}}\right )^{3/2}}}{45 b^2 \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^10/Sqrt[a - b*x^4],x]

[Out]

((-a + b*x^4)*(7*a*x^3 + 5*b*x^7) + ((21*I)*a^2*Sqrt[1 - (b*x^4)/a]*(EllipticE[I
*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1] - EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/S
qrt[a])]*x], -1]))/(-(Sqrt[b]/Sqrt[a]))^(3/2))/(45*b^2*Sqrt[a - b*x^4])

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Maple [A]  time = 0.014, size = 126, normalized size = 0.8 \[ -{\frac{{x}^{7}}{9\,b}\sqrt{-b{x}^{4}+a}}-{\frac{7\,a{x}^{3}}{45\,{b}^{2}}\sqrt{-b{x}^{4}+a}}-{\frac{7}{15}{a}^{{\frac{5}{2}}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10/(-b*x^4+a)^(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*a^(5/2)/b^(5/2)
/(1/a^(1/2)*b^(1/2))^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))
^(1/2)/(-b*x^4+a)^(1/2)*(EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)-EllipticE(x*(1
/a^(1/2)*b^(1/2))^(1/2),I))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(-b*x^4 + a),x, algorithm="maxima")

[Out]

integrate(x^10/sqrt(-b*x^4 + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(-b*x^4 + a),x, algorithm="fricas")

[Out]

integral(x^10/sqrt(-b*x^4 + a), x)

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Sympy [A]  time = 4.53067, size = 39, normalized size = 0.25 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{15}{4}\right )} \]

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(2*I*pi)/a)/(4*sqr
t(a)*gamma(15/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{10}}{\sqrt{-b x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^10/sqrt(-b*x^4 + a),x, algorithm="giac")

[Out]

integrate(x^10/sqrt(-b*x^4 + a), x)