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

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

[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*Sqr
t[a + b*x^4])/(15*b^(5/2)*(Sqrt[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]*EllipticE[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])

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Rubi [A]  time = 0.251432, 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 \[ \frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{11/4} \sqrt{a+b x^4}}-\frac{7 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{15 b^{11/4} \sqrt{a+b x^4}}+\frac{7 a^2 x \sqrt{a+b x^4}}{15 b^{5/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{7 a x^3 \sqrt{a+b x^4}}{45 b^2}+\frac{x^7 \sqrt{a+b x^4}}{9 b} \]

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*Sqr
t[a + b*x^4])/(15*b^(5/2)*(Sqrt[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]*EllipticE[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])

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Rubi in Sympy [A]  time = 28.8388, size = 238, normalized size = 0.91 \[ - \frac{7 a^{\frac{9}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{15 b^{\frac{11}{4}} \sqrt{a + b x^{4}}} + \frac{7 a^{\frac{9}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{30 b^{\frac{11}{4}} \sqrt{a + b x^{4}}} + \frac{7 a^{2} x \sqrt{a + b x^{4}}}{15 b^{\frac{5}{2}} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} - \frac{7 a x^{3} \sqrt{a + b x^{4}}}{45 b^{2}} + \frac{x^{7} \sqrt{a + b x^{4}}}{9 b} \]

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**(9/4)*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)/(15*b**(11/4)*sqrt(a + b*x**4))
+ 7*a**(9/4)*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)/(30*b**(11/4)*sqrt(a + b*x**4
)) + 7*a**2*x*sqrt(a + b*x**4)/(15*b**(5/2)*(sqrt(a) + sqrt(b)*x**2)) - 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 = 0.758686, size = 136, normalized size = 0.52 \[ \frac{\left (a+b x^4\right ) \left (5 b x^7-7 a x^3\right )+\frac{21 i a^2 \sqrt{\frac{b x^4}{a}+1} \left (E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )\right )}{\left (\frac{i \sqrt{b}}{\sqrt{a}}\right )^{3/2}}}{45 b^2 \sqrt{a+b x^4}} \]

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[(I*Sqrt[b])/Sqrt[a]]*x], -1] - EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b]
)/Sqrt[a]]*x], -1]))/((I*Sqrt[b])/Sqrt[a])^(3/2))/(45*b^2*Sqrt[a + b*x^4])

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Maple [C]  time = 0.012, size = 133, normalized size = 0.5 \[{\frac{{x}^{7}}{9\,b}\sqrt{b{x}^{4}+a}}-{\frac{7\,a{x}^{3}}{45\,{b}^{2}}\sqrt{b{x}^{4}+a}}+{{\frac{7\,i}{15}}{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){b}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

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))

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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 = 4.42504, size = 37, normalized size = 0.14 \[ \frac{x^{11} \Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{15}{4}\right )} \]

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))

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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)