3.860 \(\int \frac{x^8}{\left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ -\frac{5 a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 b^{9/4} \sqrt{a+b x^4}}+\frac{5 x \sqrt{a+b x^4}}{6 b^2}-\frac{x^5}{2 b \sqrt{a+b x^4}} \]

[Out]

-x^5/(2*b*Sqrt[a + b*x^4]) + (5*x*Sqrt[a + b*x^4])/(6*b^2) - (5*a^(3/4)*(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])/(12*b^(9/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.107616, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{5 a^{3/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{12 b^{9/4} \sqrt{a+b x^4}}+\frac{5 x \sqrt{a+b x^4}}{6 b^2}-\frac{x^5}{2 b \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^4)^(3/2),x]

[Out]

-x^5/(2*b*Sqrt[a + b*x^4]) + (5*x*Sqrt[a + b*x^4])/(6*b^2) - (5*a^(3/4)*(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])/(12*b^(9/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 11.6803, size = 116, normalized size = 0.9 \[ - \frac{5 a^{\frac{3}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{12 b^{\frac{9}{4}} \sqrt{a + b x^{4}}} - \frac{x^{5}}{2 b \sqrt{a + b x^{4}}} + \frac{5 x \sqrt{a + b x^{4}}}{6 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**4+a)**(3/2),x)

[Out]

-5*a**(3/4)*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)/(12*b**(9/4)*sqrt(a + b*x**4))
- x**5/(2*b*sqrt(a + b*x**4)) + 5*x*sqrt(a + b*x**4)/(6*b**2)

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Mathematica [C]  time = 0.217556, size = 93, normalized size = 0.72 \[ \frac{\frac{5 i a \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}}}+5 a x+2 b x^5}{6 b^2 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^4)^(3/2),x]

[Out]

(5*a*x + 2*b*x^5 + ((5*I)*a*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt
[b])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[b])/Sqrt[a]])/(6*b^2*Sqrt[a + b*x^4])

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Maple [C]  time = 0.018, size = 111, normalized size = 0.9 \[{\frac{ax}{2\,{b}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{x}{3\,{b}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{5\,a}{6\,{b}^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^4+a)^(3/2),x)

[Out]

1/2/b^2*a*x/((x^4+a/b)*b)^(1/2)+1/3*x*(b*x^4+a)^(1/2)/b^2-5/6*a/b^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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^4 + a)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^8/(b*x^4 + a)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{8}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^4 + a)^(3/2),x, algorithm="fricas")

[Out]

integral(x^8/(b*x^4 + a)^(3/2), x)

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Sympy [A]  time = 3.51584, size = 37, normalized size = 0.29 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**4+a)**(3/2),x)

[Out]

x**9*gamma(9/4)*hyper((3/2, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)
*gamma(13/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^4 + a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^8/(b*x^4 + a)^(3/2), x)