3.871 \(\int \frac{1}{\left (a+b x^4\right )^{5/2}} \, dx\)

Optimal. Leaf size=127 \[ \frac{5 \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{24 a^{9/4} \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{5 x}{12 a^2 \sqrt{a+b x^4}}+\frac{x}{6 a \left (a+b x^4\right )^{3/2}} \]

[Out]

x/(6*a*(a + b*x^4)^(3/2)) + (5*x)/(12*a^2*Sqrt[a + b*x^4]) + (5*(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])/(24*a^(9/4)*b^(1/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.0812811, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{5 \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{24 a^{9/4} \sqrt [4]{b} \sqrt{a+b x^4}}+\frac{5 x}{12 a^2 \sqrt{a+b x^4}}+\frac{x}{6 a \left (a+b x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(-5/2),x]

[Out]

x/(6*a*(a + b*x^4)^(3/2)) + (5*x)/(12*a^2*Sqrt[a + b*x^4]) + (5*(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])/(24*a^(9/4)*b^(1/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 8.11033, size = 114, normalized size = 0.9 \[ \frac{x}{6 a \left (a + b x^{4}\right )^{\frac{3}{2}}} + \frac{5 x}{12 a^{2} \sqrt{a + b x^{4}}} + \frac{5 \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{24 a^{\frac{9}{4}} \sqrt [4]{b} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**4+a)**(5/2),x)

[Out]

x/(6*a*(a + b*x**4)**(3/2)) + 5*x/(12*a**2*sqrt(a + b*x**4)) + 5*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)/(24*a**(9/4)*b**(1/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.336925, size = 99, normalized size = 0.78 \[ \frac{-\frac{5 i \left (a+b x^4\right ) \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}}}+7 a x+5 b x^5}{12 a^2 \left (a+b x^4\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(-5/2),x]

[Out]

(7*a*x + 5*b*x^5 - ((5*I)*(a + b*x^4)*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sq
rt[(I*Sqrt[b])/Sqrt[a]]*x], -1])/Sqrt[(I*Sqrt[b])/Sqrt[a]])/(12*a^2*(a + b*x^4)^
(3/2))

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Maple [C]  time = 0.019, size = 123, normalized size = 1. \[{\frac{x}{6\,a{b}^{2}}\sqrt{b{x}^{4}+a} \left ({x}^{4}+{\frac{a}{b}} \right ) ^{-2}}+{\frac{5\,x}{12\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{5}{12\,{a}^{2}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^4+a)^(5/2),x)

[Out]

1/6/a*x/b^2*(b*x^4+a)^(1/2)/(x^4+a/b)^2+5/12/a^2*x/((x^4+a/b)*b)^(1/2)+5/12/a^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. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(-5/2),x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(-5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )} \sqrt{b x^{4} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(-5/2),x, algorithm="fricas")

[Out]

integral(1/((b^2*x^8 + 2*a*b*x^4 + a^2)*sqrt(b*x^4 + a)), x)

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Sympy [A]  time = 2.80247, size = 36, normalized size = 0.28 \[ \frac{x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{2} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{2}} \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**4+a)**(5/2),x)

[Out]

x*gamma(1/4)*hyper((1/4, 5/2), (5/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(5/2)*gam
ma(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(-5/2),x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(-5/2), x)