3.869 \(\int \frac{1}{x^2 \left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=260 \[ \frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{7/4} \sqrt{a+b x^4}}-\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+b x^4}}-\frac{3 \sqrt{a+b x^4}}{2 a^2 x}+\frac{3 \sqrt{b} x \sqrt{a+b x^4}}{2 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{2 a x \sqrt{a+b x^4}} \]

[Out]

1/(2*a*x*Sqrt[a + b*x^4]) - (3*Sqrt[a + b*x^4])/(2*a^2*x) + (3*Sqrt[b]*x*Sqrt[a
+ b*x^4])/(2*a^2*(Sqrt[a] + Sqrt[b]*x^2)) - (3*b^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*S
qrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4
)], 1/2])/(2*a^(7/4)*Sqrt[a + b*x^4]) + (3*b^(1/4)*(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])/(4*a^(7/4)*Sqrt[a + b*x^4])

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Rubi [A]  time = 0.240115, antiderivative size = 260, 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 \[ \frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{4 a^{7/4} \sqrt{a+b x^4}}-\frac{3 \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{2 a^{7/4} \sqrt{a+b x^4}}-\frac{3 \sqrt{a+b x^4}}{2 a^2 x}+\frac{3 \sqrt{b} x \sqrt{a+b x^4}}{2 a^2 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{1}{2 a x \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^4)^(3/2)),x]

[Out]

1/(2*a*x*Sqrt[a + b*x^4]) - (3*Sqrt[a + b*x^4])/(2*a^2*x) + (3*Sqrt[b]*x*Sqrt[a
+ b*x^4])/(2*a^2*(Sqrt[a] + Sqrt[b]*x^2)) - (3*b^(1/4)*(Sqrt[a] + Sqrt[b]*x^2)*S
qrt[(a + b*x^4)/(Sqrt[a] + Sqrt[b]*x^2)^2]*EllipticE[2*ArcTan[(b^(1/4)*x)/a^(1/4
)], 1/2])/(2*a^(7/4)*Sqrt[a + b*x^4]) + (3*b^(1/4)*(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])/(4*a^(7/4)*Sqrt[a + b*x^4])

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Rubi in Sympy [A]  time = 29.3822, size = 235, normalized size = 0.9 \[ \frac{1}{2 a x \sqrt{a + b x^{4}}} + \frac{3 \sqrt{b} x \sqrt{a + b x^{4}}}{2 a^{2} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} - \frac{3 \sqrt{a + b x^{4}}}{2 a^{2} x} - \frac{3 \sqrt [4]{b} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{2 a^{\frac{7}{4}} \sqrt{a + b x^{4}}} + \frac{3 \sqrt [4]{b} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{7}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**4+a)**(3/2),x)

[Out]

1/(2*a*x*sqrt(a + b*x**4)) + 3*sqrt(b)*x*sqrt(a + b*x**4)/(2*a**2*(sqrt(a) + sqr
t(b)*x**2)) - 3*sqrt(a + b*x**4)/(2*a**2*x) - 3*b**(1/4)*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)/(2*a**(7/4)*sqrt(a + b*x**4)) + 3*b**(1/4)*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)/(4*a**(7/4)*sqrt(a + b*x**4))

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Mathematica [C]  time = 0.227274, size = 178, normalized size = 0.68 \[ \frac{-\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (2 a+3 b x^4\right )-3 \sqrt{a} \sqrt{b} x \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+3 \sqrt{a} \sqrt{b} x \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{2 a^2 x \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x^4)^(3/2)),x]

[Out]

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

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Maple [C]  time = 0.022, size = 137, normalized size = 0.5 \[ -{\frac{b{x}^{3}}{2\,{a}^{2}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{1}{x{a}^{2}}\sqrt{b{x}^{4}+a}}+{{\frac{3\,i}{2}}\sqrt{b}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^4+a)^(3/2),x)

[Out]

-1/2*b/a^2*x^3/((x^4+a/b)*b)^(1/2)-(b*x^4+a)^(1/2)/x/a^2+3/2*I*b^(1/2)/a^(3/2)/(
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. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(3/2)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{6} + a x^{2}\right )} \sqrt{b x^{4} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^2),x, algorithm="fricas")

[Out]

integral(1/((b*x^6 + a*x^2)*sqrt(b*x^4 + a)), x)

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Sympy [A]  time = 2.89187, size = 39, normalized size = 0.15 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{2} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**4+a)**(3/2),x)

[Out]

gamma(-1/4)*hyper((-1/4, 3/2), (3/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*x*g
amma(3/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/2)*x^2), x)