3.870 \(\int \frac{1}{x^6 \left (a+b x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=282 \[ -\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 a^{11/4} \sqrt{a+b x^4}}+\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{11/4} \sqrt{a+b x^4}}-\frac{21 b^{3/2} x \sqrt{a+b x^4}}{10 a^3 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{21 b \sqrt{a+b x^4}}{10 a^3 x}-\frac{7 \sqrt{a+b x^4}}{10 a^2 x^5}+\frac{1}{2 a x^5 \sqrt{a+b x^4}} \]

[Out]

1/(2*a*x^5*Sqrt[a + b*x^4]) - (7*Sqrt[a + b*x^4])/(10*a^2*x^5) + (21*b*Sqrt[a +
b*x^4])/(10*a^3*x) - (21*b^(3/2)*x*Sqrt[a + b*x^4])/(10*a^3*(Sqrt[a] + Sqrt[b]*x
^2)) + (21*b^(5/4)*(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])/(10*a^(11/4)*Sqrt[a + b*x^
4]) - (21*b^(5/4)*(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])/(20*a^(11/4)*Sqrt[a + b*x^4
])

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Rubi [A]  time = 0.298554, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{20 a^{11/4} \sqrt{a+b x^4}}+\frac{21 b^{5/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{10 a^{11/4} \sqrt{a+b x^4}}-\frac{21 b^{3/2} x \sqrt{a+b x^4}}{10 a^3 \left (\sqrt{a}+\sqrt{b} x^2\right )}+\frac{21 b \sqrt{a+b x^4}}{10 a^3 x}-\frac{7 \sqrt{a+b x^4}}{10 a^2 x^5}+\frac{1}{2 a x^5 \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^4)^(3/2)),x]

[Out]

1/(2*a*x^5*Sqrt[a + b*x^4]) - (7*Sqrt[a + b*x^4])/(10*a^2*x^5) + (21*b*Sqrt[a +
b*x^4])/(10*a^3*x) - (21*b^(3/2)*x*Sqrt[a + b*x^4])/(10*a^3*(Sqrt[a] + Sqrt[b]*x
^2)) + (21*b^(5/4)*(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])/(10*a^(11/4)*Sqrt[a + b*x^
4]) - (21*b^(5/4)*(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])/(20*a^(11/4)*Sqrt[a + b*x^4
])

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Rubi in Sympy [A]  time = 35.6554, size = 258, normalized size = 0.91 \[ \frac{1}{2 a x^{5} \sqrt{a + b x^{4}}} - \frac{7 \sqrt{a + b x^{4}}}{10 a^{2} x^{5}} - \frac{21 b^{\frac{3}{2}} x \sqrt{a + b x^{4}}}{10 a^{3} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} + \frac{21 b \sqrt{a + b x^{4}}}{10 a^{3} x} + \frac{21 b^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{10 a^{\frac{11}{4}} \sqrt{a + b x^{4}}} - \frac{21 b^{\frac{5}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{20 a^{\frac{11}{4}} \sqrt{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**4+a)**(3/2),x)

[Out]

1/(2*a*x**5*sqrt(a + b*x**4)) - 7*sqrt(a + b*x**4)/(10*a**2*x**5) - 21*b**(3/2)*
x*sqrt(a + b*x**4)/(10*a**3*(sqrt(a) + sqrt(b)*x**2)) + 21*b*sqrt(a + b*x**4)/(1
0*a**3*x) + 21*b**(5/4)*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)/(10*a**(11/4)*sqrt(
a + b*x**4)) - 21*b**(5/4)*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)/(20*a**(11/4)*sq
rt(a + b*x**4))

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Mathematica [C]  time = 0.269439, size = 192, normalized size = 0.68 \[ \frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (-2 a^2+14 a b x^4+21 b^2 x^8\right )+21 \sqrt{a} b^{3/2} x^5 \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )-21 \sqrt{a} b^{3/2} x^5 \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{10 a^3 x^5 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^4)^(3/2)),x]

[Out]

(Sqrt[(I*Sqrt[b])/Sqrt[a]]*(-2*a^2 + 14*a*b*x^4 + 21*b^2*x^8) - 21*Sqrt[a]*b^(3/
2)*x^5*Sqrt[1 + (b*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[b])/Sqrt[a]]*x], -1]
+ 21*Sqrt[a]*b^(3/2)*x^5*Sqrt[1 + (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[b
])/Sqrt[a]]*x], -1])/(10*a^3*Sqrt[(I*Sqrt[b])/Sqrt[a]]*x^5*Sqrt[a + b*x^4])

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Maple [C]  time = 0.027, size = 157, normalized size = 0.6 \[{\frac{{b}^{2}{x}^{3}}{2\,{a}^{3}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}-{\frac{1}{5\,{x}^{5}{a}^{2}}\sqrt{b{x}^{4}+a}}+{\frac{8\,b}{5\,{a}^{3}x}\sqrt{b{x}^{4}+a}}-{{\frac{21\,i}{10}}{b}^{{\frac{3}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){a}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^4+a)^(3/2),x)

[Out]

1/2*b^2/a^3*x^3/((x^4+a/b)*b)^(1/2)-1/5*(b*x^4+a)^(1/2)/x^5/a^2+8/5*b*(b*x^4+a)^
(1/2)/a^3/x-21/10*I/a^(5/2)*b^(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. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(3/2)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{10} + a x^{6}\right )} \sqrt{b x^{4} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^6),x, algorithm="fricas")

[Out]

integral(1/((b*x^10 + a*x^6)*sqrt(b*x^4 + a)), x)

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Sympy [A]  time = 4.90366, size = 44, normalized size = 0.16 \[ \frac{\Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{2} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**4+a)**(3/2),x)

[Out]

gamma(-5/4)*hyper((-5/4, 3/2), (-1/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(3/2)*x*
*5*gamma(-1/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{2}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/2)*x^6),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/2)*x^6), x)