3.1063 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{\left (a+b x^4\right )^{5/4}}{x}-\frac{5}{8} a \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{5}{8} a \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{5}{4} b x^3 \sqrt [4]{a+b x^4} \]

[Out]

(5*b*x^3*(a + b*x^4)^(1/4))/4 - (a + b*x^4)^(5/4)/x - (5*a*b^(1/4)*ArcTan[(b^(1/
4)*x)/(a + b*x^4)^(1/4)])/8 + (5*a*b^(1/4)*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)
])/8

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Rubi [A]  time = 0.101425, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (a+b x^4\right )^{5/4}}{x}-\frac{5}{8} a \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{5}{8} a \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{5}{4} b x^3 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^2,x]

[Out]

(5*b*x^3*(a + b*x^4)^(1/4))/4 - (a + b*x^4)^(5/4)/x - (5*a*b^(1/4)*ArcTan[(b^(1/
4)*x)/(a + b*x^4)^(1/4)])/8 + (5*a*b^(1/4)*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)
])/8

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Rubi in Sympy [A]  time = 13.2339, size = 87, normalized size = 0.93 \[ - \frac{5 a \sqrt [4]{b} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8} + \frac{5 a \sqrt [4]{b} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8} + \frac{5 b x^{3} \sqrt [4]{a + b x^{4}}}{4} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a*b**(1/4)*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/8 + 5*a*b**(1/4)*atanh(b**(1/
4)*x/(a + b*x**4)**(1/4))/8 + 5*b*x**3*(a + b*x**4)**(1/4)/4 - (a + b*x**4)**(5/
4)/x

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Mathematica [C]  time = 0.0433878, size = 79, normalized size = 0.84 \[ \frac{-12 a^2+5 a b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-9 a b x^4+3 b^2 x^8}{12 x \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4)/x^2,x]

[Out]

(-12*a^2 - 9*a*b*x^4 + 3*b^2*x^8 + 5*a*b*x^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometri
c2F1[3/4, 3/4, 7/4, -((b*x^4)/a)])/(12*x*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^2,x)

[Out]

int((b*x^4+a)^(5/4)/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 8.15099, size = 41, normalized size = 0.44 \[ \frac{a^{\frac{5}{4}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.237999, size = 309, normalized size = 3.29 \[ \frac{1}{32} \,{\left (\frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b x^{3}}{a} + 10 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right ) + 10 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right ) + 5 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right ) - 5 \, \sqrt{2} \left (-b\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right ) - \frac{32 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*(8*(b*x^4 + a)^(1/4)*b*x^3/a + 10*sqrt(2)*(-b)^(1/4)*arctan(1/2*sqrt(2)*(sq
rt(2)*(-b)^(1/4) + 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4)) + 10*sqrt(2)*(-b)^(1/4)*ar
ctan(-1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) - 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4)) + 5*s
qrt(2)*(-b)^(1/4)*ln(sqrt(-b) + sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*
x^4 + a)/x^2) - 5*sqrt(2)*(-b)^(1/4)*ln(sqrt(-b) - sqrt(2)*(b*x^4 + a)^(1/4)*(-b
)^(1/4)/x + sqrt(b*x^4 + a)/x^2) - 32*(b*x^4 + a)^(1/4)/x)*a