3.1064 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^6} \, dx\)

Optimal. Leaf size=92 \[ -\frac{1}{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{1}{2} b^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-\frac{b \sqrt [4]{a+b x^4}}{x}-\frac{\left (a+b x^4\right )^{5/4}}{5 x^5} \]

[Out]

-((b*(a + b*x^4)^(1/4))/x) - (a + b*x^4)^(5/4)/(5*x^5) - (b^(5/4)*ArcTan[(b^(1/4
)*x)/(a + b*x^4)^(1/4)])/2 + (b^(5/4)*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/2

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Rubi [A]  time = 0.0966185, antiderivative size = 92, 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 \[ -\frac{1}{2} b^{5/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{1}{2} b^{5/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-\frac{b \sqrt [4]{a+b x^4}}{x}-\frac{\left (a+b x^4\right )^{5/4}}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^6,x]

[Out]

-((b*(a + b*x^4)^(1/4))/x) - (a + b*x^4)^(5/4)/(5*x^5) - (b^(5/4)*ArcTan[(b^(1/4
)*x)/(a + b*x^4)^(1/4)])/2 + (b^(5/4)*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/2

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Rubi in Sympy [A]  time = 13.1615, size = 78, normalized size = 0.85 \[ - \frac{b^{\frac{5}{4}} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2} + \frac{b^{\frac{5}{4}} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2} - \frac{b \sqrt [4]{a + b x^{4}}}{x} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(5/4)/x**6,x)

[Out]

-b**(5/4)*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/2 + b**(5/4)*atanh(b**(1/4)*x/(a
+ b*x**4)**(1/4))/2 - b*(a + b*x**4)**(1/4)/x - (a + b*x**4)**(5/4)/(5*x**5)

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Mathematica [C]  time = 0.0556946, size = 81, normalized size = 0.88 \[ \frac{5 b^2 x^8 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-3 \left (a^2+7 a b x^4+6 b^2 x^8\right )}{15 x^5 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4)/x^6,x]

[Out]

(-3*(a^2 + 7*a*b*x^4 + 6*b^2*x^8) + 5*b^2*x^8*(1 + (b*x^4)/a)^(3/4)*Hypergeometr
ic2F1[3/4, 3/4, 7/4, -((b*x^4)/a)])/(15*x^5*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^6,x)

[Out]

int((b*x^4+a)^(5/4)/x^6,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^6,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^6,x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 9.03357, size = 46, normalized size = 0.5 \[ \frac{a^{\frac{5}{4}} \Gamma \left (- \frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{5}{4} \\ - \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{5} \Gamma \left (- \frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(5/4)/x**6,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.238053, size = 316, normalized size = 3.43 \[ \frac{1}{4} \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} b \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right ) + \frac{1}{4} \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} b \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-b\right )^{\frac{1}{4}} - \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, \left (-b\right )^{\frac{1}{4}}}\right ) + \frac{1}{8} \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} b{\rm ln}\left (\sqrt{-b} + \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right ) - \frac{1}{8} \, \sqrt{2} \left (-b\right )^{\frac{1}{4}} b{\rm ln}\left (\sqrt{-b} - \frac{\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-b\right )^{\frac{1}{4}}}{x} + \frac{\sqrt{b x^{4} + a}}{x^{2}}\right ) - \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b + \frac{a}{x^{4}}\right )}}{5 \, x} - \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^6,x, algorithm="giac")

[Out]

1/4*sqrt(2)*(-b)^(1/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(-b)^(1/4) + 2*(b*x^4 + a)^
(1/4)/x)/(-b)^(1/4)) + 1/4*sqrt(2)*(-b)^(1/4)*b*arctan(-1/2*sqrt(2)*(sqrt(2)*(-b
)^(1/4) - 2*(b*x^4 + a)^(1/4)/x)/(-b)^(1/4)) + 1/8*sqrt(2)*(-b)^(1/4)*b*ln(sqrt(
-b) + sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4 + a)/x^2) - 1/8*sqrt(2
)*(-b)^(1/4)*b*ln(sqrt(-b) - sqrt(2)*(b*x^4 + a)^(1/4)*(-b)^(1/4)/x + sqrt(b*x^4
+ a)/x^2) - 1/5*(b*x^4 + a)^(1/4)*(b + a/x^4)/x - (b*x^4 + a)^(1/4)*b/x