3.992 \(\int \frac{\sqrt [4]{a+b x^4}}{x^5} \, dx\)

Optimal. Leaf size=75 \[ -\frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.102864, antiderivative size = 75, 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 \[ -\frac{b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{3/4}}-\frac{\sqrt [4]{a+b x^4}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.8681, size = 66, normalized size = 0.88 \[ - \frac{\sqrt [4]{a + b x^{4}}}{4 x^{4}} - \frac{b \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}}} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0424096, size = 67, normalized size = 0.89 \[ \frac{-b x^4 \left (\frac{a}{b x^4}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{a}{b x^4}\right )-3 \left (a+b x^4\right )}{12 x^4 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x^5,x]

[Out]

(-3*(a + b*x^4) - b*(1 + a/(b*x^4))^(3/4)*x^4*Hypergeometric2F1[3/4, 3/4, 7/4, -
(a/(b*x^4))])/(12*x^4*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^5,x)

[Out]

int((b*x^4+a)^(1/4)/x^5,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)^(1/4)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.274834, size = 219, normalized size = 2.92 \[ \frac{4 \, \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b + \sqrt{\sqrt{b x^{4} + a} b^{2} + a^{2} \sqrt{\frac{b^{4}}{a^{3}}}}}\right ) - \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} b + a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}\right ) + \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}} x^{4} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} b - a \left (\frac{b^{4}}{a^{3}}\right )^{\frac{1}{4}}\right ) - 4 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{16 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^5,x, algorithm="fricas")

[Out]

1/16*(4*(b^4/a^3)^(1/4)*x^4*arctan(a*(b^4/a^3)^(1/4)/((b*x^4 + a)^(1/4)*b + sqrt
(sqrt(b*x^4 + a)*b^2 + a^2*sqrt(b^4/a^3)))) - (b^4/a^3)^(1/4)*x^4*log((b*x^4 + a
)^(1/4)*b + a*(b^4/a^3)^(1/4)) + (b^4/a^3)^(1/4)*x^4*log((b*x^4 + a)^(1/4)*b - a
*(b^4/a^3)^(1/4)) - 4*(b*x^4 + a)^(1/4))/x^4

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Sympy [A]  time = 5.04281, size = 41, normalized size = 0.55 \[ - \frac{\sqrt [4]{b} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(1/4)*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), a*exp_polar(I*pi)/(b*x**4))/(4*x
**3*gamma(7/4))

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GIAC/XCAS [A]  time = 0.230956, size = 277, normalized size = 3.69 \[ -\frac{1}{32} \, b{\left (\frac{2 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a} + \frac{2 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a} + \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a} - \frac{\sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a} + \frac{8 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{b x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x^5,x, algorithm="giac")

[Out]

-1/32*b*(2*sqrt(2)*(-a)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^4
+ a)^(1/4))/(-a)^(1/4))/a + 2*sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-
a)^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4))/a + sqrt(2)*(-a)^(1/4)*ln(sqrt(2)*(b
*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a - sqrt(2)*(-a)^(1/4)*
ln(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a + 8*(b*
x^4 + a)^(1/4)/(b*x^4))