3.1142 \(\int \frac{1}{x \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=70 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}+\frac{1}{a \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.108304, antiderivative size = 70, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{2 a^{5/4}}+\frac{1}{a \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x^4)^(5/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.1933, size = 60, normalized size = 0.86 \[ \frac{1}{a \sqrt [4]{a + b x^{4}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{5}{4}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0492092, size = 52, normalized size = 0.74 \[ \frac{1-\sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )}{a \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(1 - (1 + a/(b*x^4))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, -(a/(b*x^4))])/(a*(a
+ b*x^4)^(1/4))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^4+a)^(5/4),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259961, size = 203, normalized size = 2.9 \[ -\frac{4 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a \frac{1}{a^{5}}^{\frac{1}{4}} \arctan \left (\frac{a^{4} \frac{1}{a^{5}}^{\frac{3}{4}}}{\sqrt{a^{3} \sqrt{\frac{1}{a^{5}}} + \sqrt{b x^{4} + a}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\right ) +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a \frac{1}{a^{5}}^{\frac{1}{4}} \log \left (a^{4} \frac{1}{a^{5}}^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right ) -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a \frac{1}{a^{5}}^{\frac{1}{4}} \log \left (-a^{4} \frac{1}{a^{5}}^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right ) - 4}{4 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(4*(b*x^4 + a)^(1/4)*a*(a^(-5))^(1/4)*arctan(a^4*(a^(-5))^(3/4)/(sqrt(a^3*s
qrt(a^(-5)) + sqrt(b*x^4 + a)) + (b*x^4 + a)^(1/4))) + (b*x^4 + a)^(1/4)*a*(a^(-
5))^(1/4)*log(a^4*(a^(-5))^(3/4) + (b*x^4 + a)^(1/4)) - (b*x^4 + a)^(1/4)*a*(a^(
-5))^(1/4)*log(-a^4*(a^(-5))^(3/4) + (b*x^4 + a)^(1/4)) - 4)/((b*x^4 + a)^(1/4)*
a)

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Sympy [A]  time = 4.34129, size = 39, normalized size = 0.56 \[ - \frac{\Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac{5}{4}} x^{5} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-gamma(5/4)*hyper((5/4, 5/4), (9/4,), a*exp_polar(I*pi)/(b*x**4))/(4*b**(5/4)*x*
*5*gamma(9/4))

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GIAC/XCAS [A]  time = 0.224697, size = 269, normalized size = 3.84 \[ -\frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a^{2}} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{4 \, a^{2}} + \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{8 \, a^{2}} - \frac{\sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{8 \, a^{2}} + \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^4 + a)^(
1/4))/(-a)^(1/4))/a^2 - 1/4*sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)
^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4))/a^2 + 1/8*sqrt(2)*(-a)^(3/4)*ln(sqrt(2
)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a^2 - 1/8*sqrt(2)*(
-a)^(3/4)*ln(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))
/a^2 + 1/((b*x^4 + a)^(1/4)*a)