3.1143 \(\int \frac{1}{x^5 \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=96 \[ -\frac{5 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}-\frac{5 \left (a+b x^4\right )^{3/4}}{4 a^2 x^4}+\frac{1}{a x^4 \sqrt [4]{a+b x^4}} \]

[Out]

1/(a*x^4*(a + b*x^4)^(1/4)) - (5*(a + b*x^4)^(3/4))/(4*a^2*x^4) - (5*b*ArcTan[(a
+ b*x^4)^(1/4)/a^(1/4)])/(8*a^(9/4)) + (5*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])
/(8*a^(9/4))

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Rubi [A]  time = 0.142886, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{5 b \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}+\frac{5 b \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{8 a^{9/4}}-\frac{5 \left (a+b x^4\right )^{3/4}}{4 a^2 x^4}+\frac{1}{a x^4 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

1/(a*x^4*(a + b*x^4)^(1/4)) - (5*(a + b*x^4)^(3/4))/(4*a^2*x^4) - (5*b*ArcTan[(a
+ b*x^4)^(1/4)/a^(1/4)])/(8*a^(9/4)) + (5*b*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])
/(8*a^(9/4))

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Rubi in Sympy [A]  time = 15.5577, size = 90, normalized size = 0.94 \[ \frac{1}{a x^{4} \sqrt [4]{a + b x^{4}}} - \frac{5 \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 a^{2} x^{4}} - \frac{5 b \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}}} + \frac{5 b \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(a*x**4*(a + b*x**4)**(1/4)) - 5*(a + b*x**4)**(3/4)/(4*a**2*x**4) - 5*b*atan(
(a + b*x**4)**(1/4)/a**(1/4))/(8*a**(9/4)) + 5*b*atanh((a + b*x**4)**(1/4)/a**(1
/4))/(8*a**(9/4))

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Mathematica [C]  time = 0.0625522, size = 70, normalized size = 0.73 \[ \frac{5 b x^4 \sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )-a-5 b x^4}{4 a^2 x^4 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-a - 5*b*x^4 + 5*b*(1 + a/(b*x^4))^(1/4)*x^4*Hypergeometric2F1[1/4, 1/4, 5/4, -
(a/(b*x^4))])/(4*a^2*x^4*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^4+a)^(5/4),x)

[Out]

int(1/x^5/(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^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263003, size = 306, normalized size = 3.19 \[ \frac{20 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} x^{4} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{7} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3} + \sqrt{a^{5} b^{4} \sqrt{\frac{b^{4}}{a^{9}}} + \sqrt{b x^{4} + a} b^{6}}}\right ) + 5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} x^{4} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{1}{4}} \log \left (125 \, a^{7} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{3}{4}} + 125 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}\right ) - 5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} x^{4} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{1}{4}} \log \left (-125 \, a^{7} \left (\frac{b^{4}}{a^{9}}\right )^{\frac{3}{4}} + 125 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{3}\right ) - 20 \, b x^{4} - 4 \, a}{16 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*(20*(b*x^4 + a)^(1/4)*a^2*x^4*(b^4/a^9)^(1/4)*arctan(a^7*(b^4/a^9)^(3/4)/((
b*x^4 + a)^(1/4)*b^3 + sqrt(a^5*b^4*sqrt(b^4/a^9) + sqrt(b*x^4 + a)*b^6))) + 5*(
b*x^4 + a)^(1/4)*a^2*x^4*(b^4/a^9)^(1/4)*log(125*a^7*(b^4/a^9)^(3/4) + 125*(b*x^
4 + a)^(1/4)*b^3) - 5*(b*x^4 + a)^(1/4)*a^2*x^4*(b^4/a^9)^(1/4)*log(-125*a^7*(b^
4/a^9)^(3/4) + 125*(b*x^4 + a)^(1/4)*b^3) - 20*b*x^4 - 4*a)/((b*x^4 + a)^(1/4)*a
^2*x^4)

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Sympy [A]  time = 7.2587, size = 39, normalized size = 0.41 \[ - \frac{\Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac{5}{4}} x^{9} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-gamma(9/4)*hyper((5/4, 9/4), (13/4,), a*exp_polar(I*pi)/(b*x**4))/(4*b**(5/4)*x
**9*gamma(13/4))

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GIAC/XCAS [A]  time = 0.228684, size = 305, normalized size = 3.18 \[ \frac{1}{32} \, b{\left (\frac{10 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} + \frac{10 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} - \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} + \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} - \frac{8 \,{\left (5 \, b x^{4} + a\right )}}{{\left ({\left (b x^{4} + a\right )}^{\frac{5}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )} a^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32*b*(10*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^3 + 10*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^3 - 5*sqrt(2)*(-a)^(3/4)*ln(sqr
t(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a^3 + 5*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^3 - 8*(5*b*x^4 + a)/(((b*x^4 + a)^(5/4) - (b*x^4 + a)^(1/4)*a)*a^2))