3.1238 \(\int \frac{1}{x^5 \left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=81 \[ -\frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac{\sqrt [4]{a-b x^4}}{4 a x^4} \]

[Out]

-(a - b*x^4)^(1/4)/(4*a*x^4) - (3*b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(7/4
)) - (3*b*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(7/4))

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Rubi [A]  time = 0.115428, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ -\frac{3 b \tan ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac{3 b \tanh ^{-1}\left (\frac{\sqrt [4]{a-b x^4}}{\sqrt [4]{a}}\right )}{8 a^{7/4}}-\frac{\sqrt [4]{a-b x^4}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a - b*x^4)^(3/4)),x]

[Out]

-(a - b*x^4)^(1/4)/(4*a*x^4) - (3*b*ArcTan[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(7/4
)) - (3*b*ArcTanh[(a - b*x^4)^(1/4)/a^(1/4)])/(8*a^(7/4))

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Rubi in Sympy [A]  time = 12.316, size = 71, normalized size = 0.88 \[ - \frac{\sqrt [4]{a - b x^{4}}}{4 a x^{4}} - \frac{3 b \operatorname{atan}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}}} - \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt [4]{a - b x^{4}}}{\sqrt [4]{a}} \right )}}{8 a^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(-b*x**4+a)**(3/4),x)

[Out]

-(a - b*x**4)**(1/4)/(4*a*x**4) - 3*b*atan((a - b*x**4)**(1/4)/a**(1/4))/(8*a**(
7/4)) - 3*b*atanh((a - b*x**4)**(1/4)/a**(1/4))/(8*a**(7/4))

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Mathematica [C]  time = 0.0498514, size = 70, normalized size = 0.86 \[ \frac{-b x^4 \left (1-\frac{a}{b x^4}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{a}{b x^4}\right )-a+b x^4}{4 a x^4 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a - b*x^4)^(3/4)),x]

[Out]

(-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)])/(4*a*x^4*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(-b*x^4+a)^(3/4),x)

[Out]

int(1/x^5/(-b*x^4+a)^(3/4),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)^(3/4)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.243827, size = 247, normalized size = 3.05 \[ \frac{12 \, a x^{4} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b + \sqrt{a^{4} \sqrt{\frac{b^{4}}{a^{7}}} + \sqrt{-b x^{4} + a} b^{2}}}\right ) - 3 \, a x^{4} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} \log \left (3 \, a^{2} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} + 3 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b\right ) + 3 \, a x^{4} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} \log \left (-3 \, a^{2} \left (\frac{b^{4}}{a^{7}}\right )^{\frac{1}{4}} + 3 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b\right ) - 4 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{16 \, a x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^5),x, algorithm="fricas")

[Out]

1/16*(12*a*x^4*(b^4/a^7)^(1/4)*arctan(a^2*(b^4/a^7)^(1/4)/((-b*x^4 + a)^(1/4)*b
+ sqrt(a^4*sqrt(b^4/a^7) + sqrt(-b*x^4 + a)*b^2))) - 3*a*x^4*(b^4/a^7)^(1/4)*log
(3*a^2*(b^4/a^7)^(1/4) + 3*(-b*x^4 + a)^(1/4)*b) + 3*a*x^4*(b^4/a^7)^(1/4)*log(-
3*a^2*(b^4/a^7)^(1/4) + 3*(-b*x^4 + a)^(1/4)*b) - 4*(-b*x^4 + a)^(1/4))/(a*x^4)

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Sympy [A]  time = 5.9684, size = 41, normalized size = 0.51 \[ \frac{e^{- \frac{7 i \pi }{4}} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{a}{b x^{4}}} \right )}}{4 b^{\frac{3}{4}} x^{7} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(-b*x**4+a)**(3/4),x)

[Out]

exp(-7*I*pi/4)*gamma(7/4)*hyper((3/4, 7/4), (11/4,), a/(b*x**4))/(4*b**(3/4)*x**
7*gamma(11/4))

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GIAC/XCAS [A]  time = 0.221876, size = 292, normalized size = 3.6 \[ -\frac{1}{32} \, b{\left (\frac{6 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{6 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} - \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{-b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} + \frac{8 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{a b x^{4}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-b*x^4 + a)^(3/4)*x^5),x, algorithm="giac")

[Out]

-1/32*b*(6*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 + 6*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 + 3*sqrt(2)*(-a)^(1/4)*ln(sq
rt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqrt(-a))/a^2 - 3*sqrt(
2)*(-a)^(1/4)*ln(-sqrt(2)*(-b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(-b*x^4 + a) + sqr
t(-a))/a^2 + 8*(-b*x^4 + a)^(1/4)/(a*b*x^4))