3.1085 \(\int \frac{1}{x^9 \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{9/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{9/4}}+\frac{5 b \left (a+b x^4\right )^{3/4}}{32 a^2 x^4}-\frac{\left (a+b x^4\right )^{3/4}}{8 a x^8} \]

[Out]

-(a + b*x^4)^(3/4)/(8*a*x^8) + (5*b*(a + b*x^4)^(3/4))/(32*a^2*x^4) + (5*b^2*Arc
Tan[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(9/4)) - (5*b^2*ArcTanh[(a + b*x^4)^(1/4)/
a^(1/4)])/(64*a^(9/4))

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Rubi [A]  time = 0.146527, antiderivative size = 104, 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 \[ \frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{9/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{9/4}}+\frac{5 b \left (a+b x^4\right )^{3/4}}{32 a^2 x^4}-\frac{\left (a+b x^4\right )^{3/4}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(a + b*x^4)^(1/4)),x]

[Out]

-(a + b*x^4)^(3/4)/(8*a*x^8) + (5*b*(a + b*x^4)^(3/4))/(32*a^2*x^4) + (5*b^2*Arc
Tan[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(9/4)) - (5*b^2*ArcTanh[(a + b*x^4)^(1/4)/
a^(1/4)])/(64*a^(9/4))

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Rubi in Sympy [A]  time = 16.0397, size = 95, normalized size = 0.91 \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{8 a x^{8}} + \frac{5 b \left (a + b x^{4}\right )^{\frac{3}{4}}}{32 a^{2} x^{4}} + \frac{5 b^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{9}{4}}} - \frac{5 b^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(b*x**4+a)**(1/4),x)

[Out]

-(a + b*x**4)**(3/4)/(8*a*x**8) + 5*b*(a + b*x**4)**(3/4)/(32*a**2*x**4) + 5*b**
2*atan((a + b*x**4)**(1/4)/a**(1/4))/(64*a**(9/4)) - 5*b**2*atanh((a + b*x**4)**
(1/4)/a**(1/4))/(64*a**(9/4))

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Mathematica [C]  time = 0.0677167, size = 82, normalized size = 0.79 \[ \frac{-4 a^2-5 b^2 x^8 \sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )+a b x^4+5 b^2 x^8}{32 a^2 x^8 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^9*(a + b*x^4)^(1/4)),x]

[Out]

(-4*a^2 + a*b*x^4 + 5*b^2*x^8 - 5*b^2*(1 + a/(b*x^4))^(1/4)*x^8*Hypergeometric2F
1[1/4, 1/4, 5/4, -(a/(b*x^4))])/(32*a^2*x^8*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(b*x^4+a)^(1/4),x)

[Out]

int(1/x^9/(b*x^4+a)^(1/4),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)^(1/4)*x^9),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.280814, size = 274, normalized size = 2.63 \[ -\frac{20 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{7} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6} + \sqrt{a^{5} b^{8} \sqrt{\frac{b^{8}}{a^{9}}} + \sqrt{b x^{4} + a} b^{12}}}\right ) + 5 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{1}{4}} \log \left (125 \, a^{7} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{3}{4}} + 125 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6}\right ) - 5 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{1}{4}} \log \left (-125 \, a^{7} \left (\frac{b^{8}}{a^{9}}\right )^{\frac{3}{4}} + 125 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6}\right ) - 4 \,{\left (5 \, b x^{4} - 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, a^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^9),x, algorithm="fricas")

[Out]

-1/128*(20*a^2*x^8*(b^8/a^9)^(1/4)*arctan(a^7*(b^8/a^9)^(3/4)/((b*x^4 + a)^(1/4)
*b^6 + sqrt(a^5*b^8*sqrt(b^8/a^9) + sqrt(b*x^4 + a)*b^12))) + 5*a^2*x^8*(b^8/a^9
)^(1/4)*log(125*a^7*(b^8/a^9)^(3/4) + 125*(b*x^4 + a)^(1/4)*b^6) - 5*a^2*x^8*(b^
8/a^9)^(1/4)*log(-125*a^7*(b^8/a^9)^(3/4) + 125*(b*x^4 + a)^(1/4)*b^6) - 4*(5*b*
x^4 - 4*a)*(b*x^4 + a)^(3/4))/(a^2*x^8)

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Sympy [A]  time = 8.5868, size = 39, normalized size = 0.38 \[ - \frac{\Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \sqrt [4]{b} x^{9} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(b*x**4+a)**(1/4),x)

[Out]

-gamma(9/4)*hyper((1/4, 9/4), (13/4,), a*exp_polar(I*pi)/(b*x**4))/(4*b**(1/4)*x
**9*gamma(13/4))

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GIAC/XCAS [A]  time = 0.232507, size = 305, normalized size = 2.93 \[ -\frac{1}{256} \, b^{2}{\left (\frac{10 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} + \frac{10 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} - \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} + \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} - \frac{8 \,{\left (5 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} - 9 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a\right )}}{a^{2} b^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^9),x, algorithm="giac")

[Out]

-1/256*b^2*(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)*(sqr
t(2)*(-a)^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4))/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 + 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)^(7/4) - 9*(b*x^4 + a)^(3/4)*a)/(a^2*b^2*x^8))