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

Optimal. Leaf size=101 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}-\frac{b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac{\sqrt [4]{a+b x^4}}{8 x^8} \]

[Out]

-(a + b*x^4)^(1/4)/(8*x^8) - (b*(a + b*x^4)^(1/4))/(32*a*x^4) + (3*b^2*ArcTan[(a
+ b*x^4)^(1/4)/a^(1/4)])/(64*a^(7/4)) + (3*b^2*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4
)])/(64*a^(7/4))

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Rubi [A]  time = 0.138102, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ \frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{7/4}}-\frac{b \sqrt [4]{a+b x^4}}{32 a x^4}-\frac{\sqrt [4]{a+b x^4}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x^9,x]

[Out]

-(a + b*x^4)^(1/4)/(8*x^8) - (b*(a + b*x^4)^(1/4))/(32*a*x^4) + (3*b^2*ArcTan[(a
+ b*x^4)^(1/4)/a^(1/4)])/(64*a^(7/4)) + (3*b^2*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4
)])/(64*a^(7/4))

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Rubi in Sympy [A]  time = 14.6797, size = 90, normalized size = 0.89 \[ - \frac{\sqrt [4]{a + b x^{4}}}{8 x^{8}} - \frac{b \sqrt [4]{a + b x^{4}}}{32 a x^{4}} + \frac{3 b^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}}} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**4)**(1/4)/(8*x**8) - b*(a + b*x**4)**(1/4)/(32*a*x**4) + 3*b**2*atan(
(a + b*x**4)**(1/4)/a**(1/4))/(64*a**(7/4)) + 3*b**2*atanh((a + b*x**4)**(1/4)/a
**(1/4))/(64*a**(7/4))

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Mathematica [C]  time = 0.0522116, size = 82, normalized size = 0.81 \[ \frac{-4 a^2+b^2 x^8 \left (\frac{a}{b x^4}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{a}{b x^4}\right )-5 a b x^4-b^2 x^8}{32 a x^8 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^2 - 5*a*b*x^4 - b^2*x^8 + b^2*(1 + a/(b*x^4))^(3/4)*x^8*Hypergeometric2F1[
3/4, 3/4, 7/4, -(a/(b*x^4))])/(32*a*x^8*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9}}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x^9,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290427, size = 261, normalized size = 2.58 \[ -\frac{12 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \arctan \left (\frac{a^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + \sqrt{\sqrt{b x^{4} + a} b^{4} + a^{4} \sqrt{\frac{b^{8}}{a^{7}}}}}\right ) - 3 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \log \left (3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + 3 \, a^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}}\right ) + 3 \, a \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}} x^{8} \log \left (3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} - 3 \, a^{2} \left (\frac{b^{8}}{a^{7}}\right )^{\frac{1}{4}}\right ) + 4 \,{\left (b x^{4} + 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, a x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/128*(12*a*(b^8/a^7)^(1/4)*x^8*arctan(a^2*(b^8/a^7)^(1/4)/((b*x^4 + a)^(1/4)*b
^2 + sqrt(sqrt(b*x^4 + a)*b^4 + a^4*sqrt(b^8/a^7)))) - 3*a*(b^8/a^7)^(1/4)*x^8*l
og(3*(b*x^4 + a)^(1/4)*b^2 + 3*a^2*(b^8/a^7)^(1/4)) + 3*a*(b^8/a^7)^(1/4)*x^8*lo
g(3*(b*x^4 + a)^(1/4)*b^2 - 3*a^2*(b^8/a^7)^(1/4)) + 4*(b*x^4 + 4*a)*(b*x^4 + a)
^(1/4))/(a*x^8)

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Sympy [A]  time = 8.3833, size = 41, normalized size = 0.41 \[ - \frac{\sqrt [4]{b} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{7} \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(1/4)*gamma(7/4)*hyper((-1/4, 7/4), (11/4,), a*exp_polar(I*pi)/(b*x**4))/(4*
x**7*gamma(11/4))

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GIAC/XCAS [A]  time = 0.232057, size = 302, normalized size = 2.99 \[ \frac{1}{256} \, b^{2}{\left (\frac{6 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{6 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} - \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} - \frac{8 \,{\left ({\left (b x^{4} + a\right )}^{\frac{5}{4}} + 3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{a b^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256*b^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 + 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) + sqrt(-a
))/a^2 - 8*((b*x^4 + a)^(5/4) + 3*(b*x^4 + a)^(1/4)*a)/(a*b^2*x^8))