3.1022 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^9} \, dx\)

Optimal. Leaf size=101 \[ -\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}-\frac{3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac{\left (a+b x^4\right )^{3/4}}{8 x^8} \]

[Out]

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

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Rubi [A]  time = 0.147771, 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 \[ -\frac{3 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}+\frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{5/4}}-\frac{3 b \left (a+b x^4\right )^{3/4}}{32 a x^4}-\frac{\left (a+b x^4\right )^{3/4}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.6954, size = 92, normalized size = 0.91 \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{8 x^{8}} - \frac{3 b \left (a + b x^{4}\right )^{\frac{3}{4}}}{32 a x^{4}} - \frac{3 b^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{5}{4}}} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/x**9,x)

[Out]

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

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Mathematica [C]  time = 0.0561714, size = 83, normalized size = 0.82 \[ \frac{-4 a^2+3 b^2 x^8 \sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )-7 a b x^4-3 b^2 x^8}{32 a x^8 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4)/x^9,x]

[Out]

(-4*a^2 - 7*a*b*x^4 - 3*b^2*x^8 + 3*b^2*(1 + a/(b*x^4))^(1/4)*x^8*Hypergeometric
2F1[1/4, 1/4, 5/4, -(a/(b*x^4))])/(32*a*x^8*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.052, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^9,x)

[Out]

int((b*x^4+a)^(3/4)/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)^(3/4)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.421864, size = 266, normalized size = 2.63 \[ \frac{12 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{1}{4}} a x^{8} \arctan \left (\frac{\left (\frac{b^{8}}{a^{5}}\right )^{\frac{3}{4}} a^{4}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6} + \sqrt{\sqrt{b x^{4} + a} b^{12} + \sqrt{\frac{b^{8}}{a^{5}}} a^{3} b^{8}}}\right ) + 3 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{1}{4}} a x^{8} \log \left (27 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6} + 27 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{3}{4}} a^{4}\right ) - 3 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{1}{4}} a x^{8} \log \left (27 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6} - 27 \, \left (\frac{b^{8}}{a^{5}}\right )^{\frac{3}{4}} a^{4}\right ) - 4 \,{\left (3 \, b x^{4} + 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, a x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^9,x, algorithm="fricas")

[Out]

1/128*(12*(b^8/a^5)^(1/4)*a*x^8*arctan((b^8/a^5)^(3/4)*a^4/((b*x^4 + a)^(1/4)*b^
6 + sqrt(sqrt(b*x^4 + a)*b^12 + sqrt(b^8/a^5)*a^3*b^8))) + 3*(b^8/a^5)^(1/4)*a*x
^8*log(27*(b*x^4 + a)^(1/4)*b^6 + 27*(b^8/a^5)^(3/4)*a^4) - 3*(b^8/a^5)^(1/4)*a*
x^8*log(27*(b*x^4 + a)^(1/4)*b^6 - 27*(b^8/a^5)^(3/4)*a^4) - 4*(3*b*x^4 + 4*a)*(
b*x^4 + a)^(3/4))/(a*x^8)

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Sympy [A]  time = 10.8896, size = 41, normalized size = 0.41 \[ - \frac{b^{\frac{3}{4}} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{5} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/x**9,x)

[Out]

-b**(3/4)*gamma(5/4)*hyper((-3/4, 5/4), (9/4,), a*exp_polar(I*pi)/(b*x**4))/(4*x
**5*gamma(9/4))

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GIAC/XCAS [A]  time = 0.228808, size = 304, normalized size = 3.01 \[ \frac{1}{256} \, b^{2}{\left (\frac{6 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} + \frac{6 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{2}} - \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} + \frac{3 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{2}} - \frac{8 \,{\left (3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} +{\left (b x^{4} + a\right )}^{\frac{3}{4}} a\right )}}{a b^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^9,x, algorithm="giac")

[Out]

1/256*b^2*(6*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 + 6*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 - 3*sqrt(2)*(-a)^(3/4)*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)^(3/4)*ln(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a
))/a^2 - 8*(3*(b*x^4 + a)^(7/4) + (b*x^4 + a)^(3/4)*a)/(a*b^2*x^8))