3.1052 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^9} \, dx\)

Optimal. Leaf size=98 \[ -\frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8} \]

[Out]

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

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Rubi [A]  time = 0.138282, antiderivative size = 98, 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 \[ -\frac{5 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{3/4}}-\frac{5 b \sqrt [4]{a+b x^4}}{32 x^4}-\frac{\left (a+b x^4\right )^{5/4}}{8 x^8} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^9,x]

[Out]

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

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Rubi in Sympy [A]  time = 14.3535, size = 92, normalized size = 0.94 \[ - \frac{5 b \sqrt [4]{a + b x^{4}}}{32 x^{4}} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{8 x^{8}} - \frac{5 b^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{3}{4}}} - \frac{5 b^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(5/4)/x**9,x)

[Out]

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

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Mathematica [C]  time = 0.0617321, size = 85, normalized size = 0.87 \[ \left (-\frac{a}{8 x^8}-\frac{9 b}{32 x^4}\right ) \sqrt [4]{a+b x^4}-\frac{5 b^2 \left (\frac{a+b x^4}{b x^4}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{a}{b x^4}\right )}{96 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4)/x^9,x]

[Out]

(-a/(8*x^8) - (9*b)/(32*x^4))*(a + b*x^4)^(1/4) - (5*b^2*((a + b*x^4)/(b*x^4))^(
3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, -(a/(b*x^4))])/(96*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^9,x)

[Out]

int((b*x^4+a)^(5/4)/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)^(5/4)/x^9,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.30805, size = 246, normalized size = 2.51 \[ \frac{20 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} x^{8} \arctan \left (\frac{\left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} a}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + \sqrt{\sqrt{b x^{4} + a} b^{4} + \sqrt{\frac{b^{8}}{a^{3}}} a^{2}}}\right ) - 5 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} x^{8} \log \left (5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + 5 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} a\right ) + 5 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} x^{8} \log \left (5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} - 5 \, \left (\frac{b^{8}}{a^{3}}\right )^{\frac{1}{4}} a\right ) - 4 \,{\left (9 \, b x^{4} + 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^9,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 15.1545, size = 41, normalized size = 0.42 \[ - \frac{b^{\frac{5}{4}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 x^{3} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(5/4)/x**9,x)

[Out]

-b**(5/4)*gamma(3/4)*hyper((-5/4, 3/4), (7/4,), a*exp_polar(I*pi)/(b*x**4))/(4*x
**3*gamma(7/4))

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GIAC/XCAS [A]  time = 0.227978, size = 301, normalized size = 3.07 \[ -\frac{1}{256} \, b^{2}{\left (\frac{10 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a} + \frac{10 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a} + \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a} - \frac{5 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a} + \frac{8 \,{\left (9 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} - 5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{b^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^9,x, algorithm="giac")

[Out]

-1/256*b^2*(10*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 + 10*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 + 5*sqrt(2)*(-a)^(1/4)*ln(sqr
t(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a - 5*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 + 8*(9*(b*x^4 + a)^(5/4) - 5*(b*x^4 + a)^(1/4)*a)/(b^2*x^8))