3.1144 \(\int \frac{1}{x^9 \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=122 \[ \frac{45 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}}-\frac{45 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}}+\frac{45 b \left (a+b x^4\right )^{3/4}}{32 a^3 x^4}-\frac{9 \left (a+b x^4\right )^{3/4}}{8 a^2 x^8}+\frac{1}{a x^8 \sqrt [4]{a+b x^4}} \]

[Out]

1/(a*x^8*(a + b*x^4)^(1/4)) - (9*(a + b*x^4)^(3/4))/(8*a^2*x^8) + (45*b*(a + b*x
^4)^(3/4))/(32*a^3*x^4) + (45*b^2*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(13/4
)) - (45*b^2*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(13/4))

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Rubi [A]  time = 0.180038, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{45 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}}-\frac{45 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{13/4}}+\frac{45 b \left (a+b x^4\right )^{3/4}}{32 a^3 x^4}-\frac{9 \left (a+b x^4\right )^{3/4}}{8 a^2 x^8}+\frac{1}{a x^8 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(a + b*x^4)^(5/4)),x]

[Out]

1/(a*x^8*(a + b*x^4)^(1/4)) - (9*(a + b*x^4)^(3/4))/(8*a^2*x^8) + (45*b*(a + b*x
^4)^(3/4))/(32*a^3*x^4) + (45*b^2*ArcTan[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(13/4
)) - (45*b^2*ArcTanh[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(13/4))

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Rubi in Sympy [A]  time = 20.3167, size = 116, normalized size = 0.95 \[ \frac{1}{a x^{8} \sqrt [4]{a + b x^{4}}} - \frac{9 \left (a + b x^{4}\right )^{\frac{3}{4}}}{8 a^{2} x^{8}} + \frac{45 b \left (a + b x^{4}\right )^{\frac{3}{4}}}{32 a^{3} x^{4}} + \frac{45 b^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{13}{4}}} - \frac{45 b^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{13}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(b*x**4+a)**(5/4),x)

[Out]

1/(a*x**8*(a + b*x**4)**(1/4)) - 9*(a + b*x**4)**(3/4)/(8*a**2*x**8) + 45*b*(a +
b*x**4)**(3/4)/(32*a**3*x**4) + 45*b**2*atan((a + b*x**4)**(1/4)/a**(1/4))/(64*
a**(13/4)) - 45*b**2*atanh((a + b*x**4)**(1/4)/a**(1/4))/(64*a**(13/4))

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Mathematica [C]  time = 0.0748056, size = 83, normalized size = 0.68 \[ \frac{-4 a^2-45 b^2 x^8 \sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )+9 a b x^4+45 b^2 x^8}{32 a^3 x^8 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^9*(a + b*x^4)^(5/4)),x]

[Out]

(-4*a^2 + 9*a*b*x^4 + 45*b^2*x^8 - 45*b^2*(1 + a/(b*x^4))^(1/4)*x^8*Hypergeometr
ic2F1[1/4, 1/4, 5/4, -(a/(b*x^4))])/(32*a^3*x^8*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.085, 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(1/x^9/(b*x^4+a)^(5/4),x)

[Out]

int(1/x^9/(b*x^4+a)^(5/4),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)^(5/4)*x^9),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.263259, size = 321, normalized size = 2.63 \[ -\frac{180 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} x^{8} \left (\frac{b^{8}}{a^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{10} \left (\frac{b^{8}}{a^{13}}\right )^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6} + \sqrt{a^{7} b^{8} \sqrt{\frac{b^{8}}{a^{13}}} + \sqrt{b x^{4} + a} b^{12}}}\right ) + 45 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} x^{8} \left (\frac{b^{8}}{a^{13}}\right )^{\frac{1}{4}} \log \left (91125 \, a^{10} \left (\frac{b^{8}}{a^{13}}\right )^{\frac{3}{4}} + 91125 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6}\right ) - 45 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} x^{8} \left (\frac{b^{8}}{a^{13}}\right )^{\frac{1}{4}} \log \left (-91125 \, a^{10} \left (\frac{b^{8}}{a^{13}}\right )^{\frac{3}{4}} + 91125 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{6}\right ) - 180 \, b^{2} x^{8} - 36 \, a b x^{4} + 16 \, a^{2}}{128 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^9),x, algorithm="fricas")

[Out]

-1/128*(180*(b*x^4 + a)^(1/4)*a^3*x^8*(b^8/a^13)^(1/4)*arctan(a^10*(b^8/a^13)^(3
/4)/((b*x^4 + a)^(1/4)*b^6 + sqrt(a^7*b^8*sqrt(b^8/a^13) + sqrt(b*x^4 + a)*b^12)
)) + 45*(b*x^4 + a)^(1/4)*a^3*x^8*(b^8/a^13)^(1/4)*log(91125*a^10*(b^8/a^13)^(3/
4) + 91125*(b*x^4 + a)^(1/4)*b^6) - 45*(b*x^4 + a)^(1/4)*a^3*x^8*(b^8/a^13)^(1/4
)*log(-91125*a^10*(b^8/a^13)^(3/4) + 91125*(b*x^4 + a)^(1/4)*b^6) - 180*b^2*x^8
- 36*a*b*x^4 + 16*a^2)/((b*x^4 + a)^(1/4)*a^3*x^8)

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Sympy [A]  time = 14.6948, size = 39, normalized size = 0.32 \[ - \frac{\Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac{5}{4}} x^{13} \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(b*x**4+a)**(5/4),x)

[Out]

-gamma(13/4)*hyper((5/4, 13/4), (17/4,), a*exp_polar(I*pi)/(b*x**4))/(4*b**(5/4)
*x**13*gamma(17/4))

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GIAC/XCAS [A]  time = 0.233033, size = 324, normalized size = 2.66 \[ -\frac{1}{256} \, b^{2}{\left (\frac{90 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{4}} + \frac{90 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{4}} - \frac{45 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{4}} + \frac{45 \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{4}} - \frac{256}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3}} - \frac{8 \,{\left (13 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} - 17 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a\right )}}{a^{3} b^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^9),x, algorithm="giac")

[Out]

-1/256*b^2*(90*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^4 + 90*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^4 - 45*sqrt(2)*(-a)^(3/4)*l
n(sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a^4 + 45*sq
rt(2)*(-a)^(3/4)*ln(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sq
rt(-a))/a^4 - 256/((b*x^4 + a)^(1/4)*a^3) - 8*(13*(b*x^4 + a)^(7/4) - 17*(b*x^4
+ a)^(3/4)*a)/(a^3*b^2*x^8))