3.1115 \(\int \frac{1}{x^9 \left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{21 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac{21 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}+\frac{7 b \sqrt [4]{a+b x^4}}{32 a^2 x^4}-\frac{\sqrt [4]{a+b x^4}}{8 a x^8} \]

[Out]

-(a + b*x^4)^(1/4)/(8*a*x^8) + (7*b*(a + b*x^4)^(1/4))/(32*a^2*x^4) - (21*b^2*Ar
cTan[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(11/4)) - (21*b^2*ArcTanh[(a + b*x^4)^(1/
4)/a^(1/4)])/(64*a^(11/4))

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Rubi [A]  time = 0.144911, 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 \[ -\frac{21 b^2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}-\frac{21 b^2 \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )}{64 a^{11/4}}+\frac{7 b \sqrt [4]{a+b x^4}}{32 a^2 x^4}-\frac{\sqrt [4]{a+b x^4}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^9*(a + b*x^4)^(3/4)),x]

[Out]

-(a + b*x^4)^(1/4)/(8*a*x^8) + (7*b*(a + b*x^4)^(1/4))/(32*a^2*x^4) - (21*b^2*Ar
cTan[(a + b*x^4)^(1/4)/a^(1/4)])/(64*a^(11/4)) - (21*b^2*ArcTanh[(a + b*x^4)^(1/
4)/a^(1/4)])/(64*a^(11/4))

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Rubi in Sympy [A]  time = 15.0302, size = 95, normalized size = 0.91 \[ - \frac{\sqrt [4]{a + b x^{4}}}{8 a x^{8}} + \frac{7 b \sqrt [4]{a + b x^{4}}}{32 a^{2} x^{4}} - \frac{21 b^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}}} - \frac{21 b^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{64 a^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**9/(b*x**4+a)**(3/4),x)

[Out]

-(a + b*x**4)**(1/4)/(8*a*x**8) + 7*b*(a + b*x**4)**(1/4)/(32*a**2*x**4) - 21*b*
*2*atan((a + b*x**4)**(1/4)/a**(1/4))/(64*a**(11/4)) - 21*b**2*atanh((a + b*x**4
)**(1/4)/a**(1/4))/(64*a**(11/4))

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Mathematica [C]  time = 0.05794, size = 83, normalized size = 0.8 \[ \frac{-4 a^2-7 b^2 x^8 \left (\frac{a}{b x^4}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{a}{b x^4}\right )+3 a b x^4+7 b^2 x^8}{32 a^2 x^8 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^9*(a + b*x^4)^(3/4)),x]

[Out]

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

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{9}} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^9/(b*x^4+a)^(3/4),x)

[Out]

int(1/x^9/(b*x^4+a)^(3/4),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)^(3/4)*x^9),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.264016, size = 270, normalized size = 2.6 \[ \frac{84 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{3} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2} + \sqrt{a^{6} \sqrt{\frac{b^{8}}{a^{11}}} + \sqrt{b x^{4} + a} b^{4}}}\right ) - 21 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} \log \left (21 \, a^{3} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} + 21 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}\right ) + 21 \, a^{2} x^{8} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} \log \left (-21 \, a^{3} \left (\frac{b^{8}}{a^{11}}\right )^{\frac{1}{4}} + 21 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}\right ) + 4 \,{\left (7 \, b x^{4} - 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{128 \, a^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^9),x, algorithm="fricas")

[Out]

1/128*(84*a^2*x^8*(b^8/a^11)^(1/4)*arctan(a^3*(b^8/a^11)^(1/4)/((b*x^4 + a)^(1/4
)*b^2 + sqrt(a^6*sqrt(b^8/a^11) + sqrt(b*x^4 + a)*b^4))) - 21*a^2*x^8*(b^8/a^11)
^(1/4)*log(21*a^3*(b^8/a^11)^(1/4) + 21*(b*x^4 + a)^(1/4)*b^2) + 21*a^2*x^8*(b^8
/a^11)^(1/4)*log(-21*a^3*(b^8/a^11)^(1/4) + 21*(b*x^4 + a)^(1/4)*b^2) + 4*(7*b*x
^4 - 4*a)*(b*x^4 + a)^(1/4))/(a^2*x^8)

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Sympy [A]  time = 10.4396, size = 39, normalized size = 0.38 \[ - \frac{\Gamma \left (\frac{11}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{11}{4} \\ \frac{15}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 b^{\frac{3}{4}} x^{11} \Gamma \left (\frac{15}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**9/(b*x**4+a)**(3/4),x)

[Out]

-gamma(11/4)*hyper((3/4, 11/4), (15/4,), a*exp_polar(I*pi)/(b*x**4))/(4*b**(3/4)
*x**11*gamma(15/4))

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GIAC/XCAS [A]  time = 0.224345, size = 305, normalized size = 2.93 \[ -\frac{1}{256} \, b^{2}{\left (\frac{42 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} + \frac{42 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right )}{a^{3}} + \frac{21 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} - \frac{21 \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right )}{a^{3}} - \frac{8 \,{\left (7 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} - 11 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a\right )}}{a^{2} b^{2} x^{8}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^9),x, algorithm="giac")

[Out]

-1/256*b^2*(42*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^3 + 42*sqrt(2)*(-a)^(1/4)*arctan(-1/2*sqrt(2)*(sqr
t(2)*(-a)^(1/4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4))/a^3 + 21*sqrt(2)*(-a)^(1/4)*l
n(sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a))/a^3 - 21*sq
rt(2)*(-a)^(1/4)*ln(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sq
rt(-a))/a^3 - 8*(7*(b*x^4 + a)^(5/4) - 11*(b*x^4 + a)^(1/4)*a)/(a^2*b^2*x^8))