3.1152 \(\int \frac{x^{12}}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=123 \[ \frac{45 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac{45 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}-\frac{45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}-\frac{x^9}{b \sqrt [4]{a+b x^4}} \]

[Out]

-(x^9/(b*(a + b*x^4)^(1/4))) - (45*a*x*(a + b*x^4)^(3/4))/(32*b^3) + (9*x^5*(a +
b*x^4)^(3/4))/(8*b^2) + (45*a^2*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(1
3/4)) + (45*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(13/4))

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Rubi [A]  time = 0.127076, antiderivative size = 123, 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 \[ \frac{45 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}+\frac{45 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{13/4}}-\frac{45 a x \left (a+b x^4\right )^{3/4}}{32 b^3}+\frac{9 x^5 \left (a+b x^4\right )^{3/4}}{8 b^2}-\frac{x^9}{b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^12/(a + b*x^4)^(5/4),x]

[Out]

-(x^9/(b*(a + b*x^4)^(1/4))) - (45*a*x*(a + b*x^4)^(3/4))/(32*b^3) + (9*x^5*(a +
b*x^4)^(3/4))/(8*b^2) + (45*a^2*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(1
3/4)) + (45*a^2*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(13/4))

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Rubi in Sympy [A]  time = 15.8159, size = 116, normalized size = 0.94 \[ \frac{45 a^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{13}{4}}} + \frac{45 a^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{13}{4}}} - \frac{45 a x \left (a + b x^{4}\right )^{\frac{3}{4}}}{32 b^{3}} - \frac{x^{9}}{b \sqrt [4]{a + b x^{4}}} + \frac{9 x^{5} \left (a + b x^{4}\right )^{\frac{3}{4}}}{8 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12/(b*x**4+a)**(5/4),x)

[Out]

45*a**2*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(64*b**(13/4)) + 45*a**2*atanh(b**(
1/4)*x/(a + b*x**4)**(1/4))/(64*b**(13/4)) - 45*a*x*(a + b*x**4)**(3/4)/(32*b**3
) - x**9/(b*(a + b*x**4)**(1/4)) + 9*x**5*(a + b*x**4)**(3/4)/(8*b**2)

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Mathematica [A]  time = 0.295309, size = 120, normalized size = 0.98 \[ \frac{45 a^2 \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{128 b^{13/4}}+\frac{-45 a^2 x-9 a b x^5+4 b^2 x^9}{32 b^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12/(a + b*x^4)^(5/4),x]

[Out]

(-45*a^2*x - 9*a*b*x^5 + 4*b^2*x^9)/(32*b^3*(a + b*x^4)^(1/4)) + (45*a^2*(2*ArcT
an[(b^(1/4)*x)/(a + b*x^4)^(1/4)] - Log[1 - (b^(1/4)*x)/(a + b*x^4)^(1/4)] + Log
[1 + (b^(1/4)*x)/(a + b*x^4)^(1/4)]))/(128*b^(13/4))

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Maple [F]  time = 0.066, size = 0, normalized size = 0. \[ \int{{x}^{12} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12/(b*x^4+a)^(5/4),x)

[Out]

int(x^12/(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(x^12/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270845, size = 356, normalized size = 2.89 \[ \frac{180 \,{\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac{a^{8}}{b^{13}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{10} x \left (\frac{a^{8}}{b^{13}}\right )^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} + x \sqrt{\frac{a^{8} b^{7} x^{2} \sqrt{\frac{a^{8}}{b^{13}}} + \sqrt{b x^{4} + a} a^{12}}{x^{2}}}}\right ) + 45 \,{\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac{a^{8}}{b^{13}}\right )^{\frac{1}{4}} \log \left (\frac{91125 \,{\left (b^{10} x \left (\frac{a^{8}}{b^{13}}\right )^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) - 45 \,{\left (b^{4} x^{4} + a b^{3}\right )} \left (\frac{a^{8}}{b^{13}}\right )^{\frac{1}{4}} \log \left (-\frac{91125 \,{\left (b^{10} x \left (\frac{a^{8}}{b^{13}}\right )^{\frac{3}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) + 4 \,{\left (4 \, b^{2} x^{9} - 9 \, a b x^{5} - 45 \, a^{2} x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{128 \,{\left (b^{4} x^{4} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

1/128*(180*(b^4*x^4 + a*b^3)*(a^8/b^13)^(1/4)*arctan(b^10*x*(a^8/b^13)^(3/4)/((b
*x^4 + a)^(1/4)*a^6 + x*sqrt((a^8*b^7*x^2*sqrt(a^8/b^13) + sqrt(b*x^4 + a)*a^12)
/x^2))) + 45*(b^4*x^4 + a*b^3)*(a^8/b^13)^(1/4)*log(91125*(b^10*x*(a^8/b^13)^(3/
4) + (b*x^4 + a)^(1/4)*a^6)/x) - 45*(b^4*x^4 + a*b^3)*(a^8/b^13)^(1/4)*log(-9112
5*(b^10*x*(a^8/b^13)^(3/4) - (b*x^4 + a)^(1/4)*a^6)/x) + 4*(4*b^2*x^9 - 9*a*b*x^
5 - 45*a^2*x)*(b*x^4 + a)^(3/4))/(b^4*x^4 + a*b^3)

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Sympy [A]  time = 9.44878, size = 37, normalized size = 0.3 \[ \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12/(b*x**4+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{12}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x^12/(b*x^4 + a)^(5/4), x)