3.1160 \(\int \frac{x^{14}}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=129 \[ \frac{77 a^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{7/2} \sqrt [4]{a+b x^4}}+\frac{77 a^2 x^3}{120 b^3 \sqrt [4]{a+b x^4}}-\frac{11 a x^7}{60 b^2 \sqrt [4]{a+b x^4}}+\frac{x^{11}}{10 b \sqrt [4]{a+b x^4}} \]

[Out]

(77*a^2*x^3)/(120*b^3*(a + b*x^4)^(1/4)) - (11*a*x^7)/(60*b^2*(a + b*x^4)^(1/4))
+ x^11/(10*b*(a + b*x^4)^(1/4)) + (77*a^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE
[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*b^(7/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.185745, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{77 a^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 b^{7/2} \sqrt [4]{a+b x^4}}+\frac{77 a^2 x^3}{120 b^3 \sqrt [4]{a+b x^4}}-\frac{11 a x^7}{60 b^2 \sqrt [4]{a+b x^4}}+\frac{x^{11}}{10 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^14/(a + b*x^4)^(5/4),x]

[Out]

(77*a^2*x^3)/(120*b^3*(a + b*x^4)^(1/4)) - (11*a*x^7)/(60*b^2*(a + b*x^4)^(1/4))
+ x^11/(10*b*(a + b*x^4)^(1/4)) + (77*a^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE
[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*b^(7/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{77 a^{3} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{80 b^{4} \sqrt [4]{a + b x^{4}}} + \frac{77 a^{3}}{40 b^{4} x \sqrt [4]{a + b x^{4}}} + \frac{77 a^{2} x^{3}}{120 b^{3} \sqrt [4]{a + b x^{4}}} - \frac{11 a x^{7}}{60 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{11}}{10 b \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(b*x**4+a)**(5/4),x)

[Out]

-77*a**3*x*(a/(b*x**4) + 1)**(1/4)*Integral((a*x**2/b + 1)**(-1/4), (x, x**(-2))
)/(80*b**4*(a + b*x**4)**(1/4)) + 77*a**3/(40*b**4*x*(a + b*x**4)**(1/4)) + 77*a
**2*x**3/(120*b**3*(a + b*x**4)**(1/4)) - 11*a*x**7/(60*b**2*(a + b*x**4)**(1/4)
) + x**11/(10*b*(a + b*x**4)**(1/4))

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Mathematica [C]  time = 0.0700862, size = 80, normalized size = 0.62 \[ \frac{x^3 \left (77 a^2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-77 a^2-11 a b x^4+6 b^2 x^8\right )}{60 b^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/(a + b*x^4)^(5/4),x]

[Out]

(x^3*(-77*a^2 - 11*a*b*x^4 + 6*b^2*x^8 + 77*a^2*(1 + (b*x^4)/a)^(1/4)*Hypergeome
tric2F1[1/4, 3/4, 7/4, -((b*x^4)/a)]))/(60*b^3*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{{x}^{14} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(b*x^4+a)^(5/4),x)

[Out]

int(x^14/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate(x^14/(b*x^4 + a)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

integral(x^14/(b*x^4 + a)^(5/4), x)

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Sympy [A]  time = 10.3486, size = 37, normalized size = 0.29 \[ \frac{x^{15} \Gamma \left (\frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{15}{4} \\ \frac{19}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} \Gamma \left (\frac{19}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(b*x**4+a)**(5/4),x)

[Out]

x**15*gamma(15/4)*hyper((5/4, 15/4), (19/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(5
/4)*gamma(19/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x^14/(b*x^4 + a)^(5/4), x)