3.1145 \(\int \frac{x^{13}}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{8 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^4}}+\frac{4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac{2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac{x^{10}}{9 b \sqrt [4]{a+b x^4}} \]

[Out]

(4*a^2*x^2)/(3*b^3*(a + b*x^4)^(1/4)) - (2*a*x^6)/(9*b^2*(a + b*x^4)^(1/4)) + x^
10/(9*b*(a + b*x^4)^(1/4)) - (8*a^(5/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(
Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*b^(7/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.202789, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{8 a^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 b^{7/2} \sqrt [4]{a+b x^4}}+\frac{4 a^2 x^2}{3 b^3 \sqrt [4]{a+b x^4}}-\frac{2 a x^6}{9 b^2 \sqrt [4]{a+b x^4}}+\frac{x^{10}}{9 b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^13/(a + b*x^4)^(5/4),x]

[Out]

(4*a^2*x^2)/(3*b^3*(a + b*x^4)^(1/4)) - (2*a*x^6)/(9*b^2*(a + b*x^4)^(1/4)) + x^
10/(9*b*(a + b*x^4)^(1/4)) - (8*a^(5/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(
Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*b^(7/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{4 a^{2} x^{2}}{3 b^{3} \sqrt [4]{a + b x^{4}}} + \frac{4 a^{2} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{3 b^{3}} - \frac{2 a x^{6}}{9 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{10}}{9 b \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(b*x**4+a)**(5/4),x)

[Out]

-4*a**2*x**2/(3*b**3*(a + b*x**4)**(1/4)) + 4*a**2*Integral((a + b*x**2)**(-1/4)
, (x, x**2))/(3*b**3) - 2*a*x**6/(9*b**2*(a + b*x**4)**(1/4)) + x**10/(9*b*(a +
b*x**4)**(1/4))

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Mathematica [C]  time = 0.0736067, size = 79, normalized size = 0.62 \[ \frac{x^2 \left (12 a^2 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-12 a^2-2 a b x^4+b^2 x^8\right )}{9 b^3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^13/(a + b*x^4)^(5/4),x]

[Out]

(x^2*(-12*a^2 - 2*a*b*x^4 + b^2*x^8 + 12*a^2*(1 + (b*x^4)/a)^(1/4)*Hypergeometri
c2F1[1/4, 1/2, 3/2, -((b*x^4)/a)]))/(9*b^3*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.081, size = 0, normalized size = 0. \[ \int{{x}^{13} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(b*x^4+a)^(5/4),x)

[Out]

int(x^13/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

integrate(x^13/(b*x^4 + a)^(5/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

integral(x^13/(b*x^4 + a)^(5/4), x)

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Sympy [A]  time = 8.51519, size = 27, normalized size = 0.21 \[ \frac{x^{14}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{14 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13/(b*x**4+a)**(5/4),x)

[Out]

x**14*hyper((5/4, 7/2), (9/2,), b*x**4*exp_polar(I*pi)/a)/(14*a**(5/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x^13/(b*x^4 + a)^(5/4), x)