3.1056 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^3} \, dx\)

Optimal. Leaf size=98 \[ \frac{5 a^{3/2} \sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{6 \left (a+b x^4\right )^{3/4}}+\frac{5}{6} b x^2 \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{2 x^2} \]

[Out]

(5*b*x^2*(a + b*x^4)^(1/4))/6 - (a + b*x^4)^(5/4)/(2*x^2) + (5*a^(3/2)*Sqrt[b]*(
1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(6*(a + b*x^
4)^(3/4))

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Rubi [A]  time = 0.130865, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 a^{3/2} \sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{6 \left (a+b x^4\right )^{3/4}}+\frac{5}{6} b x^2 \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^3,x]

[Out]

(5*b*x^2*(a + b*x^4)^(1/4))/6 - (a + b*x^4)^(5/4)/(2*x^2) + (5*a^(3/2)*Sqrt[b]*(
1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(6*(a + b*x^
4)^(3/4))

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Rubi in Sympy [A]  time = 12.0032, size = 87, normalized size = 0.89 \[ \frac{5 a^{\frac{3}{2}} \sqrt{b} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{6 \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{5 b x^{2} \sqrt [4]{a + b x^{4}}}{6} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(5/4)/x**3,x)

[Out]

5*a**(3/2)*sqrt(b)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b)*x**2/sqrt(a))/2
, 2)/(6*(a + b*x**4)**(3/4)) + 5*b*x**2*(a + b*x**4)**(1/4)/6 - (a + b*x**4)**(5
/4)/(2*x**2)

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Mathematica [C]  time = 0.049086, size = 79, normalized size = 0.81 \[ \frac{-6 a^2+5 a b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-2 a b x^4+4 b^2 x^8}{12 x^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4)/x^3,x]

[Out]

(-6*a^2 - 2*a*b*x^4 + 4*b^2*x^8 + 5*a*b*x^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometric
2F1[1/2, 3/4, 3/2, -((b*x^4)/a)])/(12*x^2*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^3,x)

[Out]

int((b*x^4+a)^(5/4)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^3,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^3,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(5/4)/x^3, x)

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Sympy [A]  time = 6.30136, size = 32, normalized size = 0.33 \[ - \frac{a^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(5/4)/x**3,x)

[Out]

-a**(5/4)*hyper((-5/4, -1/2), (1/2,), b*x**4*exp_polar(I*pi)/a)/(2*x**2)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^3,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)/x^3, x)