3.1073 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^4} \, dx\)

Optimal. Leaf size=99 \[ -\frac{5 \sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{6 \left (a+b x^4\right )^{3/4}}+\frac{5}{6} b x \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.112679, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{5 \sqrt{a} b^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{6 \left (a+b x^4\right )^{3/4}}+\frac{5}{6} b x \sqrt [4]{a+b x^4}-\frac{\left (a+b x^4\right )^{5/4}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.0411, size = 88, normalized size = 0.89 \[ - \frac{5 \sqrt{a} b^{\frac{3}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{6 \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{5 b x \sqrt [4]{a + b x^{4}}}{6} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0488809, size = 80, normalized size = 0.81 \[ \frac{5 a b x \left (\frac{a+b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )}{6 \left (a+b x^4\right )^{3/4}}+\sqrt [4]{a+b x^4} \left (\frac{b x}{2}-\frac{a}{3 x^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-a/(3*x^3) + (b*x)/2)*(a + b*x^4)^(1/4) + (5*a*b*x*((a + b*x^4)/a)^(3/4)*Hyperg
eometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(6*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.041, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(5/4)/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(5/4)/x^4, x)

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Sympy [A]  time = 6.34283, size = 42, normalized size = 0.42 \[ \frac{a^{\frac{5}{4}} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/4)*gamma(-3/4)*hyper((-5/4, -3/4), (1/4,), b*x**4*exp_polar(I*pi)/a)/(4*x*
*3*gamma(1/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(5/4)/x^4, x)