3.1072 \(\int \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=97 \[ -\frac{5 a^{3/2} \sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}+\frac{5}{12} a x \sqrt [4]{a+b x^4} \]

[Out]

(5*a*x*(a + b*x^4)^(1/4))/12 + (x*(a + b*x^4)^(5/4))/6 - (5*a^(3/2)*Sqrt[b]*(1 +
a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*(a + b*
x^4)^(3/4))

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Rubi [A]  time = 0.107154, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{5 a^{3/2} \sqrt{b} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 \left (a+b x^4\right )^{3/4}}+\frac{1}{6} x \left (a+b x^4\right )^{5/4}+\frac{5}{12} a x \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4),x]

[Out]

(5*a*x*(a + b*x^4)^(1/4))/12 + (x*(a + b*x^4)^(5/4))/6 - (5*a^(3/2)*Sqrt[b]*(1 +
a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*(a + b*
x^4)^(3/4))

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Rubi in Sympy [A]  time = 11.4571, size = 87, normalized size = 0.9 \[ - \frac{5 a^{\frac{3}{2}} \sqrt{b} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{12 \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{5 a x \sqrt [4]{a + b x^{4}}}{12} + \frac{x \left (a + b x^{4}\right )^{\frac{5}{4}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*a**(3/2)*sqrt(b)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b
)*x**2))/2, 2)/(12*(a + b*x**4)**(3/4)) + 5*a*x*(a + b*x**4)**(1/4)/12 + x*(a +
b*x**4)**(5/4)/6

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Mathematica [C]  time = 0.0418823, size = 76, normalized size = 0.78 \[ \frac{5 a^2 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+7 a^2 x+9 a b x^5+2 b^2 x^9}{12 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4),x]

[Out]

(7*a^2*x + 9*a*b*x^5 + 2*b^2*x^9 + 5*a^2*x*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2
F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(12*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(5/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(5/4), x)

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Sympy [A]  time = 5.33077, size = 37, normalized size = 0.38 \[ \frac{a^{\frac{5}{4}} x \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(5/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(5/4), x)