3.1071 \(\int x^4 \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=123 \[ \frac{a^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{24 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4} \]

[Out]

(a^2*x*(a + b*x^4)^(1/4))/(24*b) + (a*x^5*(a + b*x^4)^(1/4))/12 + (x^5*(a + b*x^
4)^(5/4))/10 + (a^(5/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)
/Sqrt[a]]/2, 2])/(24*Sqrt[b]*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.161987, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{a^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{24 \sqrt{b} \left (a+b x^4\right )^{3/4}}+\frac{a^2 x \sqrt [4]{a+b x^4}}{24 b}+\frac{1}{10} x^5 \left (a+b x^4\right )^{5/4}+\frac{1}{12} a x^5 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^4*(a + b*x^4)^(5/4),x]

[Out]

(a^2*x*(a + b*x^4)^(1/4))/(24*b) + (a*x^5*(a + b*x^4)^(1/4))/12 + (x^5*(a + b*x^
4)^(5/4))/10 + (a^(5/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)
/Sqrt[a]]/2, 2])/(24*Sqrt[b]*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 18.5605, size = 105, normalized size = 0.85 \[ \frac{a^{\frac{5}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{24 \sqrt{b} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{a^{2} x \sqrt [4]{a + b x^{4}}}{24 b} + \frac{a x^{5} \sqrt [4]{a + b x^{4}}}{12} + \frac{x^{5} \left (a + b x^{4}\right )^{\frac{5}{4}}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2,
2)/(24*sqrt(b)*(a + b*x**4)**(3/4)) + a**2*x*(a + b*x**4)**(1/4)/(24*b) + a*x**
5*(a + b*x**4)**(1/4)/12 + x**5*(a + b*x**4)**(5/4)/10

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Mathematica [C]  time = 0.0518648, size = 90, normalized size = 0.73 \[ \frac{-5 a^3 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+5 a^3 x+27 a^2 b x^5+34 a b^2 x^9+12 b^3 x^{13}}{120 b \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(5*a^3*x + 27*a^2*b*x^5 + 34*a*b^2*x^9 + 12*b^3*x^13 - 5*a^3*x*(1 + (b*x^4)/a)^(
3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(120*b*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{x}^{4} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^4+a)^(5/4),x)

[Out]

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

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

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. \[{\rm integral}\left ({\left (b x^{8} + a x^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \]

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^8 + a*x^4)*(b*x^4 + a)^(1/4), x)

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Sympy [A]  time = 10.3812, size = 39, normalized size = 0.32 \[ \frac{a^{\frac{5}{4}} x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(5/4)*x**5*gamma(5/4)*hyper((-5/4, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4
*gamma(9/4))

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

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)