3.1007 \(\int x^{12} \sqrt [4]{a+b x^4} \, dx\)

Optimal. Leaf size=150 \[ \frac{3 a^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{112 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b} \]

[Out]

(3*a^3*x*(a + b*x^4)^(1/4))/(112*b^3) - (3*a^2*x^5*(a + b*x^4)^(1/4))/(280*b^2)
+ (a*x^9*(a + b*x^4)^(1/4))/(140*b) + (x^13*(a + b*x^4)^(1/4))/14 + (3*a^(7/2)*(
1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(112*b^(
5/2)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.21413, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{3 a^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{112 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{3 a^3 x \sqrt [4]{a+b x^4}}{112 b^3}-\frac{3 a^2 x^5 \sqrt [4]{a+b x^4}}{280 b^2}+\frac{1}{14} x^{13} \sqrt [4]{a+b x^4}+\frac{a x^9 \sqrt [4]{a+b x^4}}{140 b} \]

Antiderivative was successfully verified.

[In]  Int[x^12*(a + b*x^4)^(1/4),x]

[Out]

(3*a^3*x*(a + b*x^4)^(1/4))/(112*b^3) - (3*a^2*x^5*(a + b*x^4)^(1/4))/(280*b^2)
+ (a*x^9*(a + b*x^4)^(1/4))/(140*b) + (x^13*(a + b*x^4)^(1/4))/14 + (3*a^(7/2)*(
1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(112*b^(
5/2)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 23.3381, size = 136, normalized size = 0.91 \[ \frac{3 a^{\frac{7}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{112 b^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{3 a^{3} x \sqrt [4]{a + b x^{4}}}{112 b^{3}} - \frac{3 a^{2} x^{5} \sqrt [4]{a + b x^{4}}}{280 b^{2}} + \frac{a x^{9} \sqrt [4]{a + b x^{4}}}{140 b} + \frac{x^{13} \sqrt [4]{a + b x^{4}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12*(b*x**4+a)**(1/4),x)

[Out]

3*a**(7/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/
2, 2)/(112*b**(5/2)*(a + b*x**4)**(3/4)) + 3*a**3*x*(a + b*x**4)**(1/4)/(112*b**
3) - 3*a**2*x**5*(a + b*x**4)**(1/4)/(280*b**2) + a*x**9*(a + b*x**4)**(1/4)/(14
0*b) + x**13*(a + b*x**4)**(1/4)/14

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Mathematica [C]  time = 0.0591345, size = 101, normalized size = 0.67 \[ \frac{-15 a^4 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+15 a^4 x+9 a^3 b x^5-2 a^2 b^2 x^9+44 a b^3 x^{13}+40 b^4 x^{17}}{560 b^3 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12*(a + b*x^4)^(1/4),x]

[Out]

(15*a^4*x + 9*a^3*b*x^5 - 2*a^2*b^2*x^9 + 44*a*b^3*x^13 + 40*b^4*x^17 - 15*a^4*x
*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(560*b^3*
(a + b*x^4)^(3/4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{x}^{12}\sqrt [4]{b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12*(b*x^4+a)^(1/4),x)

[Out]

int(x^12*(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{12}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^12,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(1/4)*x^12, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{12}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^12,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(1/4)*x^12, x)

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Sympy [A]  time = 9.95379, size = 39, normalized size = 0.26 \[ \frac{\sqrt [4]{a} x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12*(b*x**4+a)**(1/4),x)

[Out]

a**(1/4)*x**13*gamma(13/4)*hyper((-1/4, 13/4), (17/4,), b*x**4*exp_polar(I*pi)/a
)/(4*gamma(17/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{12}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)*x^12,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(1/4)*x^12, x)