3.1053 \(\int x^9 \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=146 \[ \frac{4 a^{9/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac{a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4} \]

[Out]

(-2*a^3*x^2*(a + b*x^4)^(1/4))/(231*b^2) + (a^2*x^6*(a + b*x^4)^(1/4))/(231*b) +
(a*x^10*(a + b*x^4)^(1/4))/33 + (x^10*(a + b*x^4)^(5/4))/15 + (4*a^(9/2)*(1 + (
b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(231*b^(5/2)*(a +
b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.235929, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{4 a^{9/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 b^{5/2} \left (a+b x^4\right )^{3/4}}-\frac{2 a^3 x^2 \sqrt [4]{a+b x^4}}{231 b^2}+\frac{a^2 x^6 \sqrt [4]{a+b x^4}}{231 b}+\frac{1}{15} x^{10} \left (a+b x^4\right )^{5/4}+\frac{1}{33} a x^{10} \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^9*(a + b*x^4)^(5/4),x]

[Out]

(-2*a^3*x^2*(a + b*x^4)^(1/4))/(231*b^2) + (a^2*x^6*(a + b*x^4)^(1/4))/(231*b) +
(a*x^10*(a + b*x^4)^(1/4))/33 + (x^10*(a + b*x^4)^(5/4))/15 + (4*a^(9/2)*(1 + (
b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(231*b^(5/2)*(a +
b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.2114, size = 129, normalized size = 0.88 \[ \frac{4 a^{\frac{9}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{231 b^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} - \frac{2 a^{3} x^{2} \sqrt [4]{a + b x^{4}}}{231 b^{2}} + \frac{a^{2} x^{6} \sqrt [4]{a + b x^{4}}}{231 b} + \frac{a x^{10} \sqrt [4]{a + b x^{4}}}{33} + \frac{x^{10} \left (a + b x^{4}\right )^{\frac{5}{4}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9*(b*x**4+a)**(5/4),x)

[Out]

4*a**(9/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b)*x**2/sqrt(a))/2, 2)/(23
1*b**(5/2)*(a + b*x**4)**(3/4)) - 2*a**3*x**2*(a + b*x**4)**(1/4)/(231*b**2) + a
**2*x**6*(a + b*x**4)**(1/4)/(231*b) + a*x**10*(a + b*x**4)**(1/4)/33 + x**10*(a
+ b*x**4)**(5/4)/15

_______________________________________________________________________________________

Mathematica [C]  time = 0.0762219, size = 102, normalized size = 0.7 \[ \frac{x^2 \left (10 a^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-10 a^4-5 a^3 b x^4+117 a^2 b^2 x^8+189 a b^3 x^{12}+77 b^4 x^{16}\right )}{1155 b^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9*(a + b*x^4)^(5/4),x]

[Out]

(x^2*(-10*a^4 - 5*a^3*b*x^4 + 117*a^2*b^2*x^8 + 189*a*b^3*x^12 + 77*b^4*x^16 + 1
0*a^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^4)/a)]))/(11
55*b^2*(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{x}^{9} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9*(b*x^4+a)^(5/4),x)

[Out]

int(x^9*(b*x^4+a)^(5/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^9,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)*x^9, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{13} + a x^{9}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^9,x, algorithm="fricas")

[Out]

integral((b*x^13 + a*x^9)*(b*x^4 + a)^(1/4), x)

_______________________________________________________________________________________

Sympy [A]  time = 23.5186, size = 29, normalized size = 0.2 \[ \frac{a^{\frac{5}{4}} x^{10}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9*(b*x**4+a)**(5/4),x)

[Out]

a**(5/4)*x**10*hyper((-5/4, 5/2), (7/2,), b*x**4*exp_polar(I*pi)/a)/10

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{9}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^9,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)*x^9, x)