3.1137 \(\int \frac{x^{19}}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{a^4}{b^5 \sqrt [4]{a+b x^4}}-\frac{4 a^3 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac{4 a \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^5} \]

[Out]

-(a^4/(b^5*(a + b*x^4)^(1/4))) - (4*a^3*(a + b*x^4)^(3/4))/(3*b^5) + (6*a^2*(a +
b*x^4)^(7/4))/(7*b^5) - (4*a*(a + b*x^4)^(11/4))/(11*b^5) + (a + b*x^4)^(15/4)/
(15*b^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.128525, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{a^4}{b^5 \sqrt [4]{a+b x^4}}-\frac{4 a^3 \left (a+b x^4\right )^{3/4}}{3 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{7/4}}{7 b^5}-\frac{4 a \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^19/(a + b*x^4)^(5/4),x]

[Out]

-(a^4/(b^5*(a + b*x^4)^(1/4))) - (4*a^3*(a + b*x^4)^(3/4))/(3*b^5) + (6*a^2*(a +
b*x^4)^(7/4))/(7*b^5) - (4*a*(a + b*x^4)^(11/4))/(11*b^5) + (a + b*x^4)^(15/4)/
(15*b^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.3374, size = 90, normalized size = 0.91 \[ - \frac{a^{4}}{b^{5} \sqrt [4]{a + b x^{4}}} - \frac{4 a^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{3 b^{5}} + \frac{6 a^{2} \left (a + b x^{4}\right )^{\frac{7}{4}}}{7 b^{5}} - \frac{4 a \left (a + b x^{4}\right )^{\frac{11}{4}}}{11 b^{5}} + \frac{\left (a + b x^{4}\right )^{\frac{15}{4}}}{15 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**19/(b*x**4+a)**(5/4),x)

[Out]

-a**4/(b**5*(a + b*x**4)**(1/4)) - 4*a**3*(a + b*x**4)**(3/4)/(3*b**5) + 6*a**2*
(a + b*x**4)**(7/4)/(7*b**5) - 4*a*(a + b*x**4)**(11/4)/(11*b**5) + (a + b*x**4)
**(15/4)/(15*b**5)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0472433, size = 61, normalized size = 0.62 \[ \frac{-2048 a^4-512 a^3 b x^4+192 a^2 b^2 x^8-112 a b^3 x^{12}+77 b^4 x^{16}}{1155 b^5 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^19/(a + b*x^4)^(5/4),x]

[Out]

(-2048*a^4 - 512*a^3*b*x^4 + 192*a^2*b^2*x^8 - 112*a*b^3*x^12 + 77*b^4*x^16)/(11
55*b^5*(a + b*x^4)^(1/4))

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 58, normalized size = 0.6 \[ -{\frac{-77\,{x}^{16}{b}^{4}+112\,a{x}^{12}{b}^{3}-192\,{a}^{2}{x}^{8}{b}^{2}+512\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{1155\,{b}^{5}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19/(b*x^4+a)^(5/4),x)

[Out]

-1/1155*(-77*b^4*x^16+112*a*b^3*x^12-192*a^2*b^2*x^8+512*a^3*b*x^4+2048*a^4)/(b*
x^4+a)^(1/4)/b^5

_______________________________________________________________________________________

Maxima [A]  time = 1.41926, size = 109, normalized size = 1.1 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{15 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a}{11 \, b^{5}} + \frac{6 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2}}{7 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{3}}{3 \, b^{5}} - \frac{a^{4}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

1/15*(b*x^4 + a)^(15/4)/b^5 - 4/11*(b*x^4 + a)^(11/4)*a/b^5 + 6/7*(b*x^4 + a)^(7
/4)*a^2/b^5 - 4/3*(b*x^4 + a)^(3/4)*a^3/b^5 - a^4/((b*x^4 + a)^(1/4)*b^5)

_______________________________________________________________________________________

Fricas [A]  time = 0.22977, size = 77, normalized size = 0.78 \[ \frac{77 \, b^{4} x^{16} - 112 \, a b^{3} x^{12} + 192 \, a^{2} b^{2} x^{8} - 512 \, a^{3} b x^{4} - 2048 \, a^{4}}{1155 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

1/1155*(77*b^4*x^16 - 112*a*b^3*x^12 + 192*a^2*b^2*x^8 - 512*a^3*b*x^4 - 2048*a^
4)/((b*x^4 + a)^(1/4)*b^5)

_______________________________________________________________________________________

Sympy [A]  time = 50.527, size = 116, normalized size = 1.17 \[ \begin{cases} - \frac{2048 a^{4}}{1155 b^{5} \sqrt [4]{a + b x^{4}}} - \frac{512 a^{3} x^{4}}{1155 b^{4} \sqrt [4]{a + b x^{4}}} + \frac{64 a^{2} x^{8}}{385 b^{3} \sqrt [4]{a + b x^{4}}} - \frac{16 a x^{12}}{165 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{16}}{15 b \sqrt [4]{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{20}}{20 a^{\frac{5}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**19/(b*x**4+a)**(5/4),x)

[Out]

Piecewise((-2048*a**4/(1155*b**5*(a + b*x**4)**(1/4)) - 512*a**3*x**4/(1155*b**4
*(a + b*x**4)**(1/4)) + 64*a**2*x**8/(385*b**3*(a + b*x**4)**(1/4)) - 16*a*x**12
/(165*b**2*(a + b*x**4)**(1/4)) + x**16/(15*b*(a + b*x**4)**(1/4)), Ne(b, 0)), (
x**20/(20*a**(5/4)), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222133, size = 96, normalized size = 0.97 \[ \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} - 420 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a + 990 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2} - 1540 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{3} - \frac{1155 \, a^{4}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}}{1155 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

1/1155*(77*(b*x^4 + a)^(15/4) - 420*(b*x^4 + a)^(11/4)*a + 990*(b*x^4 + a)^(7/4)
*a^2 - 1540*(b*x^4 + a)^(3/4)*a^3 - 1155*a^4/(b*x^4 + a)^(1/4))/b^5