3.1078 \(\int \frac{x^{19}}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=101 \[ \frac{a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^5}-\frac{4 a \left (a+b x^4\right )^{15/4}}{15 b^5} \]

[Out]

(a^4*(a + b*x^4)^(3/4))/(3*b^5) - (4*a^3*(a + b*x^4)^(7/4))/(7*b^5) + (6*a^2*(a
+ b*x^4)^(11/4))/(11*b^5) - (4*a*(a + b*x^4)^(15/4))/(15*b^5) + (a + b*x^4)^(19/
4)/(19*b^5)

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Rubi [A]  time = 0.125861, antiderivative size = 101, 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 \[ \frac{a^4 \left (a+b x^4\right )^{3/4}}{3 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{7/4}}{7 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{11/4}}{11 b^5}+\frac{\left (a+b x^4\right )^{19/4}}{19 b^5}-\frac{4 a \left (a+b x^4\right )^{15/4}}{15 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^19/(a + b*x^4)^(1/4),x]

[Out]

(a^4*(a + b*x^4)^(3/4))/(3*b^5) - (4*a^3*(a + b*x^4)^(7/4))/(7*b^5) + (6*a^2*(a
+ b*x^4)^(11/4))/(11*b^5) - (4*a*(a + b*x^4)^(15/4))/(15*b^5) + (a + b*x^4)^(19/
4)/(19*b^5)

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Rubi in Sympy [A]  time = 17.3831, size = 92, normalized size = 0.91 \[ \frac{a^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{3 b^{5}} - \frac{4 a^{3} \left (a + b x^{4}\right )^{\frac{7}{4}}}{7 b^{5}} + \frac{6 a^{2} \left (a + b x^{4}\right )^{\frac{11}{4}}}{11 b^{5}} - \frac{4 a \left (a + b x^{4}\right )^{\frac{15}{4}}}{15 b^{5}} + \frac{\left (a + b x^{4}\right )^{\frac{19}{4}}}{19 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**19/(b*x**4+a)**(1/4),x)

[Out]

a**4*(a + b*x**4)**(3/4)/(3*b**5) - 4*a**3*(a + b*x**4)**(7/4)/(7*b**5) + 6*a**2
*(a + b*x**4)**(11/4)/(11*b**5) - 4*a*(a + b*x**4)**(15/4)/(15*b**5) + (a + b*x*
*4)**(19/4)/(19*b**5)

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Mathematica [A]  time = 0.0402484, size = 61, normalized size = 0.6 \[ \frac{\left (a+b x^4\right )^{3/4} \left (2048 a^4-1536 a^3 b x^4+1344 a^2 b^2 x^8-1232 a b^3 x^{12}+1155 b^4 x^{16}\right )}{21945 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^19/(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(3/4)*(2048*a^4 - 1536*a^3*b*x^4 + 1344*a^2*b^2*x^8 - 1232*a*b^3*x^
12 + 1155*b^4*x^16))/(21945*b^5)

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Maple [A]  time = 0.012, size = 58, normalized size = 0.6 \[{\frac{1155\,{x}^{16}{b}^{4}-1232\,a{x}^{12}{b}^{3}+1344\,{a}^{2}{x}^{8}{b}^{2}-1536\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{21945\,{b}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19/(b*x^4+a)^(1/4),x)

[Out]

1/21945*(b*x^4+a)^(3/4)*(1155*b^4*x^16-1232*a*b^3*x^12+1344*a^2*b^2*x^8-1536*a^3
*b*x^4+2048*a^4)/b^5

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Maxima [A]  time = 1.44496, size = 109, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{19}{4}}}{19 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a}{15 \, b^{5}} + \frac{6 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2}}{11 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3}}{7 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{4}}{3 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

1/19*(b*x^4 + a)^(19/4)/b^5 - 4/15*(b*x^4 + a)^(15/4)*a/b^5 + 6/11*(b*x^4 + a)^(
11/4)*a^2/b^5 - 4/7*(b*x^4 + a)^(7/4)*a^3/b^5 + 1/3*(b*x^4 + a)^(3/4)*a^4/b^5

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Fricas [A]  time = 0.301186, size = 77, normalized size = 0.76 \[ \frac{{\left (1155 \, b^{4} x^{16} - 1232 \, a b^{3} x^{12} + 1344 \, a^{2} b^{2} x^{8} - 1536 \, a^{3} b x^{4} + 2048 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21945 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

1/21945*(1155*b^4*x^16 - 1232*a*b^3*x^12 + 1344*a^2*b^2*x^8 - 1536*a^3*b*x^4 + 2
048*a^4)*(b*x^4 + a)^(3/4)/b^5

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Sympy [A]  time = 53.1374, size = 116, normalized size = 1.15 \[ \begin{cases} \frac{2048 a^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{21945 b^{5}} - \frac{512 a^{3} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7315 b^{4}} + \frac{64 a^{2} x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1045 b^{3}} - \frac{16 a x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{285 b^{2}} + \frac{x^{16} \left (a + b x^{4}\right )^{\frac{3}{4}}}{19 b} & \text{for}\: b \neq 0 \\\frac{x^{20}}{20 \sqrt [4]{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**19/(b*x**4+a)**(1/4),x)

[Out]

Piecewise((2048*a**4*(a + b*x**4)**(3/4)/(21945*b**5) - 512*a**3*x**4*(a + b*x**
4)**(3/4)/(7315*b**4) + 64*a**2*x**8*(a + b*x**4)**(3/4)/(1045*b**3) - 16*a*x**1
2*(a + b*x**4)**(3/4)/(285*b**2) + x**16*(a + b*x**4)**(3/4)/(19*b), Ne(b, 0)),
(x**20/(20*a**(1/4)), True))

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GIAC/XCAS [A]  time = 0.215521, size = 96, normalized size = 0.95 \[ \frac{1155 \,{\left (b x^{4} + a\right )}^{\frac{19}{4}} - 5852 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} a + 11970 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a^{2} - 12540 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{3} + 7315 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{4}}{21945 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

1/21945*(1155*(b*x^4 + a)^(19/4) - 5852*(b*x^4 + a)^(15/4)*a + 11970*(b*x^4 + a)
^(11/4)*a^2 - 12540*(b*x^4 + a)^(7/4)*a^3 + 7315*(b*x^4 + a)^(3/4)*a^4)/b^5