3.1173 \(\int x^{19} \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=106 \[ -\frac{a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}+\frac{4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac{6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}-\frac{\left (a-b x^4\right )^{21/4}}{21 b^5}+\frac{4 a \left (a-b x^4\right )^{17/4}}{17 b^5} \]

[Out]

-(a^4*(a - b*x^4)^(5/4))/(5*b^5) + (4*a^3*(a - b*x^4)^(9/4))/(9*b^5) - (6*a^2*(a
- b*x^4)^(13/4))/(13*b^5) + (4*a*(a - b*x^4)^(17/4))/(17*b^5) - (a - b*x^4)^(21
/4)/(21*b^5)

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Rubi [A]  time = 0.134877, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{a^4 \left (a-b x^4\right )^{5/4}}{5 b^5}+\frac{4 a^3 \left (a-b x^4\right )^{9/4}}{9 b^5}-\frac{6 a^2 \left (a-b x^4\right )^{13/4}}{13 b^5}-\frac{\left (a-b x^4\right )^{21/4}}{21 b^5}+\frac{4 a \left (a-b x^4\right )^{17/4}}{17 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^19*(a - b*x^4)^(1/4),x]

[Out]

-(a^4*(a - b*x^4)^(5/4))/(5*b^5) + (4*a^3*(a - b*x^4)^(9/4))/(9*b^5) - (6*a^2*(a
- b*x^4)^(13/4))/(13*b^5) + (4*a*(a - b*x^4)^(17/4))/(17*b^5) - (a - b*x^4)^(21
/4)/(21*b^5)

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Rubi in Sympy [A]  time = 19.0817, size = 92, normalized size = 0.87 \[ - \frac{a^{4} \left (a - b x^{4}\right )^{\frac{5}{4}}}{5 b^{5}} + \frac{4 a^{3} \left (a - b x^{4}\right )^{\frac{9}{4}}}{9 b^{5}} - \frac{6 a^{2} \left (a - b x^{4}\right )^{\frac{13}{4}}}{13 b^{5}} + \frac{4 a \left (a - b x^{4}\right )^{\frac{17}{4}}}{17 b^{5}} - \frac{\left (a - b x^{4}\right )^{\frac{21}{4}}}{21 b^{5}} \]

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)**(5/4)/(5*b**5) + 4*a**3*(a - b*x**4)**(9/4)/(9*b**5) - 6*a**
2*(a - b*x**4)**(13/4)/(13*b**5) + 4*a*(a - b*x**4)**(17/4)/(17*b**5) - (a - b*x
**4)**(21/4)/(21*b**5)

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Mathematica [A]  time = 0.0365507, size = 73, normalized size = 0.69 \[ -\frac{\sqrt [4]{a-b x^4} \left (2048 a^5+512 a^4 b x^4+320 a^3 b^2 x^8+240 a^2 b^3 x^{12}+195 a b^4 x^{16}-3315 b^5 x^{20}\right )}{69615 b^5} \]

Antiderivative was successfully verified.

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

[Out]

-((a - b*x^4)^(1/4)*(2048*a^5 + 512*a^4*b*x^4 + 320*a^3*b^2*x^8 + 240*a^2*b^3*x^
12 + 195*a*b^4*x^16 - 3315*b^5*x^20))/(69615*b^5)

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Maple [A]  time = 0.012, size = 59, normalized size = 0.6 \[ -{\frac{3315\,{x}^{16}{b}^{4}+3120\,a{x}^{12}{b}^{3}+2880\,{a}^{2}{x}^{8}{b}^{2}+2560\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{69615\,{b}^{5}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19*(-b*x^4+a)^(1/4),x)

[Out]

-1/69615*(-b*x^4+a)^(5/4)*(3315*b^4*x^16+3120*a*b^3*x^12+2880*a^2*b^2*x^8+2560*a
^3*b*x^4+2048*a^4)/b^5

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Maxima [A]  time = 1.445, size = 116, normalized size = 1.09 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{21}{4}}}{21 \, b^{5}} + \frac{4 \,{\left (-b x^{4} + a\right )}^{\frac{17}{4}} a}{17 \, b^{5}} - \frac{6 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}} a^{2}}{13 \, b^{5}} + \frac{4 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} a^{3}}{9 \, b^{5}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{4}}{5 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/21*(-b*x^4 + a)^(21/4)/b^5 + 4/17*(-b*x^4 + a)^(17/4)*a/b^5 - 6/13*(-b*x^4 +
a)^(13/4)*a^2/b^5 + 4/9*(-b*x^4 + a)^(9/4)*a^3/b^5 - 1/5*(-b*x^4 + a)^(5/4)*a^4/
b^5

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Fricas [A]  time = 0.240228, size = 93, normalized size = 0.88 \[ \frac{{\left (3315 \, b^{5} x^{20} - 195 \, a b^{4} x^{16} - 240 \, a^{2} b^{3} x^{12} - 320 \, a^{3} b^{2} x^{8} - 512 \, a^{4} b x^{4} - 2048 \, a^{5}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{69615 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/69615*(3315*b^5*x^20 - 195*a*b^4*x^16 - 240*a^2*b^3*x^12 - 320*a^3*b^2*x^8 - 5
12*a^4*b*x^4 - 2048*a^5)*(-b*x^4 + a)^(1/4)/b^5

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Sympy [A]  time = 66.8801, size = 134, normalized size = 1.26 \[ \begin{cases} - \frac{2048 a^{5} \sqrt [4]{a - b x^{4}}}{69615 b^{5}} - \frac{512 a^{4} x^{4} \sqrt [4]{a - b x^{4}}}{69615 b^{4}} - \frac{64 a^{3} x^{8} \sqrt [4]{a - b x^{4}}}{13923 b^{3}} - \frac{16 a^{2} x^{12} \sqrt [4]{a - b x^{4}}}{4641 b^{2}} - \frac{a x^{16} \sqrt [4]{a - b x^{4}}}{357 b} + \frac{x^{20} \sqrt [4]{a - b x^{4}}}{21} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{20}}{20} & \text{otherwise} \end{cases} \]

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**5*(a - b*x**4)**(1/4)/(69615*b**5) - 512*a**4*x**4*(a - b*x*
*4)**(1/4)/(69615*b**4) - 64*a**3*x**8*(a - b*x**4)**(1/4)/(13923*b**3) - 16*a**
2*x**12*(a - b*x**4)**(1/4)/(4641*b**2) - a*x**16*(a - b*x**4)**(1/4)/(357*b) +
x**20*(a - b*x**4)**(1/4)/21, Ne(b, 0)), (a**(1/4)*x**20/20, True))

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GIAC/XCAS [A]  time = 0.248959, size = 162, normalized size = 1.53 \[ \frac{3315 \,{\left (b x^{4} - a\right )}^{5}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} + 16380 \,{\left (b x^{4} - a\right )}^{4}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a + 32130 \,{\left (b x^{4} - a\right )}^{3}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + 30940 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{3} - 13923 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{4}}{69615 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/69615*(3315*(b*x^4 - a)^5*(-b*x^4 + a)^(1/4) + 16380*(b*x^4 - a)^4*(-b*x^4 + a
)^(1/4)*a + 32130*(b*x^4 - a)^3*(-b*x^4 + a)^(1/4)*a^2 + 30940*(b*x^4 - a)^2*(-b
*x^4 + a)^(1/4)*a^3 - 13923*(-b*x^4 + a)^(5/4)*a^4)/b^5