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

Optimal. Leaf size=101 \[ \frac{a^4 \left (a+b x^4\right )^{5/4}}{5 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^5}-\frac{4 a \left (a+b x^4\right )^{17/4}}{17 b^5} \]

[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 = 0.133534, 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 \[ \frac{a^4 \left (a+b x^4\right )^{5/4}}{5 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{9/4}}{9 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac{\left (a+b x^4\right )^{21/4}}{21 b^5}-\frac{4 a \left (a+b x^4\right )^{17/4}}{17 b^5} \]

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 = 17.4974, size = 92, normalized size = 0.91 \[ \frac{a^{4} \left (a + b x^{4}\right )^{\frac{5}{4}}}{5 b^{5}} - \frac{4 a^{3} \left (a + b x^{4}\right )^{\frac{9}{4}}}{9 b^{5}} + \frac{6 a^{2} \left (a + b x^{4}\right )^{\frac{13}{4}}}{13 b^{5}} - \frac{4 a \left (a + b x^{4}\right )^{\frac{17}{4}}}{17 b^{5}} + \frac{\left (a + b x^{4}\right )^{\frac{21}{4}}}{21 b^{5}} \]

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 = 0.0335333, size = 72, normalized size = 0.71 \[ \frac{\sqrt [4]{a+b x^4} \left (2048 a^5-512 a^4 b x^4+320 a^3 b^2 x^8-240 a^2 b^3 x^{12}+195 a b^4 x^{16}+3315 b^5 x^{20}\right )}{69615 b^5} \]

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^1
2 + 195*a*b^4*x^16 + 3315*b^5*x^20))/(69615*b^5)

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Maple [A]  time = 0.01, size = 58, 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 = 1.42774, size = 109, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{21}{4}}}{21 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a}{17 \, b^{5}} + \frac{6 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2}}{13 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3}}{9 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{4}}{5 \, b^{5}} \]

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 = 0.242951, size = 92, normalized size = 0.91 \[ \frac{{\left (3315 \, b^{5} x^{20} + 195 \, a b^{4} x^{16} - 240 \, a^{2} b^{3} x^{12} + 320 \, a^{3} b^{2} x^{8} - 512 \, a^{4} b x^{4} + 2048 \, a^{5}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{69615 \, b^{5}} \]

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 = 67.0538, size = 134, normalized size = 1.33 \[ \begin{cases} \frac{2048 a^{5} \sqrt [4]{a + b x^{4}}}{69615 b^{5}} - \frac{512 a^{4} x^{4} \sqrt [4]{a + b x^{4}}}{69615 b^{4}} + \frac{64 a^{3} x^{8} \sqrt [4]{a + b x^{4}}}{13923 b^{3}} - \frac{16 a^{2} x^{12} \sqrt [4]{a + b x^{4}}}{4641 b^{2}} + \frac{a x^{16} \sqrt [4]{a + b x^{4}}}{357 b} + \frac{x^{20} \sqrt [4]{a + b x^{4}}}{21} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{20}}{20} & \text{otherwise} \end{cases} \]

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 = 0.215951, size = 96, normalized size = 0.95 \[ \frac{3315 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} - 16380 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a + 32130 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2} - 30940 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3} + 13923 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{4}}{69615 \, b^{5}} \]

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)^(21/4) - 16380*(b*x^4 + a)^(17/4)*a + 32130*(b*x^4 + a
)^(13/4)*a^2 - 30940*(b*x^4 + a)^(9/4)*a^3 + 13923*(b*x^4 + a)^(5/4)*a^4)/b^5