3.1045 \(\int x^{19} \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=101 \[ \frac{a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}+\frac{\left (a+b x^4\right )^{25/4}}{25 b^5}-\frac{4 a \left (a+b x^4\right )^{21/4}}{21 b^5} \]

[Out]

(a^4*(a + b*x^4)^(9/4))/(9*b^5) - (4*a^3*(a + b*x^4)^(13/4))/(13*b^5) + (6*a^2*(
a + b*x^4)^(17/4))/(17*b^5) - (4*a*(a + b*x^4)^(21/4))/(21*b^5) + (a + b*x^4)^(2
5/4)/(25*b^5)

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Rubi [A]  time = 0.130615, 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 \[ \frac{a^4 \left (a+b x^4\right )^{9/4}}{9 b^5}-\frac{4 a^3 \left (a+b x^4\right )^{13/4}}{13 b^5}+\frac{6 a^2 \left (a+b x^4\right )^{17/4}}{17 b^5}+\frac{\left (a+b x^4\right )^{25/4}}{25 b^5}-\frac{4 a \left (a+b x^4\right )^{21/4}}{21 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^19*(a + b*x^4)^(5/4),x]

[Out]

(a^4*(a + b*x^4)^(9/4))/(9*b^5) - (4*a^3*(a + b*x^4)^(13/4))/(13*b^5) + (6*a^2*(
a + b*x^4)^(17/4))/(17*b^5) - (4*a*(a + b*x^4)^(21/4))/(21*b^5) + (a + b*x^4)^(2
5/4)/(25*b^5)

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Rubi in Sympy [A]  time = 17.647, size = 92, normalized size = 0.91 \[ \frac{a^{4} \left (a + b x^{4}\right )^{\frac{9}{4}}}{9 b^{5}} - \frac{4 a^{3} \left (a + b x^{4}\right )^{\frac{13}{4}}}{13 b^{5}} + \frac{6 a^{2} \left (a + b x^{4}\right )^{\frac{17}{4}}}{17 b^{5}} - \frac{4 a \left (a + b x^{4}\right )^{\frac{21}{4}}}{21 b^{5}} + \frac{\left (a + b x^{4}\right )^{\frac{25}{4}}}{25 b^{5}} \]

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*(a + b*x**4)**(9/4)/(9*b**5) - 4*a**3*(a + b*x**4)**(13/4)/(13*b**5) + 6*a*
*2*(a + b*x**4)**(17/4)/(17*b**5) - 4*a*(a + b*x**4)**(21/4)/(21*b**5) + (a + b*
x**4)**(25/4)/(25*b**5)

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Mathematica [A]  time = 0.0470077, size = 61, normalized size = 0.6 \[ \frac{\left (a+b x^4\right )^{9/4} \left (2048 a^4-4608 a^3 b x^4+7488 a^2 b^2 x^8-10608 a b^3 x^{12}+13923 b^4 x^{16}\right )}{348075 b^5} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^4)^(9/4)*(2048*a^4 - 4608*a^3*b*x^4 + 7488*a^2*b^2*x^8 - 10608*a*b^3*x
^12 + 13923*b^4*x^16))/(348075*b^5)

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Maple [A]  time = 0.011, size = 58, normalized size = 0.6 \[{\frac{13923\,{x}^{16}{b}^{4}-10608\,a{x}^{12}{b}^{3}+7488\,{a}^{2}{x}^{8}{b}^{2}-4608\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{348075\,{b}^{5}} \left ( b{x}^{4}+a \right ) ^{{\frac{9}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19*(b*x^4+a)^(5/4),x)

[Out]

1/348075*(b*x^4+a)^(9/4)*(13923*b^4*x^16-10608*a*b^3*x^12+7488*a^2*b^2*x^8-4608*
a^3*b*x^4+2048*a^4)/b^5

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Maxima [A]  time = 1.42591, size = 109, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{25}{4}}}{25 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} a}{21 \, b^{5}} + \frac{6 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a^{2}}{17 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{3}}{13 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{4}}{9 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/25*(b*x^4 + a)^(25/4)/b^5 - 4/21*(b*x^4 + a)^(21/4)*a/b^5 + 6/17*(b*x^4 + a)^(
17/4)*a^2/b^5 - 4/13*(b*x^4 + a)^(13/4)*a^3/b^5 + 1/9*(b*x^4 + a)^(9/4)*a^4/b^5

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Fricas [A]  time = 0.300135, size = 107, normalized size = 1.06 \[ \frac{{\left (13923 \, b^{6} x^{24} + 17238 \, a b^{5} x^{20} + 195 \, a^{2} b^{4} x^{16} - 240 \, a^{3} b^{3} x^{12} + 320 \, a^{4} b^{2} x^{8} - 512 \, a^{5} b x^{4} + 2048 \, a^{6}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{348075 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/348075*(13923*b^6*x^24 + 17238*a*b^5*x^20 + 195*a^2*b^4*x^16 - 240*a^3*b^3*x^1
2 + 320*a^4*b^2*x^8 - 512*a^5*b*x^4 + 2048*a^6)*(b*x^4 + a)^(1/4)/b^5

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220839, size = 219, normalized size = 2.17 \[ \frac{\frac{5 \,{\left (3315 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} - 16380 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a + 32130 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{2} - 30940 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{3} + 13923 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{4}\right )} a}{b^{4}} + \frac{13923 \,{\left (b x^{4} + a\right )}^{\frac{25}{4}} - 82875 \,{\left (b x^{4} + a\right )}^{\frac{21}{4}} a + 204750 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} a^{2} - 267750 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a^{3} + 193375 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{4} - 69615 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{5}}{b^{4}}}{348075 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/348075*(5*(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)*a/
b^4 + (13923*(b*x^4 + a)^(25/4) - 82875*(b*x^4 + a)^(21/4)*a + 204750*(b*x^4 + a
)^(17/4)*a^2 - 267750*(b*x^4 + a)^(13/4)*a^3 + 193375*(b*x^4 + a)^(9/4)*a^4 - 69
615*(b*x^4 + a)^(5/4)*a^5)/b^4)/b