3.1108 \(\int \frac{x^{19}}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=98 \[ \frac{a^4 \sqrt [4]{a+b x^4}}{b^5}-\frac{4 a^3 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac{2 a^2 \left (a+b x^4\right )^{9/4}}{3 b^5}+\frac{\left (a+b x^4\right )^{17/4}}{17 b^5}-\frac{4 a \left (a+b x^4\right )^{13/4}}{13 b^5} \]

[Out]

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

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Rubi [A]  time = 0.129091, antiderivative size = 98, 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 \[ \frac{a^4 \sqrt [4]{a+b x^4}}{b^5}-\frac{4 a^3 \left (a+b x^4\right )^{5/4}}{5 b^5}+\frac{2 a^2 \left (a+b x^4\right )^{9/4}}{3 b^5}+\frac{\left (a+b x^4\right )^{17/4}}{17 b^5}-\frac{4 a \left (a+b x^4\right )^{13/4}}{13 b^5} \]

Antiderivative was successfully verified.

[In]  Int[x^19/(a + b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 17.3654, size = 90, normalized size = 0.92 \[ \frac{a^{4} \sqrt [4]{a + b x^{4}}}{b^{5}} - \frac{4 a^{3} \left (a + b x^{4}\right )^{\frac{5}{4}}}{5 b^{5}} + \frac{2 a^{2} \left (a + b x^{4}\right )^{\frac{9}{4}}}{3 b^{5}} - \frac{4 a \left (a + b x^{4}\right )^{\frac{13}{4}}}{13 b^{5}} + \frac{\left (a + b x^{4}\right )^{\frac{17}{4}}}{17 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**19/(b*x**4+a)**(3/4),x)

[Out]

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

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Mathematica [A]  time = 0.0351482, size = 61, normalized size = 0.62 \[ \frac{\sqrt [4]{a+b x^4} \left (2048 a^4-512 a^3 b x^4+320 a^2 b^2 x^8-240 a b^3 x^{12}+195 b^4 x^{16}\right )}{3315 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^19/(a + b*x^4)^(3/4),x]

[Out]

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

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Maple [A]  time = 0.011, size = 58, normalized size = 0.6 \[{\frac{195\,{x}^{16}{b}^{4}-240\,a{x}^{12}{b}^{3}+320\,{a}^{2}{x}^{8}{b}^{2}-512\,{a}^{3}{x}^{4}b+2048\,{a}^{4}}{3315\,{b}^{5}}\sqrt [4]{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^19/(b*x^4+a)^(3/4),x)

[Out]

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

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Maxima [A]  time = 1.41724, size = 108, normalized size = 1.1 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{17}{4}}}{17 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a}{13 \, b^{5}} + \frac{2 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2}}{3 \, b^{5}} - \frac{4 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3}}{5 \, b^{5}} + \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4}}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.237639, size = 77, normalized size = 0.79 \[ \frac{{\left (195 \, b^{4} x^{16} - 240 \, a b^{3} x^{12} + 320 \, a^{2} b^{2} x^{8} - 512 \, a^{3} b x^{4} + 2048 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{3315 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 50.1296, size = 116, normalized size = 1.18 \[ \begin{cases} \frac{2048 a^{4} \sqrt [4]{a + b x^{4}}}{3315 b^{5}} - \frac{512 a^{3} x^{4} \sqrt [4]{a + b x^{4}}}{3315 b^{4}} + \frac{64 a^{2} x^{8} \sqrt [4]{a + b x^{4}}}{663 b^{3}} - \frac{16 a x^{12} \sqrt [4]{a + b x^{4}}}{221 b^{2}} + \frac{x^{16} \sqrt [4]{a + b x^{4}}}{17 b} & \text{for}\: b \neq 0 \\\frac{x^{20}}{20 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**19/(b*x**4+a)**(3/4),x)

[Out]

Piecewise((2048*a**4*(a + b*x**4)**(1/4)/(3315*b**5) - 512*a**3*x**4*(a + b*x**4
)**(1/4)/(3315*b**4) + 64*a**2*x**8*(a + b*x**4)**(1/4)/(663*b**3) - 16*a*x**12*
(a + b*x**4)**(1/4)/(221*b**2) + x**16*(a + b*x**4)**(1/4)/(17*b), Ne(b, 0)), (x
**20/(20*a**(3/4)), True))

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GIAC/XCAS [A]  time = 0.219883, size = 96, normalized size = 0.98 \[ \frac{195 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} - 1020 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a + 2210 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2} - 2652 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3} + 3315 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{4}}{3315 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^19/(b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

1/3315*(195*(b*x^4 + a)^(17/4) - 1020*(b*x^4 + a)^(13/4)*a + 2210*(b*x^4 + a)^(9
/4)*a^2 - 2652*(b*x^4 + a)^(5/4)*a^3 + 3315*(b*x^4 + a)^(1/4)*a^4)/b^5