3.1232 \(\int \frac{x^{19}}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=104 \[ -\frac{a^4 \sqrt [4]{a-b x^4}}{b^5}+\frac{4 a^3 \left (a-b x^4\right )^{5/4}}{5 b^5}-\frac{2 a^2 \left (a-b x^4\right )^{9/4}}{3 b^5}-\frac{\left (a-b x^4\right )^{17/4}}{17 b^5}+\frac{4 a \left (a-b x^4\right )^{13/4}}{13 b^5} \]

[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 = 0.13652, antiderivative size = 104, 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 \[ -\frac{a^4 \sqrt [4]{a-b x^4}}{b^5}+\frac{4 a^3 \left (a-b x^4\right )^{5/4}}{5 b^5}-\frac{2 a^2 \left (a-b x^4\right )^{9/4}}{3 b^5}-\frac{\left (a-b x^4\right )^{17/4}}{17 b^5}+\frac{4 a \left (a-b x^4\right )^{13/4}}{13 b^5} \]

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 = 19.6528, size = 90, normalized size = 0.87 \[ - \frac{a^{4} \sqrt [4]{a - b x^{4}}}{b^{5}} + \frac{4 a^{3} \left (a - b x^{4}\right )^{\frac{5}{4}}}{5 b^{5}} - \frac{2 a^{2} \left (a - b x^{4}\right )^{\frac{9}{4}}}{3 b^{5}} + \frac{4 a \left (a - b x^{4}\right )^{\frac{13}{4}}}{13 b^{5}} - \frac{\left (a - b x^{4}\right )^{\frac{17}{4}}}{17 b^{5}} \]

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 = 0.0373174, size = 62, normalized size = 0.6 \[ -\frac{\sqrt [4]{a-b x^4} \left (2048 a^4+512 a^3 b x^4+320 a^2 b^2 x^8+240 a b^3 x^{12}+195 b^4 x^{16}\right )}{3315 b^5} \]

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 = 0.012, size = 59, 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 = 1.43585, size = 116, normalized size = 1.12 \[ -\frac{{\left (-b x^{4} + a\right )}^{\frac{17}{4}}}{17 \, b^{5}} + \frac{4 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}} a}{13 \, b^{5}} - \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{9}{4}} a^{2}}{3 \, b^{5}} + \frac{4 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{3}}{5 \, b^{5}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{4}}{b^{5}} \]

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 = 0.220838, size = 78, normalized size = 0.75 \[ -\frac{{\left (195 \, b^{4} x^{16} + 240 \, a b^{3} x^{12} + 320 \, a^{2} b^{2} x^{8} + 512 \, a^{3} b x^{4} + 2048 \, a^{4}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{3315 \, b^{5}} \]

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 = 50.9439, size = 117, normalized size = 1.12 \[ \begin{cases} - \frac{2048 a^{4} \sqrt [4]{a - b x^{4}}}{3315 b^{5}} - \frac{512 a^{3} x^{4} \sqrt [4]{a - b x^{4}}}{3315 b^{4}} - \frac{64 a^{2} x^{8} \sqrt [4]{a - b x^{4}}}{663 b^{3}} - \frac{16 a x^{12} \sqrt [4]{a - b x^{4}}}{221 b^{2}} - \frac{x^{16} \sqrt [4]{a - b x^{4}}}{17 b} & \text{for}\: b \neq 0 \\\frac{x^{20}}{20 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \]

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 = 0.217851, size = 147, normalized size = 1.41 \[ -\frac{195 \,{\left (b x^{4} - a\right )}^{4}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} + 1020 \,{\left (b x^{4} - a\right )}^{3}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a + 2210 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2} - 2652 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{3} + 3315 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{4}}{3315 \, b^{5}} \]

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