3.1174 \(\int x^{15} \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=84 \[ -\frac{a^3 \left (a-b x^4\right )^{5/4}}{5 b^4}+\frac{a^2 \left (a-b x^4\right )^{9/4}}{3 b^4}+\frac{\left (a-b x^4\right )^{17/4}}{17 b^4}-\frac{3 a \left (a-b x^4\right )^{13/4}}{13 b^4} \]

[Out]

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

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Rubi [A]  time = 0.113796, antiderivative size = 84, 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 \[ -\frac{a^3 \left (a-b x^4\right )^{5/4}}{5 b^4}+\frac{a^2 \left (a-b x^4\right )^{9/4}}{3 b^4}+\frac{\left (a-b x^4\right )^{17/4}}{17 b^4}-\frac{3 a \left (a-b x^4\right )^{13/4}}{13 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^15*(a - b*x^4)^(1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.5245, size = 70, normalized size = 0.83 \[ - \frac{a^{3} \left (a - b x^{4}\right )^{\frac{5}{4}}}{5 b^{4}} + \frac{a^{2} \left (a - b x^{4}\right )^{\frac{9}{4}}}{3 b^{4}} - \frac{3 a \left (a - b x^{4}\right )^{\frac{13}{4}}}{13 b^{4}} + \frac{\left (a - b x^{4}\right )^{\frac{17}{4}}}{17 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**15*(-b*x**4+a)**(1/4),x)

[Out]

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

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Mathematica [A]  time = 0.0311001, size = 62, normalized size = 0.74 \[ -\frac{\sqrt [4]{a-b x^4} \left (128 a^4+32 a^3 b x^4+20 a^2 b^2 x^8+15 a b^3 x^{12}-195 b^4 x^{16}\right )}{3315 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^15*(a - b*x^4)^(1/4),x]

[Out]

-((a - b*x^4)^(1/4)*(128*a^4 + 32*a^3*b*x^4 + 20*a^2*b^2*x^8 + 15*a*b^3*x^12 - 1
95*b^4*x^16))/(3315*b^4)

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Maple [A]  time = 0.011, size = 48, normalized size = 0.6 \[ -{\frac{195\,{b}^{3}{x}^{12}+180\,a{b}^{2}{x}^{8}+160\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{3315\,{b}^{4}} \left ( -b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^15*(-b*x^4+a)^(1/4),x)

[Out]

-1/3315*(-b*x^4+a)^(5/4)*(195*b^3*x^12+180*a*b^2*x^8+160*a^2*b*x^4+128*a^3)/b^4

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Maxima [A]  time = 1.44135, size = 92, normalized size = 1.1 \[ \frac{{\left (-b x^{4} + a\right )}^{\frac{17}{4}}}{17 \, b^{4}} - \frac{3 \,{\left (-b x^{4} + a\right )}^{\frac{13}{4}} a}{13 \, b^{4}} + \frac{{\left (-b x^{4} + a\right )}^{\frac{9}{4}} a^{2}}{3 \, b^{4}} - \frac{{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{3}}{5 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^15,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.231497, size = 78, normalized size = 0.93 \[ \frac{{\left (195 \, b^{4} x^{16} - 15 \, a b^{3} x^{12} - 20 \, a^{2} b^{2} x^{8} - 32 \, a^{3} b x^{4} - 128 \, a^{4}\right )}{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{3315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^15,x, algorithm="fricas")

[Out]

1/3315*(195*b^4*x^16 - 15*a*b^3*x^12 - 20*a^2*b^2*x^8 - 32*a^3*b*x^4 - 128*a^4)*
(-b*x^4 + a)^(1/4)/b^4

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Sympy [A]  time = 36.1937, size = 110, normalized size = 1.31 \[ \begin{cases} - \frac{128 a^{4} \sqrt [4]{a - b x^{4}}}{3315 b^{4}} - \frac{32 a^{3} x^{4} \sqrt [4]{a - b x^{4}}}{3315 b^{3}} - \frac{4 a^{2} x^{8} \sqrt [4]{a - b x^{4}}}{663 b^{2}} - \frac{a x^{12} \sqrt [4]{a - b x^{4}}}{221 b} + \frac{x^{16} \sqrt [4]{a - b x^{4}}}{17} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{16}}{16} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**15*(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-128*a**4*(a - b*x**4)**(1/4)/(3315*b**4) - 32*a**3*x**4*(a - b*x**4)
**(1/4)/(3315*b**3) - 4*a**2*x**8*(a - b*x**4)**(1/4)/(663*b**2) - a*x**12*(a -
b*x**4)**(1/4)/(221*b) + x**16*(a - b*x**4)**(1/4)/17, Ne(b, 0)), (a**(1/4)*x**1
6/16, True))

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GIAC/XCAS [A]  time = 0.243589, size = 127, normalized size = 1.51 \[ \frac{195 \,{\left (b x^{4} - a\right )}^{4}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} + 765 \,{\left (b x^{4} - a\right )}^{3}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a + 1105 \,{\left (b x^{4} - a\right )}^{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{2} - 663 \,{\left (-b x^{4} + a\right )}^{\frac{5}{4}} a^{3}}{3315 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^15,x, algorithm="giac")

[Out]

1/3315*(195*(b*x^4 - a)^4*(-b*x^4 + a)^(1/4) + 765*(b*x^4 - a)^3*(-b*x^4 + a)^(1
/4)*a + 1105*(b*x^4 - a)^2*(-b*x^4 + a)^(1/4)*a^2 - 663*(-b*x^4 + a)^(5/4)*a^3)/
b^4