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

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b x^4\right )^{5/4}}{5 b^4}+\frac{a^2 \left (a+b x^4\right )^{9/4}}{3 b^4}+\frac{\left (a+b x^4\right )^{17/4}}{17 b^4}-\frac{3 a \left (a+b x^4\right )^{13/4}}{13 b^4} \]

[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 = 0.107116, antiderivative size = 80, 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 \[ -\frac{a^3 \left (a+b x^4\right )^{5/4}}{5 b^4}+\frac{a^2 \left (a+b x^4\right )^{9/4}}{3 b^4}+\frac{\left (a+b x^4\right )^{17/4}}{17 b^4}-\frac{3 a \left (a+b x^4\right )^{13/4}}{13 b^4} \]

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 = 14.2401, size = 70, normalized size = 0.88 \[ - \frac{a^{3} \left (a + b x^{4}\right )^{\frac{5}{4}}}{5 b^{4}} + \frac{a^{2} \left (a + b x^{4}\right )^{\frac{9}{4}}}{3 b^{4}} - \frac{3 a \left (a + b x^{4}\right )^{\frac{13}{4}}}{13 b^{4}} + \frac{\left (a + b x^{4}\right )^{\frac{17}{4}}}{17 b^{4}} \]

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 = 0.0295268, size = 61, normalized size = 0.76 \[ \frac{\sqrt [4]{a+b x^4} \left (-128 a^4+32 a^3 b x^4-20 a^2 b^2 x^8+15 a b^3 x^{12}+195 b^4 x^{16}\right )}{3315 b^4} \]

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 = 0.009, size = 47, 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 = 1.43967, size = 86, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{17}{4}}}{17 \, b^{4}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a}{13 \, b^{4}} + \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2}}{3 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3}}{5 \, b^{4}} \]

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 = 0.265648, size = 77, normalized size = 0.96 \[ \frac{{\left (195 \, b^{4} x^{16} + 15 \, a b^{3} x^{12} - 20 \, a^{2} b^{2} x^{8} + 32 \, a^{3} b x^{4} - 128 \, a^{4}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{3315 \, b^{4}} \]

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 = 34.8031, size = 110, normalized size = 1.38 \[ \begin{cases} - \frac{128 a^{4} \sqrt [4]{a + b x^{4}}}{3315 b^{4}} + \frac{32 a^{3} x^{4} \sqrt [4]{a + b x^{4}}}{3315 b^{3}} - \frac{4 a^{2} x^{8} \sqrt [4]{a + b x^{4}}}{663 b^{2}} + \frac{a x^{12} \sqrt [4]{a + b x^{4}}}{221 b} + \frac{x^{16} \sqrt [4]{a + b x^{4}}}{17} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{16}}{16} & \text{otherwise} \end{cases} \]

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 = 0.215606, size = 77, normalized size = 0.96 \[ \frac{195 \,{\left (b x^{4} + a\right )}^{\frac{17}{4}} - 765 \,{\left (b x^{4} + a\right )}^{\frac{13}{4}} a + 1105 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} a^{2} - 663 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a^{3}}{3315 \, b^{4}} \]

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)^(17/4) - 765*(b*x^4 + a)^(13/4)*a + 1105*(b*x^4 + a)^(9/
4)*a^2 - 663*(b*x^4 + a)^(5/4)*a^3)/b^4