3.1079 \(\int \frac{x^{15}}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{a^3 \left (a+b x^4\right )^{3/4}}{3 b^4}+\frac{3 a^2 \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^4}-\frac{3 a \left (a+b x^4\right )^{11/4}}{11 b^4} \]

[Out]

-(a^3*(a + b*x^4)^(3/4))/(3*b^4) + (3*a^2*(a + b*x^4)^(7/4))/(7*b^4) - (3*a*(a +
b*x^4)^(11/4))/(11*b^4) + (a + b*x^4)^(15/4)/(15*b^4)

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Rubi [A]  time = 0.108409, 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 \[ -\frac{a^3 \left (a+b x^4\right )^{3/4}}{3 b^4}+\frac{3 a^2 \left (a+b x^4\right )^{7/4}}{7 b^4}+\frac{\left (a+b x^4\right )^{15/4}}{15 b^4}-\frac{3 a \left (a+b x^4\right )^{11/4}}{11 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^15/(a + b*x^4)^(1/4),x]

[Out]

-(a^3*(a + b*x^4)^(3/4))/(3*b^4) + (3*a^2*(a + b*x^4)^(7/4))/(7*b^4) - (3*a*(a +
b*x^4)^(11/4))/(11*b^4) + (a + b*x^4)^(15/4)/(15*b^4)

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Rubi in Sympy [A]  time = 14.198, size = 71, normalized size = 0.89 \[ - \frac{a^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{3 b^{4}} + \frac{3 a^{2} \left (a + b x^{4}\right )^{\frac{7}{4}}}{7 b^{4}} - \frac{3 a \left (a + b x^{4}\right )^{\frac{11}{4}}}{11 b^{4}} + \frac{\left (a + b x^{4}\right )^{\frac{15}{4}}}{15 b^{4}} \]

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)**(3/4)/(3*b**4) + 3*a**2*(a + b*x**4)**(7/4)/(7*b**4) - 3*a*(
a + b*x**4)**(11/4)/(11*b**4) + (a + b*x**4)**(15/4)/(15*b**4)

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Mathematica [A]  time = 0.0360742, size = 50, normalized size = 0.62 \[ \frac{\left (a+b x^4\right )^{3/4} \left (-128 a^3+96 a^2 b x^4-84 a b^2 x^8+77 b^3 x^{12}\right )}{1155 b^4} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^4)^(3/4)*(-128*a^3 + 96*a^2*b*x^4 - 84*a*b^2*x^8 + 77*b^3*x^12))/(1155
*b^4)

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Maple [A]  time = 0.009, size = 47, normalized size = 0.6 \[ -{\frac{-77\,{b}^{3}{x}^{12}+84\,a{b}^{2}{x}^{8}-96\,{a}^{2}b{x}^{4}+128\,{a}^{3}}{1155\,{b}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^15/(b*x^4+a)^(1/4),x)

[Out]

-1/1155*(b*x^4+a)^(3/4)*(-77*b^3*x^12+84*a*b^2*x^8-96*a^2*b*x^4+128*a^3)/b^4

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Maxima [A]  time = 1.44371, size = 86, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{15}{4}}}{15 \, b^{4}} - \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a}{11 \, b^{4}} + \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2}}{7 \, b^{4}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{3}}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(b*x^4 + a)^(15/4)/b^4 - 3/11*(b*x^4 + a)^(11/4)*a/b^4 + 3/7*(b*x^4 + a)^(7
/4)*a^2/b^4 - 1/3*(b*x^4 + a)^(3/4)*a^3/b^4

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Fricas [A]  time = 0.272414, size = 62, normalized size = 0.78 \[ \frac{{\left (77 \, b^{3} x^{12} - 84 \, a b^{2} x^{8} + 96 \, a^{2} b x^{4} - 128 \, a^{3}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{1155 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1155*(77*b^3*x^12 - 84*a*b^2*x^8 + 96*a^2*b*x^4 - 128*a^3)*(b*x^4 + a)^(3/4)/b
^4

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Sympy [A]  time = 25.2298, size = 92, normalized size = 1.15 \[ \begin{cases} - \frac{128 a^{3} \left (a + b x^{4}\right )^{\frac{3}{4}}}{1155 b^{4}} + \frac{32 a^{2} x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{385 b^{3}} - \frac{4 a x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{55 b^{2}} + \frac{x^{12} \left (a + b x^{4}\right )^{\frac{3}{4}}}{15 b} & \text{for}\: b \neq 0 \\\frac{x^{16}}{16 \sqrt [4]{a}} & \text{otherwise} \end{cases} \]

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**3*(a + b*x**4)**(3/4)/(1155*b**4) + 32*a**2*x**4*(a + b*x**4)
**(3/4)/(385*b**3) - 4*a*x**8*(a + b*x**4)**(3/4)/(55*b**2) + x**12*(a + b*x**4)
**(3/4)/(15*b), Ne(b, 0)), (x**16/(16*a**(1/4)), True))

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GIAC/XCAS [A]  time = 0.21663, size = 77, normalized size = 0.96 \[ \frac{77 \,{\left (b x^{4} + a\right )}^{\frac{15}{4}} - 315 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} a + 495 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a^{2} - 385 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a^{3}}{1155 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1155*(77*(b*x^4 + a)^(15/4) - 315*(b*x^4 + a)^(11/4)*a + 495*(b*x^4 + a)^(7/4)
*a^2 - 385*(b*x^4 + a)^(3/4)*a^3)/b^4