3.1081 \(\int \frac{x^7}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{7/4}}{7 b^2}-\frac{a \left (a+b x^4\right )^{3/4}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0592666, antiderivative size = 38, 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 \[ \frac{\left (a+b x^4\right )^{7/4}}{7 b^2}-\frac{a \left (a+b x^4\right )^{3/4}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^4)^(1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.03994, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + b x^{4}\right )^{\frac{3}{4}}}{3 b^{2}} + \frac{\left (a + b x^{4}\right )^{\frac{7}{4}}}{7 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**4+a)**(1/4),x)

[Out]

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

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Mathematica [A]  time = 0.0243971, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^4\right )^{3/4} \left (3 b x^4-4 a\right )}{21 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^4)^(1/4),x]

[Out]

((a + b*x^4)^(3/4)*(-4*a + 3*b*x^4))/(21*b^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-3\,b{x}^{4}+4\,a}{21\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^4+a)^(1/4),x)

[Out]

-1/21*(b*x^4+a)^(3/4)*(-3*b*x^4+4*a)/b^2

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Maxima [A]  time = 1.44955, size = 41, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}}}{7 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} a}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.269957, size = 32, normalized size = 0.84 \[ \frac{{\left (3 \, b x^{4} - 4 \, a\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{21 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

1/21*(3*b*x^4 - 4*a)*(b*x^4 + a)^(3/4)/b^2

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Sympy [A]  time = 4.23873, size = 44, normalized size = 1.16 \[ \begin{cases} - \frac{4 a \left (a + b x^{4}\right )^{\frac{3}{4}}}{21 b^{2}} + \frac{x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{7 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 \sqrt [4]{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**4+a)**(1/4),x)

[Out]

Piecewise((-4*a*(a + b*x**4)**(3/4)/(21*b**2) + x**4*(a + b*x**4)**(3/4)/(7*b),
Ne(b, 0)), (x**8/(8*a**(1/4)), True))

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GIAC/XCAS [A]  time = 0.215455, size = 39, normalized size = 1.03 \[ \frac{3 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} - 7 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} a}{21 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

1/21*(3*(b*x^4 + a)^(7/4) - 7*(b*x^4 + a)^(3/4)*a)/b^2