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

Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac{a \left (a+b x^4\right )^{5/4}}{5 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0590743, 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 \[ \frac{\left (a+b x^4\right )^{9/4}}{9 b^2}-\frac{a \left (a+b x^4\right )^{5/4}}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x^4)^(1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.05372, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + b x^{4}\right )^{\frac{5}{4}}}{5 b^{2}} + \frac{\left (a + b x^{4}\right )^{\frac{9}{4}}}{9 b^{2}} \]

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)**(5/4)/(5*b**2) + (a + b*x**4)**(9/4)/(9*b**2)

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Mathematica [A]  time = 0.0214011, size = 38, normalized size = 1. \[ \frac{\sqrt [4]{a+b x^4} \left (-4 a^2+a b x^4+5 b^2 x^8\right )}{45 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((a + b*x^4)^(1/4)*(-4*a^2 + a*b*x^4 + 5*b^2*x^8))/(45*b^2)

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Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{-5\,b{x}^{4}+4\,a}{45\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x^4+a)^(1/4),x)

[Out]

-1/45*(b*x^4+a)^(5/4)*(-5*b*x^4+4*a)/b^2

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Maxima [A]  time = 1.58795, size = 41, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{9}{4}}}{9 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}} a}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.254994, size = 46, normalized size = 1.21 \[ \frac{{\left (5 \, b^{2} x^{8} + a b x^{4} - 4 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{45 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*(5*b^2*x^8 + a*b*x^4 - 4*a^2)*(b*x^4 + a)^(1/4)/b^2

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Sympy [A]  time = 5.22519, size = 63, normalized size = 1.66 \[ \begin{cases} - \frac{4 a^{2} \sqrt [4]{a + b x^{4}}}{45 b^{2}} + \frac{a x^{4} \sqrt [4]{a + b x^{4}}}{45 b} + \frac{x^{8} \sqrt [4]{a + b x^{4}}}{9} & \text{for}\: b \neq 0 \\\frac{\sqrt [4]{a} x^{8}}{8} & \text{otherwise} \end{cases} \]

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**2*(a + b*x**4)**(1/4)/(45*b**2) + a*x**4*(a + b*x**4)**(1/4)/(4
5*b) + x**8*(a + b*x**4)**(1/4)/9, Ne(b, 0)), (a**(1/4)*x**8/8, True))

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GIAC/XCAS [A]  time = 0.21503, size = 39, normalized size = 1.03 \[ \frac{5 \,{\left (b x^{4} + a\right )}^{\frac{9}{4}} - 9 \,{\left (b x^{4} + a\right )}^{\frac{5}{4}} a}{45 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/45*(5*(b*x^4 + a)^(9/4) - 9*(b*x^4 + a)^(5/4)*a)/b^2