3.1018 \(\int x^7 \left (a+b x^4\right )^{3/4} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{11/4}}{11 b^2}-\frac{a \left (a+b x^4\right )^{7/4}}{7 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0575112, 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 \[ \frac{\left (a+b x^4\right )^{11/4}}{11 b^2}-\frac{a \left (a+b x^4\right )^{7/4}}{7 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.035, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + b x^{4}\right )^{\frac{7}{4}}}{7 b^{2}} + \frac{\left (a + b x^{4}\right )^{\frac{11}{4}}}{11 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x**4+a)**(3/4),x)

[Out]

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

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Mathematica [A]  time = 0.0236039, size = 39, normalized size = 1.03 \[ \frac{\left (a+b x^4\right )^{3/4} \left (-4 a^2+3 a b x^4+7 b^2 x^8\right )}{77 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + b*x^4)^(3/4),x]

[Out]

((a + b*x^4)^(3/4)*(-4*a^2 + 3*a*b*x^4 + 7*b^2*x^8))/(77*b^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-7\,b{x}^{4}+4\,a}{77\,{b}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{7}{4}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x^4+a)^(3/4),x)

[Out]

-1/77*(b*x^4+a)^(7/4)*(-7*b*x^4+4*a)/b^2

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Maxima [A]  time = 1.4398, size = 41, normalized size = 1.08 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{11}{4}}}{11 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{7}{4}} a}{7 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^7,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.367529, size = 47, normalized size = 1.24 \[ \frac{{\left (7 \, b^{2} x^{8} + 3 \, a b x^{4} - 4 \, a^{2}\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{77 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^7,x, algorithm="fricas")

[Out]

1/77*(7*b^2*x^8 + 3*a*b*x^4 - 4*a^2)*(b*x^4 + a)^(3/4)/b^2

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Sympy [A]  time = 14.5384, size = 65, normalized size = 1.71 \[ \begin{cases} - \frac{4 a^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{77 b^{2}} + \frac{3 a x^{4} \left (a + b x^{4}\right )^{\frac{3}{4}}}{77 b} + \frac{x^{8} \left (a + b x^{4}\right )^{\frac{3}{4}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{3}{4}} x^{8}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x**4+a)**(3/4),x)

[Out]

Piecewise((-4*a**2*(a + b*x**4)**(3/4)/(77*b**2) + 3*a*x**4*(a + b*x**4)**(3/4)/
(77*b) + x**8*(a + b*x**4)**(3/4)/11, Ne(b, 0)), (a**(3/4)*x**8/8, True))

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GIAC/XCAS [A]  time = 0.218457, size = 39, normalized size = 1.03 \[ \frac{7 \,{\left (b x^{4} + a\right )}^{\frac{11}{4}} - 11 \,{\left (b x^{4} + a\right )}^{\frac{7}{4}} a}{77 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)*x^7,x, algorithm="giac")

[Out]

1/77*(7*(b*x^4 + a)^(11/4) - 11*(b*x^4 + a)^(7/4)*a)/b^2