3.1111 \(\int \frac{x^7}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=36 \[ \frac{\left (a+b x^4\right )^{5/4}}{5 b^2}-\frac{a \sqrt [4]{a+b x^4}}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0598471, antiderivative size = 36, 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 \[ \frac{\left (a+b x^4\right )^{5/4}}{5 b^2}-\frac{a \sqrt [4]{a+b x^4}}{b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.01839, size = 29, normalized size = 0.81 \[ - \frac{a \sqrt [4]{a + b x^{4}}}{b^{2}} + \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{5 b^{2}} \]

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

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Mathematica [A]  time = 0.0218177, size = 27, normalized size = 0.75 \[ \frac{\left (b x^4-4 a\right ) \sqrt [4]{a+b x^4}}{5 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-b{x}^{4}+4\,a}{5\,{b}^{2}}\sqrt [4]{b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^4+a)^(3/4),x)

[Out]

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

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Maxima [A]  time = 1.43938, size = 41, normalized size = 1.14 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{5 \, b^{2}} - \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.227554, size = 31, normalized size = 0.86 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{1}{4}}{\left (b x^{4} - 4 \, a\right )}}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.84424, size = 44, normalized size = 1.22 \[ \begin{cases} - \frac{4 a \sqrt [4]{a + b x^{4}}}{5 b^{2}} + \frac{x^{4} \sqrt [4]{a + b x^{4}}}{5 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{3}{4}}} & \text{otherwise} \end{cases} \]

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*(a + b*x**4)**(1/4)/(5*b**2) + x**4*(a + b*x**4)**(1/4)/(5*b), N
e(b, 0)), (x**8/(8*a**(3/4)), True))

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GIAC/XCAS [A]  time = 0.213693, size = 36, normalized size = 1. \[ \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}} - 5 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} a}{5 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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