3.1140 \(\int \frac{x^7}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=35 \[ \frac{a}{b^2 \sqrt [4]{a+b x^4}}+\frac{\left (a+b x^4\right )^{3/4}}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0588314, antiderivative size = 35, 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 \[ \frac{a}{b^2 \sqrt [4]{a+b x^4}}+\frac{\left (a+b x^4\right )^{3/4}}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^4)^(5/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.0686, size = 29, normalized size = 0.83 \[ \frac{a}{b^{2} \sqrt [4]{a + b x^{4}}} + \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b*x**4+a)**(5/4),x)

[Out]

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

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Mathematica [A]  time = 0.0253759, size = 27, normalized size = 0.77 \[ \frac{4 a+b x^4}{3 b^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^4)^(5/4),x]

[Out]

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

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Maple [A]  time = 0.007, size = 24, normalized size = 0.7 \[{\frac{b{x}^{4}+4\,a}{3\,{b}^{2}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b*x^4+a)^(5/4),x)

[Out]

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

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Maxima [A]  time = 1.44187, size = 39, normalized size = 1.11 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, b^{2}} + \frac{a}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(5/4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.233416, size = 31, normalized size = 0.89 \[ \frac{b x^{4} + 4 \, a}{3 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.05691, size = 44, normalized size = 1.26 \[ \begin{cases} \frac{4 a}{3 b^{2} \sqrt [4]{a + b x^{4}}} + \frac{x^{4}}{3 b \sqrt [4]{a + b x^{4}}} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 a^{\frac{5}{4}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b*x**4+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214989, size = 36, normalized size = 1.03 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}} + \frac{3 \, a}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}}{3 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

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