3.830 \(\int \frac{x^7}{\sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (a-b x^4\right )^{3/2}}{6 b^2}-\frac{a \sqrt{a-b x^4}}{2 b^2} \]

[Out]

-(a*Sqrt[a - b*x^4])/(2*b^2) + (a - b*x^4)^(3/2)/(6*b^2)

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Rubi [A]  time = 0.064627, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (a-b x^4\right )^{3/2}}{6 b^2}-\frac{a \sqrt{a-b x^4}}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[a - b*x^4],x]

[Out]

-(a*Sqrt[a - b*x^4])/(2*b^2) + (a - b*x^4)^(3/2)/(6*b^2)

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Rubi in Sympy [A]  time = 7.90742, size = 31, normalized size = 0.78 \[ - \frac{a \sqrt{a - b x^{4}}}{2 b^{2}} + \frac{\left (a - b x^{4}\right )^{\frac{3}{2}}}{6 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(-b*x**4+a)**(1/2),x)

[Out]

-a*sqrt(a - b*x**4)/(2*b**2) + (a - b*x**4)**(3/2)/(6*b**2)

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Mathematica [A]  time = 0.0235731, size = 28, normalized size = 0.7 \[ -\frac{\sqrt{a-b x^4} \left (2 a+b x^4\right )}{6 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/Sqrt[a - b*x^4],x]

[Out]

-(Sqrt[a - b*x^4]*(2*a + b*x^4))/(6*b^2)

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Maple [A]  time = 0.007, size = 25, normalized size = 0.6 \[ -{\frac{b{x}^{4}+2\,a}{6\,{b}^{2}}\sqrt{-b{x}^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(-b*x^4+a)^(1/2),x)

[Out]

-1/6*(-b*x^4+a)^(1/2)*(b*x^4+2*a)/b^2

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Maxima [A]  time = 1.42636, size = 43, normalized size = 1.08 \[ \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{2}}}{6 \, b^{2}} - \frac{\sqrt{-b x^{4} + a} a}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(-b*x^4 + a)^(3/2)/b^2 - 1/2*sqrt(-b*x^4 + a)*a/b^2

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Fricas [A]  time = 0.270373, size = 32, normalized size = 0.8 \[ -\frac{{\left (b x^{4} + 2 \, a\right )} \sqrt{-b x^{4} + a}}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(b*x^4 + 2*a)*sqrt(-b*x^4 + a)/b^2

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Sympy [A]  time = 3.58123, size = 44, normalized size = 1.1 \[ \begin{cases} - \frac{a \sqrt{a - b x^{4}}}{3 b^{2}} - \frac{x^{4} \sqrt{a - b x^{4}}}{6 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 \sqrt{a}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(-b*x**4+a)**(1/2),x)

[Out]

Piecewise((-a*sqrt(a - b*x**4)/(3*b**2) - x**4*sqrt(a - b*x**4)/(6*b), Ne(b, 0))
, (x**8/(8*sqrt(a)), True))

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GIAC/XCAS [A]  time = 0.218204, size = 39, normalized size = 0.98 \[ \frac{{\left (-b x^{4} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{-b x^{4} + a} a}{6 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*((-b*x^4 + a)^(3/2) - 3*sqrt(-b*x^4 + a)*a)/b^2