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

Optimal. Leaf size=38 \[ \frac{\left (a+b x^4\right )^{3/2}}{6 b^2}-\frac{a \sqrt{a+b x^4}}{2 b^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0629535, 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 \[ \frac{\left (a+b x^4\right )^{3/2}}{6 b^2}-\frac{a \sqrt{a+b x^4}}{2 b^2} \]

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)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 7.13061, size = 31, normalized size = 0.82 \[ - \frac{a \sqrt{a + b x^{4}}}{2 b^{2}} + \frac{\left (a + b x^{4}\right )^{\frac{3}{2}}}{6 b^{2}} \]

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)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0221623, size = 27, normalized size = 0.71 \[ \frac{\left (b x^4-2 a\right ) \sqrt{a+b x^4}}{6 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 25, normalized size = 0.7 \[ -{\frac{-b{x}^{4}+2\,a}{6\,{b}^{2}}\sqrt{b{x}^{4}+a}} \]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.43814, size = 41, 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

_______________________________________________________________________________________

Fricas [A]  time = 0.235908, size = 31, normalized size = 0.82 \[ \frac{\sqrt{b x^{4} + a}{\left (b x^{4} - 2 \, a\right )}}{6 \, b^{2}} \]

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*sqrt(b*x^4 + a)*(b*x^4 - 2*a)/b^2

_______________________________________________________________________________________

Sympy [A]  time = 3.58189, size = 42, normalized size = 1.11 \[ \begin{cases} - \frac{a \sqrt{a + b x^{4}}}{3 b^{2}} + \frac{x^{4} \sqrt{a + b x^{4}}}{6 b} & \text{for}\: b \neq 0 \\\frac{x^{8}}{8 \sqrt{a}} & \text{otherwise} \end{cases} \]

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))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214511, size = 36, normalized size = 0.95 \[ \frac{{\left (b x^{4} + a\right )}^{\frac{3}{2}} - 3 \, \sqrt{b x^{4} + a} a}{6 \, b^{2}} \]

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