3.763 \(\int x^7 \sqrt{a+c x^4} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c^2}-\frac{a \left (a+c x^4\right )^{3/2}}{6 c^2} \]

[Out]

-(a*(a + c*x^4)^(3/2))/(6*c^2) + (a + c*x^4)^(5/2)/(10*c^2)

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Rubi [A]  time = 0.05825, 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 \[ \frac{\left (a+c x^4\right )^{5/2}}{10 c^2}-\frac{a \left (a+c x^4\right )^{3/2}}{6 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7*Sqrt[a + c*x^4],x]

[Out]

-(a*(a + c*x^4)^(3/2))/(6*c^2) + (a + c*x^4)^(5/2)/(10*c^2)

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Rubi in Sympy [A]  time = 7.08941, size = 31, normalized size = 0.82 \[ - \frac{a \left (a + c x^{4}\right )^{\frac{3}{2}}}{6 c^{2}} + \frac{\left (a + c x^{4}\right )^{\frac{5}{2}}}{10 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a*(a + c*x**4)**(3/2)/(6*c**2) + (a + c*x**4)**(5/2)/(10*c**2)

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Mathematica [A]  time = 0.0221895, size = 38, normalized size = 1. \[ \frac{\sqrt{a+c x^4} \left (-2 a^2+a c x^4+3 c^2 x^8\right )}{30 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + c*x^4]*(-2*a^2 + a*c*x^4 + 3*c^2*x^8))/(30*c^2)

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Maple [A]  time = 0.008, size = 25, normalized size = 0.7 \[ -{\frac{-3\,c{x}^{4}+2\,a}{30\,{c}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(c*x^4+a)^(1/2),x)

[Out]

-1/30*(c*x^4+a)^(3/2)*(-3*c*x^4+2*a)/c^2

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Maxima [A]  time = 1.44381, size = 41, normalized size = 1.08 \[ \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}}}{10 \, c^{2}} - \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}} a}{6 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(c*x^4 + a)^(5/2)/c^2 - 1/6*(c*x^4 + a)^(3/2)*a/c^2

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Fricas [A]  time = 0.230666, size = 46, normalized size = 1.21 \[ \frac{{\left (3 \, c^{2} x^{8} + a c x^{4} - 2 \, a^{2}\right )} \sqrt{c x^{4} + a}}{30 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(3*c^2*x^8 + a*c*x^4 - 2*a^2)*sqrt(c*x^4 + a)/c^2

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Sympy [A]  time = 2.93014, size = 61, normalized size = 1.61 \[ \begin{cases} - \frac{a^{2} \sqrt{a + c x^{4}}}{15 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{4}}}{30 c} + \frac{x^{8} \sqrt{a + c x^{4}}}{10} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{8}}{8} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a**2*sqrt(a + c*x**4)/(15*c**2) + a*x**4*sqrt(a + c*x**4)/(30*c) + x
**8*sqrt(a + c*x**4)/10, Ne(c, 0)), (sqrt(a)*x**8/8, True))

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GIAC/XCAS [A]  time = 0.2133, size = 39, normalized size = 1.03 \[ \frac{3 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} - 5 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a}{30 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(3*(c*x^4 + a)^(5/2) - 5*(c*x^4 + a)^(3/2)*a)/c^2