3.762 \(\int x^{11} \sqrt{a+c x^4} \, dx\)

Optimal. Leaf size=59 \[ \frac{a^2 \left (a+c x^4\right )^{3/2}}{6 c^3}+\frac{\left (a+c x^4\right )^{7/2}}{14 c^3}-\frac{a \left (a+c x^4\right )^{5/2}}{5 c^3} \]

[Out]

(a^2*(a + c*x^4)^(3/2))/(6*c^3) - (a*(a + c*x^4)^(5/2))/(5*c^3) + (a + c*x^4)^(7
/2)/(14*c^3)

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Rubi [A]  time = 0.0815291, antiderivative size = 59, 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 \[ \frac{a^2 \left (a+c x^4\right )^{3/2}}{6 c^3}+\frac{\left (a+c x^4\right )^{7/2}}{14 c^3}-\frac{a \left (a+c x^4\right )^{5/2}}{5 c^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11*Sqrt[a + c*x^4],x]

[Out]

(a^2*(a + c*x^4)^(3/2))/(6*c^3) - (a*(a + c*x^4)^(5/2))/(5*c^3) + (a + c*x^4)^(7
/2)/(14*c^3)

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Rubi in Sympy [A]  time = 10.5893, size = 49, normalized size = 0.83 \[ \frac{a^{2} \left (a + c x^{4}\right )^{\frac{3}{2}}}{6 c^{3}} - \frac{a \left (a + c x^{4}\right )^{\frac{5}{2}}}{5 c^{3}} + \frac{\left (a + c x^{4}\right )^{\frac{7}{2}}}{14 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(c*x**4+a)**(1/2),x)

[Out]

a**2*(a + c*x**4)**(3/2)/(6*c**3) - a*(a + c*x**4)**(5/2)/(5*c**3) + (a + c*x**4
)**(7/2)/(14*c**3)

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Mathematica [A]  time = 0.0280059, size = 50, normalized size = 0.85 \[ \frac{\sqrt{a+c x^4} \left (8 a^3-4 a^2 c x^4+3 a c^2 x^8+15 c^3 x^{12}\right )}{210 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*Sqrt[a + c*x^4],x]

[Out]

(Sqrt[a + c*x^4]*(8*a^3 - 4*a^2*c*x^4 + 3*a*c^2*x^8 + 15*c^3*x^12))/(210*c^3)

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Maple [A]  time = 0.008, size = 36, normalized size = 0.6 \[{\frac{15\,{x}^{8}{c}^{2}-12\,a{x}^{4}c+8\,{a}^{2}}{210\,{c}^{3}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(c*x^4+a)^(1/2),x)

[Out]

1/210*(c*x^4+a)^(3/2)*(15*c^2*x^8-12*a*c*x^4+8*a^2)/c^3

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Maxima [A]  time = 1.4301, size = 63, normalized size = 1.07 \[ \frac{{\left (c x^{4} + a\right )}^{\frac{7}{2}}}{14 \, c^{3}} - \frac{{\left (c x^{4} + a\right )}^{\frac{5}{2}} a}{5 \, c^{3}} + \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}} a^{2}}{6 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*x^11,x, algorithm="maxima")

[Out]

1/14*(c*x^4 + a)^(7/2)/c^3 - 1/5*(c*x^4 + a)^(5/2)*a/c^3 + 1/6*(c*x^4 + a)^(3/2)
*a^2/c^3

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Fricas [A]  time = 0.225684, size = 62, normalized size = 1.05 \[ \frac{{\left (15 \, c^{3} x^{12} + 3 \, a c^{2} x^{8} - 4 \, a^{2} c x^{4} + 8 \, a^{3}\right )} \sqrt{c x^{4} + a}}{210 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*x^11,x, algorithm="fricas")

[Out]

1/210*(15*c^3*x^12 + 3*a*c^2*x^8 - 4*a^2*c*x^4 + 8*a^3)*sqrt(c*x^4 + a)/c^3

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Sympy [A]  time = 9.76098, size = 87, normalized size = 1.47 \[ \begin{cases} \frac{4 a^{3} \sqrt{a + c x^{4}}}{105 c^{3}} - \frac{2 a^{2} x^{4} \sqrt{a + c x^{4}}}{105 c^{2}} + \frac{a x^{8} \sqrt{a + c x^{4}}}{70 c} + \frac{x^{12} \sqrt{a + c x^{4}}}{14} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{12}}{12} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(c*x**4+a)**(1/2),x)

[Out]

Piecewise((4*a**3*sqrt(a + c*x**4)/(105*c**3) - 2*a**2*x**4*sqrt(a + c*x**4)/(10
5*c**2) + a*x**8*sqrt(a + c*x**4)/(70*c) + x**12*sqrt(a + c*x**4)/14, Ne(c, 0)),
(sqrt(a)*x**12/12, True))

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GIAC/XCAS [A]  time = 0.21862, size = 58, normalized size = 0.98 \[ \frac{15 \,{\left (c x^{4} + a\right )}^{\frac{7}{2}} - 42 \,{\left (c x^{4} + a\right )}^{\frac{5}{2}} a + 35 \,{\left (c x^{4} + a\right )}^{\frac{3}{2}} a^{2}}{210 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*x^11,x, algorithm="giac")

[Out]

1/210*(15*(c*x^4 + a)^(7/2) - 42*(c*x^4 + a)^(5/2)*a + 35*(c*x^4 + a)^(3/2)*a^2)
/c^3