3.186 \(\int x^{11} \sqrt{a+b x^3+c x^6} \, dx\)

Optimal. Leaf size=171 \[ \frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{768 c^{9/2}}-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{384 c^4}+\frac{\left (-32 a c+35 b^2-42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 c^3}+\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c} \]

[Out]

-(b*(7*b^2 - 12*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(384*c^4) + (x^6*(a
+ b*x^3 + c*x^6)^(3/2))/(15*c) + ((35*b^2 - 32*a*c - 42*b*c*x^3)*(a + b*x^3 + c*
x^6)^(3/2))/(720*c^3) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/
(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(768*c^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.341052, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{768 c^{9/2}}-\frac{b \left (7 b^2-12 a c\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{384 c^4}+\frac{\left (-32 a c+35 b^2-42 b c x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{720 c^3}+\frac{x^6 \left (a+b x^3+c x^6\right )^{3/2}}{15 c} \]

Antiderivative was successfully verified.

[In]  Int[x^11*Sqrt[a + b*x^3 + c*x^6],x]

[Out]

-(b*(7*b^2 - 12*a*c)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(384*c^4) + (x^6*(a
+ b*x^3 + c*x^6)^(3/2))/(15*c) + ((35*b^2 - 32*a*c - 42*b*c*x^3)*(a + b*x^3 + c*
x^6)^(3/2))/(720*c^3) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*ArcTanh[(b + 2*c*x^3)/
(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(768*c^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 32.0307, size = 163, normalized size = 0.95 \[ - \frac{b \left (b + 2 c x^{3}\right ) \left (- 12 a c + 7 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{384 c^{4}} + \frac{b \left (- 12 a c + 7 b^{2}\right ) \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{768 c^{\frac{9}{2}}} + \frac{x^{6} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{15 c} + \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}} \left (- 8 a c + \frac{35 b^{2}}{4} - \frac{21 b c x^{3}}{2}\right )}{180 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(c*x**6+b*x**3+a)**(1/2),x)

[Out]

-b*(b + 2*c*x**3)*(-12*a*c + 7*b**2)*sqrt(a + b*x**3 + c*x**6)/(384*c**4) + b*(-
12*a*c + 7*b**2)*(-4*a*c + b**2)*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3
+ c*x**6)))/(768*c**(9/2)) + x**6*(a + b*x**3 + c*x**6)**(3/2)/(15*c) + (a + b*
x**3 + c*x**6)**(3/2)*(-8*a*c + 35*b**2/4 - 21*b*c*x**3/2)/(180*c**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.119824, size = 159, normalized size = 0.93 \[ \frac{\sqrt{a+b x^3+c x^6} \left (128 c^2 \left (-2 a^2+a c x^6+3 c^2 x^{12}\right )+4 b^2 c \left (115 a-14 c x^6\right )+8 b c^2 x^3 \left (6 c x^6-29 a\right )-105 b^4+70 b^3 c x^3\right )}{5760 c^4}+\frac{b \left (7 b^2-12 a c\right ) \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{768 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11*Sqrt[a + b*x^3 + c*x^6],x]

[Out]

(Sqrt[a + b*x^3 + c*x^6]*(-105*b^4 + 70*b^3*c*x^3 + 4*b^2*c*(115*a - 14*c*x^6) +
8*b*c^2*x^3*(-29*a + 6*c*x^6) + 128*c^2*(-2*a^2 + a*c*x^6 + 3*c^2*x^12)))/(5760
*c^4) + (b*(7*b^2 - 12*a*c)*(b^2 - 4*a*c)*Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b
*x^3 + c*x^6]])/(768*c^(9/2))

_______________________________________________________________________________________

Maple [F]  time = 0.031, size = 0, normalized size = 0. \[ \int{x}^{11}\sqrt{c{x}^{6}+b{x}^{3}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(x^11*(c*x^6+b*x^3+a)^(1/2),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^6 + b*x^3 + a)*x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.288798, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (384 \, c^{4} x^{12} + 48 \, b c^{3} x^{9} - 8 \,{\left (7 \, b^{2} c^{2} - 16 \, a c^{3}\right )} x^{6} - 105 \, b^{4} + 460 \, a b^{2} c - 256 \, a^{2} c^{2} + 2 \,{\left (35 \, b^{3} c - 116 \, a b c^{2}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} + 15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{23040 \, c^{\frac{9}{2}}}, \frac{2 \,{\left (384 \, c^{4} x^{12} + 48 \, b c^{3} x^{9} - 8 \,{\left (7 \, b^{2} c^{2} - 16 \, a c^{3}\right )} x^{6} - 105 \, b^{4} + 460 \, a b^{2} c - 256 \, a^{2} c^{2} + 2 \,{\left (35 \, b^{3} c - 116 \, a b c^{2}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} + 15 \,{\left (7 \, b^{5} - 40 \, a b^{3} c + 48 \, a^{2} b c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{11520 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^6 + b*x^3 + a)*x^11,x, algorithm="fricas")

[Out]

[1/23040*(4*(384*c^4*x^12 + 48*b*c^3*x^9 - 8*(7*b^2*c^2 - 16*a*c^3)*x^6 - 105*b^
4 + 460*a*b^2*c - 256*a^2*c^2 + 2*(35*b^3*c - 116*a*b*c^2)*x^3)*sqrt(c*x^6 + b*x
^3 + a)*sqrt(c) + 15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*log(-4*sqrt(c*x^6 + b*x
^3 + a)*(2*c^2*x^3 + b*c) - (8*c^2*x^6 + 8*b*c*x^3 + b^2 + 4*a*c)*sqrt(c)))/c^(9
/2), 1/11520*(2*(384*c^4*x^12 + 48*b*c^3*x^9 - 8*(7*b^2*c^2 - 16*a*c^3)*x^6 - 10
5*b^4 + 460*a*b^2*c - 256*a^2*c^2 + 2*(35*b^3*c - 116*a*b*c^2)*x^3)*sqrt(c*x^6 +
b*x^3 + a)*sqrt(-c) + 15*(7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*arctan(1/2*(2*c*x^
3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*c)))/(sqrt(-c)*c^4)]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{11} \sqrt{a + b x^{3} + c x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(x**11*sqrt(a + b*x**3 + c*x**6), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{6} + b x^{3} + a} x^{11}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^6 + b*x^3 + a)*x^11,x, algorithm="giac")

[Out]

integrate(sqrt(c*x^6 + b*x^3 + a)*x^11, x)