3.219 \(\int \frac{x^{14}}{\sqrt{a+b x^3+c x^6}} \, dx\)

Optimal. Leaf size=171 \[ \frac{\left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{384 c^{9/2}}-\frac{\left (5 b \left (21 b^2-44 a c\right )-2 c x^3 \left (35 b^2-36 a c\right )\right ) \sqrt{a+b x^3+c x^6}}{576 c^4}-\frac{7 b x^6 \sqrt{a+b x^3+c x^6}}{72 c^2}+\frac{x^9 \sqrt{a+b x^3+c x^6}}{12 c} \]

[Out]

(-7*b*x^6*Sqrt[a + b*x^3 + c*x^6])/(72*c^2) + (x^9*Sqrt[a + b*x^3 + c*x^6])/(12*
c) - ((5*b*(21*b^2 - 44*a*c) - 2*c*(35*b^2 - 36*a*c)*x^3)*Sqrt[a + b*x^3 + c*x^6
])/(576*c^4) + ((35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*ArcTanh[(b + 2*c*x^3)/(2*Sqr
t[c]*Sqrt[a + b*x^3 + c*x^6])])/(384*c^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.497091, 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 \[ \frac{\left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{384 c^{9/2}}-\frac{\left (5 b \left (21 b^2-44 a c\right )-2 c x^3 \left (35 b^2-36 a c\right )\right ) \sqrt{a+b x^3+c x^6}}{576 c^4}-\frac{7 b x^6 \sqrt{a+b x^3+c x^6}}{72 c^2}+\frac{x^9 \sqrt{a+b x^3+c x^6}}{12 c} \]

Antiderivative was successfully verified.

[In]  Int[x^14/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

(-7*b*x^6*Sqrt[a + b*x^3 + c*x^6])/(72*c^2) + (x^9*Sqrt[a + b*x^3 + c*x^6])/(12*
c) - ((5*b*(21*b^2 - 44*a*c) - 2*c*(35*b^2 - 36*a*c)*x^3)*Sqrt[a + b*x^3 + c*x^6
])/(576*c^4) + ((35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*ArcTanh[(b + 2*c*x^3)/(2*Sqr
t[c]*Sqrt[a + b*x^3 + c*x^6])])/(384*c^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.0434, size = 162, normalized size = 0.95 \[ - \frac{7 b x^{6} \sqrt{a + b x^{3} + c x^{6}}}{72 c^{2}} + \frac{x^{9} \sqrt{a + b x^{3} + c x^{6}}}{12 c} - \frac{\left (\frac{5 b \left (- 44 a c + 21 b^{2}\right )}{8} - \frac{c x^{3} \left (- 36 a c + 35 b^{2}\right )}{4}\right ) \sqrt{a + b x^{3} + c x^{6}}}{72 c^{4}} + \frac{\left (48 a^{2} c^{2} - 120 a b^{2} c + 35 b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{384 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

-7*b*x**6*sqrt(a + b*x**3 + c*x**6)/(72*c**2) + x**9*sqrt(a + b*x**3 + c*x**6)/(
12*c) - (5*b*(-44*a*c + 21*b**2)/8 - c*x**3*(-36*a*c + 35*b**2)/4)*sqrt(a + b*x*
*3 + c*x**6)/(72*c**4) + (48*a**2*c**2 - 120*a*b**2*c + 35*b**4)*atanh((b + 2*c*
x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(384*c**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.185408, size = 135, normalized size = 0.79 \[ \frac{3 \left (48 a^2 c^2-120 a b^2 c+35 b^4\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )+2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (4 b c \left (55 a-14 c x^6\right )+24 c^2 x^3 \left (2 c x^6-3 a\right )-105 b^3+70 b^2 c x^3\right )}{1152 c^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/Sqrt[a + b*x^3 + c*x^6],x]

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(-105*b^3 + 70*b^2*c*x^3 + 4*b*c*(55*a - 14*c
*x^6) + 24*c^2*x^3*(-3*a + 2*c*x^6)) + 3*(35*b^4 - 120*a*b^2*c + 48*a^2*c^2)*Log
[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(1152*c^(9/2))

_______________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{14}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(x^14/(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(x^14/sqrt(c*x^6 + b*x^3 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.290877, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, c^{3} x^{9} - 56 \, b c^{2} x^{6} + 2 \,{\left (35 \, b^{2} c - 36 \, a c^{2}\right )} x^{3} - 105 \, b^{3} + 220 \, a b c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} + 3 \,{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2304 \, c^{\frac{9}{2}}}, \frac{2 \,{\left (48 \, c^{3} x^{9} - 56 \, b c^{2} x^{6} + 2 \,{\left (35 \, b^{2} c - 36 \, a c^{2}\right )} x^{3} - 105 \, b^{3} + 220 \, a b c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} + 3 \,{\left (35 \, b^{4} - 120 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{1152 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/sqrt(c*x^6 + b*x^3 + a),x, algorithm="fricas")

[Out]

[1/2304*(4*(48*c^3*x^9 - 56*b*c^2*x^6 + 2*(35*b^2*c - 36*a*c^2)*x^3 - 105*b^3 +
220*a*b*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c) + 3*(35*b^4 - 120*a*b^2*c + 48*a^2*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), 1/1152*(2*(48*c^3*x^9 - 56*b*c^2*x^6 + 2*(35*b^2*
c - 36*a*c^2)*x^3 - 105*b^3 + 220*a*b*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) + 3*(3
5*b^4 - 120*a*b^2*c + 48*a^2*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. \[ \int \frac{x^{14}}{\sqrt{a + b x^{3} + c x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(x**14/sqrt(a + b*x**3 + c*x**6), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{\sqrt{c x^{6} + b x^{3} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/sqrt(c*x^6 + b*x^3 + a),x, algorithm="giac")

[Out]

integrate(x^14/sqrt(c*x^6 + b*x^3 + a), x)