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

Optimal. Leaf size=231 \[ -\frac{\left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{11/2}}+\frac{\left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^5}-\frac{\left (7 b \left (15 b^2-28 a c\right )-6 c x^3 \left (21 b^2-20 a c\right )\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c} \]

[Out]

((21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536
*c^5) - (b*x^6*(a + b*x^3 + c*x^6)^(3/2))/(20*c^2) + (x^9*(a + b*x^3 + c*x^6)^(3
/2))/(18*c) - ((7*b*(15*b^2 - 28*a*c) - 6*c*(21*b^2 - 20*a*c)*x^3)*(a + b*x^3 +
c*x^6)^(3/2))/(2880*c^4) - ((b^2 - 4*a*c)*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*Arc
Tanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(3072*c^(11/2))

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Rubi [A]  time = 0.661407, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ -\frac{\left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{3072 c^{11/2}}+\frac{\left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \left (b+2 c x^3\right ) \sqrt{a+b x^3+c x^6}}{1536 c^5}-\frac{\left (7 b \left (15 b^2-28 a c\right )-6 c x^3 \left (21 b^2-20 a c\right )\right ) \left (a+b x^3+c x^6\right )^{3/2}}{2880 c^4}-\frac{b x^6 \left (a+b x^3+c x^6\right )^{3/2}}{20 c^2}+\frac{x^9 \left (a+b x^3+c x^6\right )^{3/2}}{18 c} \]

Antiderivative was successfully verified.

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

[Out]

((21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*(b + 2*c*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1536
*c^5) - (b*x^6*(a + b*x^3 + c*x^6)^(3/2))/(20*c^2) + (x^9*(a + b*x^3 + c*x^6)^(3
/2))/(18*c) - ((7*b*(15*b^2 - 28*a*c) - 6*c*(21*b^2 - 20*a*c)*x^3)*(a + b*x^3 +
c*x^6)^(3/2))/(2880*c^4) - ((b^2 - 4*a*c)*(21*b^4 - 56*a*b^2*c + 16*a^2*c^2)*Arc
Tanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(3072*c^(11/2))

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Rubi in Sympy [A]  time = 51.703, size = 221, normalized size = 0.96 \[ - \frac{b x^{6} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{20 c^{2}} + \frac{x^{9} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{18 c} - \frac{\left (\frac{21 b \left (- 28 a c + 15 b^{2}\right )}{8} - \frac{9 c x^{3} \left (- 20 a c + 21 b^{2}\right )}{4}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{1080 c^{4}} + \frac{\left (b + 2 c x^{3}\right ) \sqrt{a + b x^{3} + c x^{6}} \left (16 a^{2} c^{2} - 56 a b^{2} c + 21 b^{4}\right )}{1536 c^{5}} - \frac{\left (- 4 a c + b^{2}\right ) \left (16 a^{2} c^{2} - 56 a b^{2} c + 21 b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3072 c^{\frac{11}{2}}} \]

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]

-b*x**6*(a + b*x**3 + c*x**6)**(3/2)/(20*c**2) + x**9*(a + b*x**3 + c*x**6)**(3/
2)/(18*c) - (21*b*(-28*a*c + 15*b**2)/8 - 9*c*x**3*(-20*a*c + 21*b**2)/4)*(a + b
*x**3 + c*x**6)**(3/2)/(1080*c**4) + (b + 2*c*x**3)*sqrt(a + b*x**3 + c*x**6)*(1
6*a**2*c**2 - 56*a*b**2*c + 21*b**4)/(1536*c**5) - (-4*a*c + b**2)*(16*a**2*c**2
- 56*a*b**2*c + 21*b**4)*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x*
*6)))/(3072*c**(11/2))

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Mathematica [A]  time = 0.203254, size = 203, normalized size = 0.88 \[ \frac{2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (16 b c^2 \left (113 a^2-34 a c x^6+8 c^2 x^{12}\right )+160 c^3 x^3 \left (-3 a^2+2 a c x^6+8 c^2 x^{12}\right )+168 b^3 c \left (c x^6-10 a\right )+16 b^2 c^2 x^3 \left (56 a-9 c x^6\right )+315 b^5-210 b^4 c x^3\right )-15 \left (b^2-4 a c\right ) \left (16 a^2 c^2-56 a b^2 c+21 b^4\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{46080 c^{11/2}} \]

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]*(315*b^5 - 210*b^4*c*x^3 + 16*b^2*c^2*x^3*(56
*a - 9*c*x^6) + 168*b^3*c*(-10*a + c*x^6) + 16*b*c^2*(113*a^2 - 34*a*c*x^6 + 8*c
^2*x^12) + 160*c^3*x^3*(-3*a^2 + 2*a*c*x^6 + 8*c^2*x^12)) - 15*(b^2 - 4*a*c)*(21
*b^4 - 56*a*b^2*c + 16*a^2*c^2)*Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x
^6]])/(46080*c^(11/2))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{x}^{14}\sqrt{c{x}^{6}+b{x}^{3}+a}\, dx \]

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)

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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^14,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294425, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (1280 \, c^{5} x^{15} + 128 \, b c^{4} x^{12} - 16 \,{\left (9 \, b^{2} c^{3} - 20 \, a c^{4}\right )} x^{9} + 8 \,{\left (21 \, b^{3} c^{2} - 68 \, a b c^{3}\right )} x^{6} + 315 \, b^{5} - 1680 \, a b^{3} c + 1808 \, a^{2} b c^{2} - 2 \,{\left (105 \, b^{4} c - 448 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} - 15 \,{\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{92160 \, c^{\frac{11}{2}}}, \frac{2 \,{\left (1280 \, c^{5} x^{15} + 128 \, b c^{4} x^{12} - 16 \,{\left (9 \, b^{2} c^{3} - 20 \, a c^{4}\right )} x^{9} + 8 \,{\left (21 \, b^{3} c^{2} - 68 \, a b c^{3}\right )} x^{6} + 315 \, b^{5} - 1680 \, a b^{3} c + 1808 \, a^{2} b c^{2} - 2 \,{\left (105 \, b^{4} c - 448 \, a b^{2} c^{2} + 240 \, a^{2} c^{3}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} - 15 \,{\left (21 \, b^{6} - 140 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{46080 \, \sqrt{-c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/92160*(4*(1280*c^5*x^15 + 128*b*c^4*x^12 - 16*(9*b^2*c^3 - 20*a*c^4)*x^9 + 8*
(21*b^3*c^2 - 68*a*b*c^3)*x^6 + 315*b^5 - 1680*a*b^3*c + 1808*a^2*b*c^2 - 2*(105
*b^4*c - 448*a*b^2*c^2 + 240*a^2*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c) - 15*
(21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*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^(11/2
), 1/46080*(2*(1280*c^5*x^15 + 128*b*c^4*x^12 - 16*(9*b^2*c^3 - 20*a*c^4)*x^9 +
8*(21*b^3*c^2 - 68*a*b*c^3)*x^6 + 315*b^5 - 1680*a*b^3*c + 1808*a^2*b*c^2 - 2*(1
05*b^4*c - 448*a*b^2*c^2 + 240*a^2*c^3)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) -
15*(21*b^6 - 140*a*b^4*c + 240*a^2*b^2*c^2 - 64*a^3*c^3)*arctan(1/2*(2*c*x^3 + b
)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*c)))/(sqrt(-c)*c^5)]

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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