3.234 \(\int \frac{x^{14}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=195 \[ \frac{\left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{8 c^{7/2}}-\frac{\left (b \left (15 b^2-52 a c\right )-2 c x^3 \left (5 b^2-12 a c\right )\right ) \sqrt{a+b x^3+c x^6}}{12 c^3 \left (b^2-4 a c\right )}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

[Out]

(2*x^9*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) - (2*b*x^6*Sqrt[
a + b*x^3 + c*x^6])/(3*c*(b^2 - 4*a*c)) - ((b*(15*b^2 - 52*a*c) - 2*c*(5*b^2 - 1
2*a*c)*x^3)*Sqrt[a + b*x^3 + c*x^6])/(12*c^3*(b^2 - 4*a*c)) + ((5*b^2 - 4*a*c)*A
rcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(8*c^(7/2))

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Rubi [A]  time = 0.546425, antiderivative size = 195, 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 \[ \frac{\left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{8 c^{7/2}}-\frac{\left (b \left (15 b^2-52 a c\right )-2 c x^3 \left (5 b^2-12 a c\right )\right ) \sqrt{a+b x^3+c x^6}}{12 c^3 \left (b^2-4 a c\right )}-\frac{2 b x^6 \sqrt{a+b x^3+c x^6}}{3 c \left (b^2-4 a c\right )}+\frac{2 x^9 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^9*(2*a + b*x^3))/(3*(b^2 - 4*a*c)*Sqrt[a + b*x^3 + c*x^6]) - (2*b*x^6*Sqrt[
a + b*x^3 + c*x^6])/(3*c*(b^2 - 4*a*c)) - ((b*(15*b^2 - 52*a*c) - 2*c*(5*b^2 - 1
2*a*c)*x^3)*Sqrt[a + b*x^3 + c*x^6])/(12*c^3*(b^2 - 4*a*c)) + ((5*b^2 - 4*a*c)*A
rcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6])])/(8*c^(7/2))

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Rubi in Sympy [A]  time = 47.4993, size = 182, normalized size = 0.93 \[ - \frac{2 b x^{6} \sqrt{a + b x^{3} + c x^{6}}}{3 c \left (- 4 a c + b^{2}\right )} + \frac{2 x^{9} \left (2 a + b x^{3}\right )}{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}} - \frac{\left (b \left (- 39 a c + \frac{45 b^{2}}{4}\right ) - \frac{3 c x^{3} \left (- 12 a c + 5 b^{2}\right )}{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{9 c^{3} \left (- 4 a c + b^{2}\right )} + \frac{\left (- 4 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{8 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*b*x**6*sqrt(a + b*x**3 + c*x**6)/(3*c*(-4*a*c + b**2)) + 2*x**9*(2*a + b*x**3
)/(3*(-4*a*c + b**2)*sqrt(a + b*x**3 + c*x**6)) - (b*(-39*a*c + 45*b**2/4) - 3*c
*x**3*(-12*a*c + 5*b**2)/2)*sqrt(a + b*x**3 + c*x**6)/(9*c**3*(-4*a*c + b**2)) +
(-4*a*c + 5*b**2)*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(
8*c**(7/2))

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Mathematica [A]  time = 0.268702, size = 147, normalized size = 0.75 \[ \frac{\sqrt{a+b x^3+c x^6} \left (-\frac{8 \left (a^2 c \left (2 c x^3-3 b\right )+a b^2 \left (b-4 c x^3\right )+b^4 x^3\right )}{\left (b^2-4 a c\right ) \left (a+b x^3+c x^6\right )}-7 b+2 c x^3\right )}{12 c^3}+\frac{\left (5 b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{8 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^3 + c*x^6]*(-7*b + 2*c*x^3 - (8*(b^4*x^3 + a*b^2*(b - 4*c*x^3) + a
^2*c*(-3*b + 2*c*x^3)))/((b^2 - 4*a*c)*(a + b*x^3 + c*x^6))))/(12*c^3) + ((5*b^2
- 4*a*c)*Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(8*c^(7/2))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{{x}^{14} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^14/(c*x^6+b*x^3+a)^(3/2),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(x^14/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.351939, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{9} - 5 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} - 15 \, a b^{3} + 52 \, a^{2} b c -{\left (15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} - 3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{6} + 5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{3}\right )} \log \left (4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{48 \,{\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3}\right )} \sqrt{c}}, \frac{2 \,{\left (2 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{9} - 5 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} - 15 \, a b^{3} + 52 \, a^{2} b c -{\left (15 \, b^{4} - 62 \, a b^{2} c + 24 \, a^{2} c^{2}\right )} x^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} + 3 \,{\left ({\left (5 \, b^{4} c - 24 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} x^{6} + 5 \, a b^{4} - 24 \, a^{2} b^{2} c + 16 \, a^{3} c^{2} +{\left (5 \, b^{5} - 24 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{3}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{24 \,{\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{6} + a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/48*(4*(2*(b^2*c^2 - 4*a*c^3)*x^9 - 5*(b^3*c - 4*a*b*c^2)*x^6 - 15*a*b^3 + 52*
a^2*b*c - (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(c
) - 3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^6 + 5*a*b^4 - 24*a^2*b^2*c + 16*a
^3*c^2 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*x^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)))/(((b^2*c^4 -
4*a*c^5)*x^6 + a*b^2*c^3 - 4*a^2*c^4 + (b^3*c^3 - 4*a*b*c^4)*x^3)*sqrt(c)), 1/2
4*(2*(2*(b^2*c^2 - 4*a*c^3)*x^9 - 5*(b^3*c - 4*a*b*c^2)*x^6 - 15*a*b^3 + 52*a^2*
b*c - (15*b^4 - 62*a*b^2*c + 24*a^2*c^2)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) +
3*((5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*x^6 + 5*a*b^4 - 24*a^2*b^2*c + 16*a^3*
c^2 + (5*b^5 - 24*a*b^3*c + 16*a^2*b*c^2)*x^3)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)
/(sqrt(c*x^6 + b*x^3 + a)*c)))/(((b^2*c^4 - 4*a*c^5)*x^6 + a*b^2*c^3 - 4*a^2*c^4
+ (b^3*c^3 - 4*a*b*c^4)*x^3)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**14/(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="giac")

[Out]

integrate(x^14/(c*x^6 + b*x^3 + a)^(3/2), x)