3.235 \(\int \frac{x^{11}}{\left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=137 \[ \frac{\left (-8 a c+3 b^2-2 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{2 c^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.256959, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\left (-8 a c+3 b^2-2 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{3 c^2 \left (b^2-4 a c\right )}+\frac{2 x^6 \left (2 a+b x^3\right )}{3 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 29.1089, size = 126, normalized size = 0.92 \[ - \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{2 c^{\frac{5}{2}}} + \frac{2 x^{6} \left (2 a + b x^{3}\right )}{3 \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}} + \frac{\sqrt{a + b x^{3} + c x^{6}} \left (- 8 a c + 3 b^{2} - 2 b c x^{3}\right )}{3 c^{2} \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6)))/(2*c**(5/2)) + 2*
x**6*(2*a + b*x**3)/(3*(-4*a*c + b**2)*sqrt(a + b*x**3 + c*x**6)) + sqrt(a + b*x
**3 + c*x**6)*(-8*a*c + 3*b**2 - 2*b*c*x**3)/(3*c**2*(-4*a*c + b**2))

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Mathematica [A]  time = 0.19916, size = 127, normalized size = 0.93 \[ \frac{1}{3} \sqrt{a+b x^3+c x^6} \left (\frac{2 \left (2 a^2 c-a b^2+3 a b c x^3-b^3 x^3\right )}{c^2 \left (4 a c-b^2\right ) \left (a+b x^3+c x^6\right )}+\frac{1}{c^2}\right )-\frac{b \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )}{2 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{{x}^{11} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

int(x^11/(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^11/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.302761, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (3 \, b^{3} - 10 \, a b c\right )} x^{3} + 3 \, a b^{2} - 8 \, a^{2} c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} + 3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} + a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{3}\right )} \log \left (4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{12 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} + a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{3}\right )} \sqrt{c}}, \frac{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{6} +{\left (3 \, b^{3} - 10 \, a b c\right )} x^{3} + 3 \, a b^{2} - 8 \, a^{2} c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} - 3 \,{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{6} + a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{3}\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{6 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{6} + a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{3}\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="fricas")

[Out]

[1/12*(4*((b^2*c - 4*a*c^2)*x^6 + (3*b^3 - 10*a*b*c)*x^3 + 3*a*b^2 - 8*a^2*c)*sq
rt(c*x^6 + b*x^3 + a)*sqrt(c) + 3*((b^3*c - 4*a*b*c^2)*x^6 + a*b^3 - 4*a^2*b*c +
(b^4 - 4*a*b^2*c)*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^3 - 4*a*c^4)*x^6 + a*b^2*c^2
- 4*a^2*c^3 + (b^3*c^2 - 4*a*b*c^3)*x^3)*sqrt(c)), 1/6*(2*((b^2*c - 4*a*c^2)*x^6
+ (3*b^3 - 10*a*b*c)*x^3 + 3*a*b^2 - 8*a^2*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c)
- 3*((b^3*c - 4*a*b*c^2)*x^6 + a*b^3 - 4*a^2*b*c + (b^4 - 4*a*b^2*c)*x^3)*arctan
(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*c)))/(((b^2*c^3 - 4*a*c^4)*
x^6 + a*b^2*c^2 - 4*a^2*c^3 + (b^3*c^2 - 4*a*b*c^3)*x^3)*sqrt(-c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{11}}{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x**11/(a + b*x**3 + c*x**6)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{11}}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^6 + b*x^3 + a)^(3/2),x, algorithm="giac")

[Out]

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