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

Optimal. Leaf size=121 \[ -\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{7/2}}+\frac{\left (-16 a c+15 b^2-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}+\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c} \]

[Out]

(x^6*Sqrt[a + b*x^3 + c*x^6])/(9*c) + ((15*b^2 - 16*a*c - 10*b*c*x^3)*Sqrt[a + b
*x^3 + c*x^6])/(72*c^3) - (b*(5*b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*S
qrt[a + b*x^3 + c*x^6])])/(48*c^(7/2))

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Rubi [A]  time = 0.253063, antiderivative size = 121, 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 \[ -\frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )}{48 c^{7/2}}+\frac{\left (-16 a c+15 b^2-10 b c x^3\right ) \sqrt{a+b x^3+c x^6}}{72 c^3}+\frac{x^6 \sqrt{a+b x^3+c x^6}}{9 c} \]

Antiderivative was successfully verified.

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

[Out]

(x^6*Sqrt[a + b*x^3 + c*x^6])/(9*c) + ((15*b^2 - 16*a*c - 10*b*c*x^3)*Sqrt[a + b
*x^3 + c*x^6])/(72*c^3) - (b*(5*b^2 - 12*a*c)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*S
qrt[a + b*x^3 + c*x^6])])/(48*c^(7/2))

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Rubi in Sympy [A]  time = 26.0097, size = 114, normalized size = 0.94 \[ - \frac{b \left (- 12 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{48 c^{\frac{7}{2}}} + \frac{x^{6} \sqrt{a + b x^{3} + c x^{6}}}{9 c} + \frac{\sqrt{a + b x^{3} + c x^{6}} \left (- 4 a c + \frac{15 b^{2}}{4} - \frac{5 b c x^{3}}{2}\right )}{18 c^{3}} \]

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*(-12*a*c + 5*b**2)*atanh((b + 2*c*x**3)/(2*sqrt(c)*sqrt(a + b*x**3 + c*x**6))
)/(48*c**(7/2)) + x**6*sqrt(a + b*x**3 + c*x**6)/(9*c) + sqrt(a + b*x**3 + c*x**
6)*(-4*a*c + 15*b**2/4 - 5*b*c*x**3/2)/(18*c**3)

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Mathematica [A]  time = 0.100495, size = 102, normalized size = 0.84 \[ \frac{\left (36 a b c-15 b^3\right ) \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )+2 \sqrt{c} \sqrt{a+b x^3+c x^6} \left (8 c \left (c x^6-2 a\right )+15 b^2-10 b c x^3\right )}{144 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]*(15*b^2 - 10*b*c*x^3 + 8*c*(-2*a + c*x^6)) +
(-15*b^3 + 36*a*b*c)*Log[b + 2*c*x^3 + 2*Sqrt[c]*Sqrt[a + b*x^3 + c*x^6]])/(144*
c^(7/2))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{{x}^{11}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

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)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28028, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (8 \, c^{2} x^{6} - 10 \, b c x^{3} + 15 \, b^{2} - 16 \, a c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{c} - 3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} \log \left (-4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (2 \, c^{2} x^{3} + b c\right )} -{\left (8 \, c^{2} x^{6} + 8 \, b c x^{3} + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{288 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (8 \, c^{2} x^{6} - 10 \, b c x^{3} + 15 \, b^{2} - 16 \, a c\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-c} - 3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} \arctan \left (\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{6} + b x^{3} + a} c}\right )}{144 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/288*(4*(8*c^2*x^6 - 10*b*c*x^3 + 15*b^2 - 16*a*c)*sqrt(c*x^6 + b*x^3 + a)*sqr
t(c) - 3*(5*b^3 - 12*a*b*c)*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^(7/2), 1/144*(2*(8*c^2*x^6 - 10
*b*c*x^3 + 15*b^2 - 16*a*c)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-c) - 3*(5*b^3 - 12*a*b
*c)*arctan(1/2*(2*c*x^3 + b)*sqrt(-c)/(sqrt(c*x^6 + b*x^3 + a)*c)))/(sqrt(-c)*c^
3)]

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

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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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