3.227 \(\int \frac{1}{x^{10} \sqrt{a+b x^3+c x^6}} \, dx\)

Optimal. Leaf size=145 \[ \frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{7/2}}-\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{72 a^3 x^3}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9} \]

[Out]

-Sqrt[a + b*x^3 + c*x^6]/(9*a*x^9) + (5*b*Sqrt[a + b*x^3 + c*x^6])/(36*a^2*x^6)
- ((15*b^2 - 16*a*c)*Sqrt[a + b*x^3 + c*x^6])/(72*a^3*x^3) + (b*(5*b^2 - 12*a*c)
*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(48*a^(7/2))

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Rubi [A]  time = 0.394019, antiderivative size = 145, 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 \[ \frac{b \left (5 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{7/2}}-\frac{\left (15 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{72 a^3 x^3}+\frac{5 b \sqrt{a+b x^3+c x^6}}{36 a^2 x^6}-\frac{\sqrt{a+b x^3+c x^6}}{9 a x^9} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

-Sqrt[a + b*x^3 + c*x^6]/(9*a*x^9) + (5*b*Sqrt[a + b*x^3 + c*x^6])/(36*a^2*x^6)
- ((15*b^2 - 16*a*c)*Sqrt[a + b*x^3 + c*x^6])/(72*a^3*x^3) + (b*(5*b^2 - 12*a*c)
*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(48*a^(7/2))

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Rubi in Sympy [A]  time = 44.584, size = 133, normalized size = 0.92 \[ - \frac{\sqrt{a + b x^{3} + c x^{6}}}{9 a x^{9}} + \frac{5 b \sqrt{a + b x^{3} + c x^{6}}}{36 a^{2} x^{6}} - \frac{\left (- 16 a c + 15 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{72 a^{3} x^{3}} + \frac{b \left (- 12 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{48 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

-sqrt(a + b*x**3 + c*x**6)/(9*a*x**9) + 5*b*sqrt(a + b*x**3 + c*x**6)/(36*a**2*x
**6) - (-16*a*c + 15*b**2)*sqrt(a + b*x**3 + c*x**6)/(72*a**3*x**3) + b*(-12*a*c
+ 5*b**2)*atanh((2*a + b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/(48*a**(7
/2))

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Mathematica [A]  time = 0.488732, size = 117, normalized size = 0.81 \[ \frac{b \left (12 a c-5 b^2\right ) \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )}{48 a^{7/2}}-\frac{\sqrt{a+b x^3+c x^6} \left (8 a^2-2 a \left (5 b x^3+8 c x^6\right )+15 b^2 x^6\right )}{72 a^3 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*Sqrt[a + b*x^3 + c*x^6]),x]

[Out]

-(Sqrt[a + b*x^3 + c*x^6]*(8*a^2 + 15*b^2*x^6 - 2*a*(5*b*x^3 + 8*c*x^6)))/(72*a^
3*x^9) + (b*(-5*b^2 + 12*a*c)*(Log[x^3] - Log[2*a + b*x^3 + 2*Sqrt[a]*Sqrt[a + b
*x^3 + c*x^6]]))/(48*a^(7/2))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10}}{\frac{1}{\sqrt{c{x}^{6}+b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(c*x^6+b*x^3+a)^(1/2),x)

[Out]

int(1/x^10/(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(1/(sqrt(c*x^6 + b*x^3 + a)*x^10),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.291736, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} x^{9} \log \left (\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) + 4 \,{\left ({\left (15 \, b^{2} - 16 \, a c\right )} x^{6} - 10 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{a}}{288 \, a^{\frac{7}{2}} x^{9}}, \frac{3 \,{\left (5 \, b^{3} - 12 \, a b c\right )} x^{9} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \,{\left ({\left (15 \, b^{2} - 16 \, a c\right )} x^{6} - 10 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a}}{144 \, \sqrt{-a} a^{3} x^{9}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^10),x, algorithm="fricas")

[Out]

[-1/288*(3*(5*b^3 - 12*a*b*c)*x^9*log((4*sqrt(c*x^6 + b*x^3 + a)*(a*b*x^3 + 2*a^
2) - ((b^2 + 4*a*c)*x^6 + 8*a*b*x^3 + 8*a^2)*sqrt(a))/x^6) + 4*((15*b^2 - 16*a*c
)*x^6 - 10*a*b*x^3 + 8*a^2)*sqrt(c*x^6 + b*x^3 + a)*sqrt(a))/(a^(7/2)*x^9), 1/14
4*(3*(5*b^3 - 12*a*b*c)*x^9*arctan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^
3 + a)*a)) - 2*((15*b^2 - 16*a*c)*x^6 - 10*a*b*x^3 + 8*a^2)*sqrt(c*x^6 + b*x^3 +
a)*sqrt(-a))/(sqrt(-a)*a^3*x^9)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{10} \sqrt{a + b x^{3} + c x^{6}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(c*x**6+b*x**3+a)**(1/2),x)

[Out]

Integral(1/(x**10*sqrt(a + b*x**3 + c*x**6)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{6} + b x^{3} + a} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^10),x, algorithm="giac")

[Out]

integrate(1/(sqrt(c*x^6 + b*x^3 + a)*x^10), x)