3.242 \(\int \frac{1}{x^{10} \left (a+b x^3+c x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=256 \[ \frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{9/2}}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a+b x^3+c x^6}}{36 a^3 x^6 \left (b^2-4 a c\right )}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 x^9 \left (b^2-4 a c\right )}-\frac{\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt{a+b x^3+c x^6}}{72 a^4 x^3 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^9 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^9*Sqrt[a + b*x^3 + c*x^6]) - ((
7*b^2 - 16*a*c)*Sqrt[a + b*x^3 + c*x^6])/(9*a^2*(b^2 - 4*a*c)*x^9) + (b*(35*b^2
- 116*a*c)*Sqrt[a + b*x^3 + c*x^6])/(36*a^3*(b^2 - 4*a*c)*x^6) - ((105*b^4 - 460
*a*b^2*c + 256*a^2*c^2)*Sqrt[a + b*x^3 + c*x^6])/(72*a^4*(b^2 - 4*a*c)*x^3) + (5
*b*(7*b^2 - 12*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/
(48*a^(9/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.683328, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{5 b \left (7 b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{9/2}}+\frac{b \left (35 b^2-116 a c\right ) \sqrt{a+b x^3+c x^6}}{36 a^3 x^6 \left (b^2-4 a c\right )}-\frac{\left (7 b^2-16 a c\right ) \sqrt{a+b x^3+c x^6}}{9 a^2 x^9 \left (b^2-4 a c\right )}-\frac{\left (256 a^2 c^2-460 a b^2 c+105 b^4\right ) \sqrt{a+b x^3+c x^6}}{72 a^4 x^3 \left (b^2-4 a c\right )}+\frac{2 \left (-2 a c+b^2+b c x^3\right )}{3 a x^9 \left (b^2-4 a c\right ) \sqrt{a+b x^3+c x^6}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b^2 - 2*a*c + b*c*x^3))/(3*a*(b^2 - 4*a*c)*x^9*Sqrt[a + b*x^3 + c*x^6]) - ((
7*b^2 - 16*a*c)*Sqrt[a + b*x^3 + c*x^6])/(9*a^2*(b^2 - 4*a*c)*x^9) + (b*(35*b^2
- 116*a*c)*Sqrt[a + b*x^3 + c*x^6])/(36*a^3*(b^2 - 4*a*c)*x^6) - ((105*b^4 - 460
*a*b^2*c + 256*a^2*c^2)*Sqrt[a + b*x^3 + c*x^6])/(72*a^4*(b^2 - 4*a*c)*x^3) + (5
*b*(7*b^2 - 12*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/
(48*a^(9/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 94.4524, size = 241, normalized size = 0.94 \[ \frac{2 \left (- 2 a c + b^{2} + b c x^{3}\right )}{3 a x^{9} \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}} - \frac{\left (- 16 a c + 7 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{9 a^{2} x^{9} \left (- 4 a c + b^{2}\right )} + \frac{b \left (- 116 a c + 35 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{36 a^{3} x^{6} \left (- 4 a c + b^{2}\right )} - \frac{\sqrt{a + b x^{3} + c x^{6}} \left (256 a^{2} c^{2} - 460 a b^{2} c + 105 b^{4}\right )}{72 a^{4} x^{3} \left (- 4 a c + b^{2}\right )} + \frac{5 b \left (- 12 a c + 7 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{48 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-2*a*c + b**2 + b*c*x**3)/(3*a*x**9*(-4*a*c + b**2)*sqrt(a + b*x**3 + c*x**6)
) - (-16*a*c + 7*b**2)*sqrt(a + b*x**3 + c*x**6)/(9*a**2*x**9*(-4*a*c + b**2)) +
b*(-116*a*c + 35*b**2)*sqrt(a + b*x**3 + c*x**6)/(36*a**3*x**6*(-4*a*c + b**2))
- sqrt(a + b*x**3 + c*x**6)*(256*a**2*c**2 - 460*a*b**2*c + 105*b**4)/(72*a**4*
x**3*(-4*a*c + b**2)) + 5*b*(-12*a*c + 7*b**2)*atanh((2*a + b*x**3)/(2*sqrt(a)*s
qrt(a + b*x**3 + c*x**6)))/(48*a**(9/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.522135, size = 191, normalized size = 0.75 \[ \frac{5 b \left (12 a c-7 b^2\right ) \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )}{48 a^{9/2}}+\frac{\sqrt{a+b x^3+c x^6} \left (-\frac{48 \left (5 a^2 b c^2+2 a^2 c^3 x^3-5 a b^3 c-4 a b^2 c^2 x^3+b^5+b^4 c x^3\right )}{\left (b^2-4 a c\right ) \left (a+b x^3+c x^6\right )}-\frac{8 a^2}{x^9}+\frac{40 a c-57 b^2}{x^3}+\frac{22 a b}{x^6}\right )}{72 a^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b*x^3 + c*x^6]*((-8*a^2)/x^9 + (22*a*b)/x^6 + (-57*b^2 + 40*a*c)/x^3 -
(48*(b^5 - 5*a*b^3*c + 5*a^2*b*c^2 + b^4*c*x^3 - 4*a*b^2*c^2*x^3 + 2*a^2*c^3*x^
3))/((b^2 - 4*a*c)*(a + b*x^3 + c*x^6))))/(72*a^4) + (5*b*(-7*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^(9/2))

_______________________________________________________________________________________

Maple [F]  time = 0.091, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{-{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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/((c*x^6 + b*x^3 + a)^(3/2)*x^10),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.388079, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^10),x, algorithm="fricas")

[Out]

[-1/288*(4*((105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*x^12 + (105*b^5 - 530*a*b^
3*c + 488*a^2*b*c^2)*x^9 + (35*a*b^4 - 172*a^2*b^2*c + 128*a^3*c^2)*x^6 + 8*a^3*
b^2 - 32*a^4*c - 14*(a^2*b^3 - 4*a^3*b*c)*x^3)*sqrt(c*x^6 + b*x^3 + a)*sqrt(a) +
15*((7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*x^15 + (7*b^6 - 40*a*b^4*c + 48*a^2
*b^2*c^2)*x^12 + (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2)*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))/(((a^4*b^2*c - 4*a^5*c^2)*x^15 + (a^4*b^3 - 4*a^5*b*c)*x^12 + (a^5*b^2 -
4*a^6*c)*x^9)*sqrt(a)), -1/144*(2*((105*b^4*c - 460*a*b^2*c^2 + 256*a^2*c^3)*x^
12 + (105*b^5 - 530*a*b^3*c + 488*a^2*b*c^2)*x^9 + (35*a*b^4 - 172*a^2*b^2*c + 1
28*a^3*c^2)*x^6 + 8*a^3*b^2 - 32*a^4*c - 14*(a^2*b^3 - 4*a^3*b*c)*x^3)*sqrt(c*x^
6 + b*x^3 + a)*sqrt(-a) - 15*((7*b^5*c - 40*a*b^3*c^2 + 48*a^2*b*c^3)*x^15 + (7*
b^6 - 40*a*b^4*c + 48*a^2*b^2*c^2)*x^12 + (7*a*b^5 - 40*a^2*b^3*c + 48*a^3*b*c^2
)*x^9)*arctan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a)*a)))/(((a^4*b^
2*c - 4*a^5*c^2)*x^15 + (a^4*b^3 - 4*a^5*b*c)*x^12 + (a^5*b^2 - 4*a^6*c)*x^9)*sq
rt(-a))]

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{10} \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**10*(a + b*x**3 + c*x**6)**(3/2)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^6 + b*x^3 + a)^(3/2)*x^10),x, algorithm="giac")

[Out]

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