3.210 \(\int \frac{\left (a+b x^3+c x^6\right )^{3/2}}{x^{13}} \, dx\)

Optimal. Leaf size=133 \[ -\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{128 a^{5/2}}+\frac{\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{64 a^2 x^6}-\frac{\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}} \]

[Out]

((b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(64*a^2*x^6) - ((2*a + b*x
^3)*(a + b*x^3 + c*x^6)^(3/2))/(24*a*x^12) - ((b^2 - 4*a*c)^2*ArcTanh[(2*a + b*x
^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(128*a^(5/2))

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Rubi [A]  time = 0.23522, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{128 a^{5/2}}+\frac{\left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{64 a^2 x^6}-\frac{\left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{24 a x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)^(3/2)/x^13,x]

[Out]

((b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(64*a^2*x^6) - ((2*a + b*x
^3)*(a + b*x^3 + c*x^6)^(3/2))/(24*a*x^12) - ((b^2 - 4*a*c)^2*ArcTanh[(2*a + b*x
^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6])])/(128*a^(5/2))

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Rubi in Sympy [A]  time = 28.0324, size = 119, normalized size = 0.89 \[ - \frac{\left (2 a + b x^{3}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{24 a x^{12}} + \frac{\left (2 a + b x^{3}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{64 a^{2} x^{6}} - \frac{\left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{128 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)**(3/2)/x**13,x)

[Out]

-(2*a + b*x**3)*(a + b*x**3 + c*x**6)**(3/2)/(24*a*x**12) + (2*a + b*x**3)*(-4*a
*c + b**2)*sqrt(a + b*x**3 + c*x**6)/(64*a**2*x**6) - (-4*a*c + b**2)**2*atanh((
2*a + b*x**3)/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/(128*a**(5/2))

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Mathematica [A]  time = 0.233658, size = 125, normalized size = 0.94 \[ \frac{3 \left (b^2-4 a c\right )^2 \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )-\frac{2 \sqrt{a} \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6} \left (8 a^2+8 a b x^3+20 a c x^6-3 b^2 x^6\right )}{x^{12}}}{384 a^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^13,x]

[Out]

((-2*Sqrt[a]*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6]*(8*a^2 + 8*a*b*x^3 - 3*b^2*x^
6 + 20*a*c*x^6))/x^12 + 3*(b^2 - 4*a*c)^2*(Log[x^3] - Log[2*a + b*x^3 + 2*Sqrt[a
]*Sqrt[a + b*x^3 + c*x^6]]))/(384*a^(5/2))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{13}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)^(3/2)/x^13,x)

[Out]

int((c*x^6+b*x^3+a)^(3/2)/x^13,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((c*x^6 + b*x^3 + a)^(3/2)/x^13,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.305055, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} x^{12} \log \left (\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) + 4 \,{\left ({\left (3 \, b^{3} - 20 \, a b c\right )} x^{9} - 2 \,{\left (a b^{2} + 20 \, a^{2} c\right )} x^{6} - 24 \, a^{2} b x^{3} - 16 \, a^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{a}}{768 \, a^{\frac{5}{2}} x^{12}}, -\frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} x^{12} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \,{\left ({\left (3 \, b^{3} - 20 \, a b c\right )} x^{9} - 2 \,{\left (a b^{2} + 20 \, a^{2} c\right )} x^{6} - 24 \, a^{2} b x^{3} - 16 \, a^{3}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a}}{384 \, \sqrt{-a} a^{2} x^{12}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x^13,x, algorithm="fricas")

[Out]

[1/768*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*x^12*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*((3*b
^3 - 20*a*b*c)*x^9 - 2*(a*b^2 + 20*a^2*c)*x^6 - 24*a^2*b*x^3 - 16*a^3)*sqrt(c*x^
6 + b*x^3 + a)*sqrt(a))/(a^(5/2)*x^12), -1/384*(3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)
*x^12*arctan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a)*a)) - 2*((3*b^3
- 20*a*b*c)*x^9 - 2*(a*b^2 + 20*a^2*c)*x^6 - 24*a^2*b*x^3 - 16*a^3)*sqrt(c*x^6
+ b*x^3 + a)*sqrt(-a))/(sqrt(-a)*a^2*x^12)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{x^{13}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**6+b*x**3+a)**(3/2)/x**13,x)

[Out]

Integral((a + b*x**3 + c*x**6)**(3/2)/x**13, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x^{13}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x^13,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)/x^13, x)