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

Optimal. Leaf size=163 \[ \frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{3/2}}-\frac{\left (x^3 \left (8 a c+b^2\right )+2 a b\right ) \sqrt{a+b x^3+c x^6}}{24 a x^6}+\frac{1}{3} c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9} \]

[Out]

-((2*a*b + (b^2 + 8*a*c)*x^3)*Sqrt[a + b*x^3 + c*x^6])/(24*a*x^6) - (a + b*x^3 +
c*x^6)^(3/2)/(9*x^9) + (b*(b^2 - 12*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[
a + b*x^3 + c*x^6])])/(48*a^(3/2)) + (c^(3/2)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*S
qrt[a + b*x^3 + c*x^6])])/3

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Rubi [A]  time = 0.427679, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35 \[ \frac{b \left (b^2-12 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{48 a^{3/2}}-\frac{\left (x^3 \left (8 a c+b^2\right )+2 a b\right ) \sqrt{a+b x^3+c x^6}}{24 a x^6}+\frac{1}{3} c^{3/2} \tanh ^{-1}\left (\frac{b+2 c x^3}{2 \sqrt{c} \sqrt{a+b x^3+c x^6}}\right )-\frac{\left (a+b x^3+c x^6\right )^{3/2}}{9 x^9} \]

Antiderivative was successfully verified.

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

[Out]

-((2*a*b + (b^2 + 8*a*c)*x^3)*Sqrt[a + b*x^3 + c*x^6])/(24*a*x^6) - (a + b*x^3 +
c*x^6)^(3/2)/(9*x^9) + (b*(b^2 - 12*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[
a + b*x^3 + c*x^6])])/(48*a^(3/2)) + (c^(3/2)*ArcTanh[(b + 2*c*x^3)/(2*Sqrt[c]*S
qrt[a + b*x^3 + c*x^6])])/3

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Rubi in Sympy [A]  time = 45.878, size = 146, normalized size = 0.9 \[ \frac{c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x^{3}}{2 \sqrt{c} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{3} - \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{9 x^{9}} - \frac{\left (a b + x^{3} \left (4 a c + \frac{b^{2}}{2}\right )\right ) \sqrt{a + b x^{3} + c x^{6}}}{12 a x^{6}} + \frac{b \left (- 12 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{48 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.471896, size = 174, normalized size = 1.07 \[ \frac{-2 \sqrt{a} \left (\sqrt{a+b x^3+c x^6} \left (8 a^2+14 a b x^3+32 a c x^6+3 b^2 x^6\right )-24 a c^{3/2} x^9 \log \left (2 \sqrt{c} \sqrt{a+b x^3+c x^6}+b+2 c x^3\right )\right )-3 b x^9 \log \left (x^3\right ) \left (b^2-12 a c\right )+3 b x^9 \left (b^2-12 a c\right ) \log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )}{144 a^{3/2} x^9} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*(b^2 - 12*a*c)*x^9*Log[x^3] + 3*b*(b^2 - 12*a*c)*x^9*Log[2*a + b*x^3 + 2*S
qrt[a]*Sqrt[a + b*x^3 + c*x^6]] - 2*Sqrt[a]*(Sqrt[a + b*x^3 + c*x^6]*(8*a^2 + 14
*a*b*x^3 + 3*b^2*x^6 + 32*a*c*x^6) - 24*a*c^(3/2)*x^9*Log[b + 2*c*x^3 + 2*Sqrt[c
]*Sqrt[a + b*x^3 + c*x^6]]))/(144*a^(3/2)*x^9)

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Maple [F]  time = 0.066, 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((c*x^6+b*x^3+a)^(3/2)/x^10,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.361038, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/288*(48*a^(3/2)*c^(3/2)*x^9*log(-8*c^2*x^6 - 8*b*c*x^3 - b^2 - 4*sqrt(c*x^6 +
b*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*a*c) - 3*(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)*s
qrt(a))/x^6) - 4*((3*b^2 + 32*a*c)*x^6 + 14*a*b*x^3 + 8*a^2)*sqrt(c*x^6 + b*x^3
+ a)*sqrt(a))/(a^(3/2)*x^9), 1/288*(96*a^(3/2)*sqrt(-c)*c*x^9*arctan(1/2*(2*c*x^
3 + b)/(sqrt(c*x^6 + b*x^3 + a)*sqrt(-c))) - 3*(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)*s
qrt(a))/x^6) - 4*((3*b^2 + 32*a*c)*x^6 + 14*a*b*x^3 + 8*a^2)*sqrt(c*x^6 + b*x^3
+ a)*sqrt(a))/(a^(3/2)*x^9), 1/144*(24*sqrt(-a)*a*c^(3/2)*x^9*log(-8*c^2*x^6 - 8
*b*c*x^3 - b^2 - 4*sqrt(c*x^6 + b*x^3 + a)*(2*c*x^3 + b)*sqrt(c) - 4*a*c) + 3*(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*((3*b^2 + 32*a*c)*x^6 + 14*a*b*x^3 + 8*a^2)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-
a))/(sqrt(-a)*a*x^9), 1/144*(48*sqrt(-a)*a*sqrt(-c)*c*x^9*arctan(1/2*(2*c*x^3 +
b)/(sqrt(c*x^6 + b*x^3 + a)*sqrt(-c))) + 3*(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*((3*b^2 + 32*a*c)*x^6 + 14*a*
b*x^3 + 8*a^2)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-a))/(sqrt(-a)*a*x^9)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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