3.211 \(\int \frac{\left (a+b x^3+c x^6\right )^{3/2}}{x^{16}} \, dx\)

Optimal. Leaf size=162 \[ \frac{b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{256 a^{7/2}}-\frac{b \left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{128 a^3 x^6}+\frac{b \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 a^2 x^{12}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}} \]

[Out]

-(b*(b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(128*a^3*x^6) + (b*(2*a
+ b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(48*a^2*x^12) - (a + b*x^3 + c*x^6)^(5/2)/(
15*a*x^15) + (b*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3
+ c*x^6])])/(256*a^(7/2))

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Rubi [A]  time = 0.296424, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{256 a^{7/2}}-\frac{b \left (b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{128 a^3 x^6}+\frac{b \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{48 a^2 x^{12}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{15 a x^{15}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)^(3/2)/x^16,x]

[Out]

-(b*(b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(128*a^3*x^6) + (b*(2*a
+ b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(48*a^2*x^12) - (a + b*x^3 + c*x^6)^(5/2)/(
15*a*x^15) + (b*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3
+ c*x^6])])/(256*a^(7/2))

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Rubi in Sympy [A]  time = 34.3938, size = 148, normalized size = 0.91 \[ - \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{15 a x^{15}} + \frac{b \left (2 a + b x^{3}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{48 a^{2} x^{12}} - \frac{b \left (2 a + b x^{3}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{128 a^{3} x^{6}} + \frac{b \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{256 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)**(3/2)/x**16,x)

[Out]

-(a + b*x**3 + c*x**6)**(5/2)/(15*a*x**15) + b*(2*a + b*x**3)*(a + b*x**3 + c*x*
*6)**(3/2)/(48*a**2*x**12) - b*(2*a + b*x**3)*(-4*a*c + b**2)*sqrt(a + b*x**3 +
c*x**6)/(128*a**3*x**6) + b*(-4*a*c + b**2)**2*atanh((2*a + b*x**3)/(2*sqrt(a)*s
qrt(a + b*x**3 + c*x**6)))/(256*a**(7/2))

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Mathematica [A]  time = 0.271081, size = 163, normalized size = 1.01 \[ -\frac{b \left (b^2-4 a c\right )^2 \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )}{256 a^{7/2}}-\frac{\sqrt{a+b x^3+c x^6} \left (128 a^4+16 a^3 \left (11 b x^3+16 c x^6\right )+8 a^2 x^6 \left (b^2+7 b c x^3+16 c^2 x^6\right )-10 a b^2 x^9 \left (b+10 c x^3\right )+15 b^4 x^{12}\right )}{1920 a^3 x^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^16,x]

[Out]

-(Sqrt[a + b*x^3 + c*x^6]*(128*a^4 + 15*b^4*x^12 - 10*a*b^2*x^9*(b + 10*c*x^3) +
16*a^3*(11*b*x^3 + 16*c*x^6) + 8*a^2*x^6*(b^2 + 7*b*c*x^3 + 16*c^2*x^6)))/(1920
*a^3*x^15) - (b*(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]]))/(256*a^(7/2))

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{16}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)^(3/2)/x^16,x)

[Out]

int((c*x^6+b*x^3+a)^(3/2)/x^16,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^16,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.303809, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{15} \log \left (-\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) - 4 \,{\left ({\left (15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2}\right )} x^{12} - 2 \,{\left (5 \, a b^{3} - 28 \, a^{2} b c\right )} x^{9} + 176 \, a^{3} b x^{3} + 8 \,{\left (a^{2} b^{2} + 32 \, a^{3} c\right )} x^{6} + 128 \, a^{4}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{a}}{7680 \, a^{\frac{7}{2}} x^{15}}, \frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{15} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \,{\left ({\left (15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2}\right )} x^{12} - 2 \,{\left (5 \, a b^{3} - 28 \, a^{2} b c\right )} x^{9} + 176 \, a^{3} b x^{3} + 8 \,{\left (a^{2} b^{2} + 32 \, a^{3} c\right )} x^{6} + 128 \, a^{4}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a}}{3840 \, \sqrt{-a} a^{3} x^{15}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x^16,x, algorithm="fricas")

[Out]

[1/7680*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^15*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^4 - 100*a*b^2*c + 128*a^2*c^2)*x^12 - 2*(5*a*b^3 - 28*a^2*b*c)*x^9 + 176*
a^3*b*x^3 + 8*(a^2*b^2 + 32*a^3*c)*x^6 + 128*a^4)*sqrt(c*x^6 + b*x^3 + a)*sqrt(a
))/(a^(7/2)*x^15), 1/3840*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^15*arctan(1/2*(
b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a)*a)) - 2*((15*b^4 - 100*a*b^2*c +
128*a^2*c^2)*x^12 - 2*(5*a*b^3 - 28*a^2*b*c)*x^9 + 176*a^3*b*x^3 + 8*(a^2*b^2 +
32*a^3*c)*x^6 + 128*a^4)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^15)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{x^{16}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**6+b*x**3+a)**(3/2)/x**16,x)

[Out]

Integral((a + b*x**3 + c*x**6)**(3/2)/x**16, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x^16,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)/x^16, x)