3.213 \(\int \frac{\left (a+b x^3+c x^6\right )^{3/2}}{x^{22}} \, dx\)

Optimal. Leaf size=255 \[ \frac{b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{2048 a^{11/2}}-\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 a^5 x^6}+\frac{b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac{\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}+\frac{b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}} \]

[Out]

-(b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1024*a
^5*x^6) + (b*(3*b^2 - 4*a*c)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*a^4*x
^12) - (a + b*x^3 + c*x^6)^(5/2)/(21*a*x^21) + (b*(a + b*x^3 + c*x^6)^(5/2))/(28
*a^2*x^18) - ((21*b^2 - 16*a*c)*(a + b*x^3 + c*x^6)^(5/2))/(840*a^3*x^15) + (b*(
b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 +
c*x^6])])/(2048*a^(11/2))

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Rubi [A]  time = 0.670914, antiderivative size = 255, 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 \[ \frac{b \left (b^2-4 a c\right )^2 \left (3 b^2-4 a c\right ) \tanh ^{-1}\left (\frac{2 a+b x^3}{2 \sqrt{a} \sqrt{a+b x^3+c x^6}}\right )}{2048 a^{11/2}}-\frac{b \left (b^2-4 a c\right ) \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \sqrt{a+b x^3+c x^6}}{1024 a^5 x^6}+\frac{b \left (3 b^2-4 a c\right ) \left (2 a+b x^3\right ) \left (a+b x^3+c x^6\right )^{3/2}}{384 a^4 x^{12}}-\frac{\left (21 b^2-16 a c\right ) \left (a+b x^3+c x^6\right )^{5/2}}{840 a^3 x^{15}}+\frac{b \left (a+b x^3+c x^6\right )^{5/2}}{28 a^2 x^{18}}-\frac{\left (a+b x^3+c x^6\right )^{5/2}}{21 a x^{21}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3 + c*x^6)^(3/2)/x^22,x]

[Out]

-(b*(b^2 - 4*a*c)*(3*b^2 - 4*a*c)*(2*a + b*x^3)*Sqrt[a + b*x^3 + c*x^6])/(1024*a
^5*x^6) + (b*(3*b^2 - 4*a*c)*(2*a + b*x^3)*(a + b*x^3 + c*x^6)^(3/2))/(384*a^4*x
^12) - (a + b*x^3 + c*x^6)^(5/2)/(21*a*x^21) + (b*(a + b*x^3 + c*x^6)^(5/2))/(28
*a^2*x^18) - ((21*b^2 - 16*a*c)*(a + b*x^3 + c*x^6)^(5/2))/(840*a^3*x^15) + (b*(
b^2 - 4*a*c)^2*(3*b^2 - 4*a*c)*ArcTanh[(2*a + b*x^3)/(2*Sqrt[a]*Sqrt[a + b*x^3 +
c*x^6])])/(2048*a^(11/2))

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Rubi in Sympy [A]  time = 70.7257, size = 238, normalized size = 0.93 \[ - \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{21 a x^{21}} + \frac{b \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{28 a^{2} x^{18}} - \frac{\left (- 16 a c + 21 b^{2}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{5}{2}}}{840 a^{3} x^{15}} + \frac{b \left (2 a + b x^{3}\right ) \left (- 4 a c + 3 b^{2}\right ) \left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{384 a^{4} x^{12}} - \frac{b \left (2 a + b x^{3}\right ) \left (- 4 a c + b^{2}\right ) \left (- 4 a c + 3 b^{2}\right ) \sqrt{a + b x^{3} + c x^{6}}}{1024 a^{5} x^{6}} + \frac{b \left (- 4 a c + b^{2}\right )^{2} \left (- 4 a c + 3 b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x^{3}}{2 \sqrt{a} \sqrt{a + b x^{3} + c x^{6}}} \right )}}{2048 a^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**6+b*x**3+a)**(3/2)/x**22,x)

[Out]

-(a + b*x**3 + c*x**6)**(5/2)/(21*a*x**21) + b*(a + b*x**3 + c*x**6)**(5/2)/(28*
a**2*x**18) - (-16*a*c + 21*b**2)*(a + b*x**3 + c*x**6)**(5/2)/(840*a**3*x**15)
+ b*(2*a + b*x**3)*(-4*a*c + 3*b**2)*(a + b*x**3 + c*x**6)**(3/2)/(384*a**4*x**1
2) - b*(2*a + b*x**3)*(-4*a*c + b**2)*(-4*a*c + 3*b**2)*sqrt(a + b*x**3 + c*x**6
)/(1024*a**5*x**6) + b*(-4*a*c + b**2)**2*(-4*a*c + 3*b**2)*atanh((2*a + b*x**3)
/(2*sqrt(a)*sqrt(a + b*x**3 + c*x**6)))/(2048*a**(11/2))

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Mathematica [A]  time = 0.401172, size = 247, normalized size = 0.97 \[ -\frac{b \left (3 b^2-4 a c\right ) \left (b^2-4 a c\right )^2 \left (\log \left (x^3\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^3+c x^6}+2 a+b x^3\right )\right )}{2048 a^{11/2}}-\frac{\sqrt{a+b x^3+c x^6} \left (5120 a^6+256 a^5 \left (25 b x^3+32 c x^6\right )+64 a^4 x^6 \left (2 b^2+11 b c x^3+16 c^2 x^6\right )-16 a^3 x^9 \left (9 b^3+62 b^2 c x^3+146 b c^2 x^6+128 c^3 x^9\right )+56 a^2 b^2 x^{12} \left (3 b^2+26 b c x^3+98 c^2 x^6\right )-210 a b^4 x^{15} \left (b+12 c x^3\right )+315 b^6 x^{18}\right )}{107520 a^5 x^{21}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3 + c*x^6)^(3/2)/x^22,x]

[Out]

-(Sqrt[a + b*x^3 + c*x^6]*(5120*a^6 + 315*b^6*x^18 - 210*a*b^4*x^15*(b + 12*c*x^
3) + 256*a^5*(25*b*x^3 + 32*c*x^6) + 64*a^4*x^6*(2*b^2 + 11*b*c*x^3 + 16*c^2*x^6
) + 56*a^2*b^2*x^12*(3*b^2 + 26*b*c*x^3 + 98*c^2*x^6) - 16*a^3*x^9*(9*b^3 + 62*b
^2*c*x^3 + 146*b*c^2*x^6 + 128*c^3*x^9)))/(107520*a^5*x^21) - (b*(b^2 - 4*a*c)^2
*(3*b^2 - 4*a*c)*(Log[x^3] - Log[2*a + b*x^3 + 2*Sqrt[a]*Sqrt[a + b*x^3 + c*x^6]
]))/(2048*a^(11/2))

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Maple [F]  time = 0.106, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{22}} \left ( c{x}^{6}+b{x}^{3}+a \right ) ^{{\frac{3}{2}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^6+b*x^3+a)^(3/2)/x^22,x)

[Out]

int((c*x^6+b*x^3+a)^(3/2)/x^22,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^22,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.372054, size = 1, normalized size = 0. \[ \left [-\frac{105 \,{\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} x^{21} \log \left (\frac{4 \, \sqrt{c x^{6} + b x^{3} + a}{\left (a b x^{3} + 2 \, a^{2}\right )} -{\left ({\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{3} + 8 \, a^{2}\right )} \sqrt{a}}{x^{6}}\right ) + 4 \,{\left ({\left (315 \, b^{6} - 2520 \, a b^{4} c + 5488 \, a^{2} b^{2} c^{2} - 2048 \, a^{3} c^{3}\right )} x^{18} - 2 \,{\left (105 \, a b^{5} - 728 \, a^{2} b^{3} c + 1168 \, a^{3} b c^{2}\right )} x^{15} + 8 \,{\left (21 \, a^{2} b^{4} - 124 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} x^{12} - 16 \,{\left (9 \, a^{3} b^{3} - 44 \, a^{4} b c\right )} x^{9} + 6400 \, a^{5} b x^{3} + 128 \,{\left (a^{4} b^{2} + 64 \, a^{5} c\right )} x^{6} + 5120 \, a^{6}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{a}}{430080 \, a^{\frac{11}{2}} x^{21}}, \frac{105 \,{\left (3 \, b^{7} - 28 \, a b^{5} c + 80 \, a^{2} b^{3} c^{2} - 64 \, a^{3} b c^{3}\right )} x^{21} \arctan \left (\frac{{\left (b x^{3} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{6} + b x^{3} + a} a}\right ) - 2 \,{\left ({\left (315 \, b^{6} - 2520 \, a b^{4} c + 5488 \, a^{2} b^{2} c^{2} - 2048 \, a^{3} c^{3}\right )} x^{18} - 2 \,{\left (105 \, a b^{5} - 728 \, a^{2} b^{3} c + 1168 \, a^{3} b c^{2}\right )} x^{15} + 8 \,{\left (21 \, a^{2} b^{4} - 124 \, a^{3} b^{2} c + 128 \, a^{4} c^{2}\right )} x^{12} - 16 \,{\left (9 \, a^{3} b^{3} - 44 \, a^{4} b c\right )} x^{9} + 6400 \, a^{5} b x^{3} + 128 \,{\left (a^{4} b^{2} + 64 \, a^{5} c\right )} x^{6} + 5120 \, a^{6}\right )} \sqrt{c x^{6} + b x^{3} + a} \sqrt{-a}}{215040 \, \sqrt{-a} a^{5} x^{21}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x^22,x, algorithm="fricas")

[Out]

[-1/430080*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*c^3)*x^21*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*((315*b^6 - 2520*a*b^4*c + 5488*a^2*b^2*c^2 - 2048*a^3*c^
3)*x^18 - 2*(105*a*b^5 - 728*a^2*b^3*c + 1168*a^3*b*c^2)*x^15 + 8*(21*a^2*b^4 -
124*a^3*b^2*c + 128*a^4*c^2)*x^12 - 16*(9*a^3*b^3 - 44*a^4*b*c)*x^9 + 6400*a^5*b
*x^3 + 128*(a^4*b^2 + 64*a^5*c)*x^6 + 5120*a^6)*sqrt(c*x^6 + b*x^3 + a)*sqrt(a))
/(a^(11/2)*x^21), 1/215040*(105*(3*b^7 - 28*a*b^5*c + 80*a^2*b^3*c^2 - 64*a^3*b*
c^3)*x^21*arctan(1/2*(b*x^3 + 2*a)*sqrt(-a)/(sqrt(c*x^6 + b*x^3 + a)*a)) - 2*((3
15*b^6 - 2520*a*b^4*c + 5488*a^2*b^2*c^2 - 2048*a^3*c^3)*x^18 - 2*(105*a*b^5 - 7
28*a^2*b^3*c + 1168*a^3*b*c^2)*x^15 + 8*(21*a^2*b^4 - 124*a^3*b^2*c + 128*a^4*c^
2)*x^12 - 16*(9*a^3*b^3 - 44*a^4*b*c)*x^9 + 6400*a^5*b*x^3 + 128*(a^4*b^2 + 64*a
^5*c)*x^6 + 5120*a^6)*sqrt(c*x^6 + b*x^3 + a)*sqrt(-a))/(sqrt(-a)*a^5*x^21)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{3} + c x^{6}\right )^{\frac{3}{2}}}{x^{22}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**6+b*x**3+a)**(3/2)/x**22,x)

[Out]

Integral((a + b*x**3 + c*x**6)**(3/2)/x**22, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{6} + b x^{3} + a\right )}^{\frac{3}{2}}}{x^{22}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^6 + b*x^3 + a)^(3/2)/x^22,x, algorithm="giac")

[Out]

integrate((c*x^6 + b*x^3 + a)^(3/2)/x^22, x)