3.945 \(\int \frac{\left (a+b x^2+c x^4\right )^{3/2}}{x^{11}} \, dx\)

Optimal. Leaf size=162 \[ \frac{3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{512 a^{7/2}}-\frac{3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{256 a^3 x^4}+\frac{b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}} \]

[Out]

(-3*b*(b^2 - 4*a*c)*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*a^3*x^4) + (b*(2
*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(32*a^2*x^8) - (a + b*x^2 + c*x^4)^(5/2)/
(10*a*x^10) + (3*b*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x
^2 + c*x^4])])/(512*a^(7/2))

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Rubi [A]  time = 0.327918, 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 \[ \frac{3 b \left (b^2-4 a c\right )^2 \tanh ^{-1}\left (\frac{2 a+b x^2}{2 \sqrt{a} \sqrt{a+b x^2+c x^4}}\right )}{512 a^{7/2}}-\frac{3 b \left (b^2-4 a c\right ) \left (2 a+b x^2\right ) \sqrt{a+b x^2+c x^4}}{256 a^3 x^4}+\frac{b \left (2 a+b x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}}{32 a^2 x^8}-\frac{\left (a+b x^2+c x^4\right )^{5/2}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2 + c*x^4)^(3/2)/x^11,x]

[Out]

(-3*b*(b^2 - 4*a*c)*(2*a + b*x^2)*Sqrt[a + b*x^2 + c*x^4])/(256*a^3*x^4) + (b*(2
*a + b*x^2)*(a + b*x^2 + c*x^4)^(3/2))/(32*a^2*x^8) - (a + b*x^2 + c*x^4)^(5/2)/
(10*a*x^10) + (3*b*(b^2 - 4*a*c)^2*ArcTanh[(2*a + b*x^2)/(2*Sqrt[a]*Sqrt[a + b*x
^2 + c*x^4])])/(512*a^(7/2))

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Rubi in Sympy [A]  time = 30.1936, size = 151, normalized size = 0.93 \[ - \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{5}{2}}}{10 a x^{10}} + \frac{b \left (2 a + b x^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{32 a^{2} x^{8}} - \frac{3 b \left (2 a + b x^{2}\right ) \left (- 4 a c + b^{2}\right ) \sqrt{a + b x^{2} + c x^{4}}}{256 a^{3} x^{4}} + \frac{3 b \left (- 4 a c + b^{2}\right )^{2} \operatorname{atanh}{\left (\frac{2 a + b x^{2}}{2 \sqrt{a} \sqrt{a + b x^{2} + c x^{4}}} \right )}}{512 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+b*x**2+a)**(3/2)/x**11,x)

[Out]

-(a + b*x**2 + c*x**4)**(5/2)/(10*a*x**10) + b*(2*a + b*x**2)*(a + b*x**2 + c*x*
*4)**(3/2)/(32*a**2*x**8) - 3*b*(2*a + b*x**2)*(-4*a*c + b**2)*sqrt(a + b*x**2 +
c*x**4)/(256*a**3*x**4) + 3*b*(-4*a*c + b**2)**2*atanh((2*a + b*x**2)/(2*sqrt(a
)*sqrt(a + b*x**2 + c*x**4)))/(512*a**(7/2))

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Mathematica [A]  time = 0.218082, size = 163, normalized size = 1.01 \[ -\frac{3 b \left (b^2-4 a c\right )^2 \left (\log \left (x^2\right )-\log \left (2 \sqrt{a} \sqrt{a+b x^2+c x^4}+2 a+b x^2\right )\right )}{512 a^{7/2}}-\frac{\sqrt{a+b x^2+c x^4} \left (128 a^4+16 a^3 \left (11 b x^2+16 c x^4\right )+8 a^2 x^4 \left (b^2+7 b c x^2+16 c^2 x^4\right )-10 a b^2 x^6 \left (b+10 c x^2\right )+15 b^4 x^8\right )}{1280 a^3 x^{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2 + c*x^4)^(3/2)/x^11,x]

[Out]

-(Sqrt[a + b*x^2 + c*x^4]*(128*a^4 + 15*b^4*x^8 - 10*a*b^2*x^6*(b + 10*c*x^2) +
16*a^3*(11*b*x^2 + 16*c*x^4) + 8*a^2*x^4*(b^2 + 7*b*c*x^2 + 16*c^2*x^4)))/(1280*
a^3*x^10) - (3*b*(b^2 - 4*a*c)^2*(Log[x^2] - Log[2*a + b*x^2 + 2*Sqrt[a]*Sqrt[a
+ b*x^2 + c*x^4]]))/(512*a^(7/2))

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Maple [B]  time = 0.028, size = 337, normalized size = 2.1 \[ -{\frac{a}{10\,{x}^{10}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{11\,b}{80\,{x}^{8}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{b}^{2}}{160\,a{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{{b}^{3}}{128\,{a}^{2}{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{3\,{b}^{4}}{256\,{a}^{3}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{b}^{5}}{512}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{3\,{b}^{3}c}{64}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{5\,{b}^{2}c}{64\,{a}^{2}{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{7\,bc}{160\,a{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}+a}}+{\frac{3\,{c}^{2}b}{32}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+b{x}^{2}+2\,\sqrt{a}\sqrt{c{x}^{4}+b{x}^{2}+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{c}{5\,{x}^{6}}\sqrt{c{x}^{4}+b{x}^{2}+a}}-{\frac{{c}^{2}}{10\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+b*x^2+a)^(3/2)/x^11,x)

[Out]

-1/10*a/x^10*(c*x^4+b*x^2+a)^(1/2)-11/80*b/x^8*(c*x^4+b*x^2+a)^(1/2)-1/160/a*b^2
/x^6*(c*x^4+b*x^2+a)^(1/2)+1/128/a^2*b^3/x^4*(c*x^4+b*x^2+a)^(1/2)-3/256/a^3*b^4
/x^2*(c*x^4+b*x^2+a)^(1/2)+3/512/a^(7/2)*b^5*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^
2+a)^(1/2))/x^2)-3/64/a^(5/2)*b^3*c*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2
))/x^2)+5/64/a^2*b^2*c/x^2*(c*x^4+b*x^2+a)^(1/2)-7/160/a*b*c/x^4*(c*x^4+b*x^2+a)
^(1/2)+3/32/a^(3/2)*b*c^2*ln((2*a+b*x^2+2*a^(1/2)*(c*x^4+b*x^2+a)^(1/2))/x^2)-1/
5*c/x^6*(c*x^4+b*x^2+a)^(1/2)-1/10/a*c^2/x^2*(c*x^4+b*x^2+a)^(1/2)

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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^4 + b*x^2 + a)^(3/2)/x^11,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.324237, size = 1, normalized size = 0.01 \[ \left [\frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{10} \log \left (-\frac{4 \, \sqrt{c x^{4} + b x^{2} + a}{\left (a b x^{2} + 2 \, a^{2}\right )} +{\left ({\left (b^{2} + 4 \, a c\right )} x^{4} + 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{4}}\right ) - 4 \,{\left ({\left (15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2}\right )} x^{8} - 2 \,{\left (5 \, a b^{3} - 28 \, a^{2} b c\right )} x^{6} + 176 \, a^{3} b x^{2} + 8 \,{\left (a^{2} b^{2} + 32 \, a^{3} c\right )} x^{4} + 128 \, a^{4}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{a}}{5120 \, a^{\frac{7}{2}} x^{10}}, \frac{15 \,{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} x^{10} \arctan \left (\frac{{\left (b x^{2} + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{4} + b x^{2} + a} a}\right ) - 2 \,{\left ({\left (15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2}\right )} x^{8} - 2 \,{\left (5 \, a b^{3} - 28 \, a^{2} b c\right )} x^{6} + 176 \, a^{3} b x^{2} + 8 \,{\left (a^{2} b^{2} + 32 \, a^{3} c\right )} x^{4} + 128 \, a^{4}\right )} \sqrt{c x^{4} + b x^{2} + a} \sqrt{-a}}{2560 \, \sqrt{-a} a^{3} x^{10}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^11,x, algorithm="fricas")

[Out]

[1/5120*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^10*log(-(4*sqrt(c*x^4 + b*x^2 + a
)*(a*b*x^2 + 2*a^2) + ((b^2 + 4*a*c)*x^4 + 8*a*b*x^2 + 8*a^2)*sqrt(a))/x^4) - 4*
((15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*x^8 - 2*(5*a*b^3 - 28*a^2*b*c)*x^6 + 176*a
^3*b*x^2 + 8*(a^2*b^2 + 32*a^3*c)*x^4 + 128*a^4)*sqrt(c*x^4 + b*x^2 + a)*sqrt(a)
)/(a^(7/2)*x^10), 1/2560*(15*(b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*x^10*arctan(1/2*(b
*x^2 + 2*a)*sqrt(-a)/(sqrt(c*x^4 + b*x^2 + a)*a)) - 2*((15*b^4 - 100*a*b^2*c + 1
28*a^2*c^2)*x^8 - 2*(5*a*b^3 - 28*a^2*b*c)*x^6 + 176*a^3*b*x^2 + 8*(a^2*b^2 + 32
*a^3*c)*x^4 + 128*a^4)*sqrt(c*x^4 + b*x^2 + a)*sqrt(-a))/(sqrt(-a)*a^3*x^10)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{x^{11}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+b*x**2+a)**(3/2)/x**11,x)

[Out]

Integral((a + b*x**2 + c*x**4)**(3/2)/x**11, x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (c x^{4} + b x^{2} + a\right )}^{\frac{3}{2}}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^4 + b*x^2 + a)^(3/2)/x^11,x, algorithm="giac")

[Out]

integrate((c*x^4 + b*x^2 + a)^(3/2)/x^11, x)