3.89 \(\int \frac{x^{11}}{\left (a x+b x^3+c x^5\right )^2} \, dx\)

Optimal. Leaf size=166 \[ -\frac{\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}-\frac{b x^4}{2 c \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \log \left (a+b x^2+c x^4\right )}{2 c^3} \]

[Out]

((b^2 - 3*a*c)*x^2)/(c^2*(b^2 - 4*a*c)) - (b*x^4)/(2*c*(b^2 - 4*a*c)) + (x^6*(2*
a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((b^4 - 6*a*b^2*c + 6*a^2*c^
2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) - (b*Log[
a + b*x^2 + c*x^4])/(2*c^3)

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Rubi [A]  time = 0.422914, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (b^2-3 a c\right )}{c^2 \left (b^2-4 a c\right )}-\frac{b x^4}{2 c \left (b^2-4 a c\right )}+\frac{x^6 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \log \left (a+b x^2+c x^4\right )}{2 c^3} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(a*x + b*x^3 + c*x^5)^2,x]

[Out]

((b^2 - 3*a*c)*x^2)/(c^2*(b^2 - 4*a*c)) - (b*x^4)/(2*c*(b^2 - 4*a*c)) + (x^6*(2*
a + b*x^2))/(2*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((b^4 - 6*a*b^2*c + 6*a^2*c^
2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) - (b*Log[
a + b*x^2 + c*x^4])/(2*c^3)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{b \int ^{x^{2}} x\, dx}{c \left (- 4 a c + b^{2}\right )} - \frac{b \log{\left (a + b x^{2} + c x^{4} \right )}}{2 c^{3}} + \frac{x^{6} \left (2 a + b x^{2}\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} + \frac{x^{2} \left (- 3 a c + b^{2}\right )}{c^{2} \left (- 4 a c + b^{2}\right )} - \frac{\left (6 a^{2} c^{2} - 6 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{3} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(c*x**5+b*x**3+a*x)**2,x)

[Out]

-b*Integral(x, (x, x**2))/(c*(-4*a*c + b**2)) - b*log(a + b*x**2 + c*x**4)/(2*c*
*3) + x**6*(2*a + b*x**2)/(2*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)) + x**2*(-3*a
*c + b**2)/(c**2*(-4*a*c + b**2)) - (6*a**2*c**2 - 6*a*b**2*c + b**4)*atanh((b +
2*c*x**2)/sqrt(-4*a*c + b**2))/(c**3*(-4*a*c + b**2)**(3/2))

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Mathematica [A]  time = 0.305693, size = 151, normalized size = 0.91 \[ \frac{-\frac{2 \left (6 a^2 c^2-6 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{a^2 c \left (3 b-2 c x^2\right )-a b^2 \left (b-4 c x^2\right )+b^4 \left (-x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-b \log \left (a+b x^2+c x^4\right )+c x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(a*x + b*x^3 + c*x^5)^2,x]

[Out]

(c*x^2 + (-(b^4*x^2) - a*b^2*(b - 4*c*x^2) + a^2*c*(3*b - 2*c*x^2))/((b^2 - 4*a*
c)*(a + b*x^2 + c*x^4)) - (2*(b^4 - 6*a*b^2*c + 6*a^2*c^2)*ArcTan[(b + 2*c*x^2)/
Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) - b*Log[a + b*x^2 + c*x^4])/(2*c^3)

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Maple [B]  time = 0.018, size = 600, normalized size = 3.6 \[{\frac{{x}^{2}}{2\,{c}^{2}}}+{\frac{{a}^{2}{x}^{2}}{c \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{a{x}^{2}{b}^{2}}{{c}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{{x}^{2}{b}^{4}}{2\,{c}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{3\,{a}^{2}b}{2\,{c}^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a{b}^{3}}{2\,{c}^{3} \left ( c{x}^{4}+b{x}^{2}+a \right ) \left ( 4\,ac-{b}^{2} \right ) }}-2\,{\frac{\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) \right ) ab}{ \left ( 4\,ac-{b}^{2} \right ){c}^{2}}}+{\frac{\ln \left ( \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) \right ){b}^{3}}{2\,{c}^{3} \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{{a}^{2}}{c\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) c{x}^{2}+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }+6\,{\frac{a{b}^{2}}{{c}^{2}\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}\arctan \left ({\frac{2\, \left ( 4\,ac-{b}^{2} \right ) c{x}^{2}+ \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}} \right ) }-{\frac{{b}^{4}}{{c}^{3}}\arctan \left ({(2\, \left ( 4\,ac-{b}^{2} \right ) c{x}^{2}+ \left ( 4\,ac-{b}^{2} \right ) b){\frac{1}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}}} \right ){\frac{1}{\sqrt{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(c*x^5+b*x^3+a*x)^2,x)

[Out]

1/2*x^2/c^2+1/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a^2-2/c^2/(c*x^4+b*x^2+a)/(4*a*c
-b^2)*x^2*a*b^2+1/2/c^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b^4-3/2/c^2/(c*x^4+b*x^2
+a)*a^2*b/(4*a*c-b^2)+1/2/c^3/(c*x^4+b*x^2+a)*a*b^3/(4*a*c-b^2)-2/c^2/(4*a*c-b^2
)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*a*b+1/2/c^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+
b*x^2+a))*b^3-6/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*
a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*
a^2+6/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)
*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b^2-1/
c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2
+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\frac{a b^{3} - 3 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} x^{2}}{2 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )}} + \frac{x^{2}}{2 \, c^{2}} + \frac{-2 \, \int \frac{{\left (b^{3} - 4 \, a b c\right )} x^{3} +{\left (a b^{2} - 3 \, a^{2} c\right )} x}{c x^{4} + b x^{2} + a}\,{d x}}{b^{2} c^{2} - 4 \, a c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="maxima")

[Out]

-1/2*(a*b^3 - 3*a^2*b*c + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*x^2)/(a*b^2*c^3 - 4*a^2*
c^4 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2) + 1/2*x^2/c^2 + 2*int
egrate(-((b^3 - 4*a*b*c)*x^3 + (a*b^2 - 3*a^2*c)*x)/(c*x^4 + b*x^2 + a), x)/(b^2
*c^2 - 4*a*c^3)

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Fricas [A]  time = 0.299428, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} +{\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{4} +{\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x^{2}\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} - a b^{3} + 3 \, a^{2} b c -{\left (b^{4} - 5 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{2} -{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (a b^{4} - 6 \, a^{2} b^{2} c + 6 \, a^{3} c^{2} +{\left (b^{4} c - 6 \, a b^{2} c^{2} + 6 \, a^{2} c^{3}\right )} x^{4} +{\left (b^{5} - 6 \, a b^{3} c + 6 \, a^{2} b c^{2}\right )} x^{2}\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{6} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} - a b^{3} + 3 \, a^{2} b c -{\left (b^{4} - 5 \, a b^{2} c + 6 \, a^{2} c^{2}\right )} x^{2} -{\left ({\left (b^{3} c - 4 \, a b c^{2}\right )} x^{4} + a b^{3} - 4 \, a^{2} b c +{\left (b^{4} - 4 \, a b^{2} c\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a b^{2} c^{3} - 4 \, a^{2} c^{4} +{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} +{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="fricas")

[Out]

[-1/2*((a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*x^4
+ (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*
x^2 + (2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 +
a)) - ((b^2*c^2 - 4*a*c^3)*x^6 + (b^3*c - 4*a*b*c^2)*x^4 - a*b^3 + 3*a^2*b*c - (
b^4 - 5*a*b^2*c + 6*a^2*c^2)*x^2 - ((b^3*c - 4*a*b*c^2)*x^4 + a*b^3 - 4*a^2*b*c
+ (b^4 - 4*a*b^2*c)*x^2)*log(c*x^4 + b*x^2 + a))*sqrt(b^2 - 4*a*c))/((a*b^2*c^3
- 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2)*sqrt(b^2 - 4*
a*c)), 1/2*(2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2 + (b^4*c - 6*a*b^2*c^2 + 6*a^2*c^
3)*x^4 + (b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 +
4*a*c)/(b^2 - 4*a*c)) + ((b^2*c^2 - 4*a*c^3)*x^6 + (b^3*c - 4*a*b*c^2)*x^4 - a*b
^3 + 3*a^2*b*c - (b^4 - 5*a*b^2*c + 6*a^2*c^2)*x^2 - ((b^3*c - 4*a*b*c^2)*x^4 +
a*b^3 - 4*a^2*b*c + (b^4 - 4*a*b^2*c)*x^2)*log(c*x^4 + b*x^2 + a))*sqrt(-b^2 + 4
*a*c))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)
*x^2)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 17.1066, size = 877, normalized size = 5.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(c*x**5+b*x**3+a*x)**2,x)

[Out]

(-b/(2*c**3) - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**
3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x**2 + (-5*a**2*
b*c - 16*a**2*c**4*(-b/(2*c**3) - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b*
*2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) +
a*b**3 + 8*a*b**2*c**3*(-b/(2*c**3) - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6
*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6
))) - b**4*c**2*(-b/(2*c**3) - sqrt(-(4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*
c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))))/(6*
a**2*c**2 - 6*a*b**2*c + b**4)) + (-b/(2*c**3) + sqrt(-(4*a*c - b**2)**3)*(6*a**
2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**
4*c - b**6)))*log(x**2 + (-5*a**2*b*c - 16*a**2*c**4*(-b/(2*c**3) + sqrt(-(4*a*c
- b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b
**2*c**2 + 12*a*b**4*c - b**6))) + a*b**3 + 8*a*b**2*c**3*(-b/(2*c**3) + sqrt(-(
4*a*c - b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a
**2*b**2*c**2 + 12*a*b**4*c - b**6))) - b**4*c**2*(-b/(2*c**3) + sqrt(-(4*a*c -
b**2)**3)*(6*a**2*c**2 - 6*a*b**2*c + b**4)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6))))/(6*a**2*c**2 - 6*a*b**2*c + b**4)) + (-3*a**2*b*c
+ a*b**3 + x**2*(2*a**2*c**2 - 4*a*b**2*c + b**4))/(8*a**2*c**4 - 2*a*b**2*c**3
+ x**4*(8*a*c**5 - 2*b**2*c**4) + x**2*(8*a*b*c**4 - 2*b**3*c**3)) + x**2/(2*c*
*2)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError