3.861 \(\int \frac{x^7}{\left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=132 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}-\frac{b x^2}{2 c \left (b^2-4 a c\right )}+\frac{x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\log \left (a+b x^2+c x^4\right )}{4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.372598, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389 \[ \frac{b \left (b^2-6 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}-\frac{b x^2}{2 c \left (b^2-4 a c\right )}+\frac{x^4 \left (2 a+b x^2\right )}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\log \left (a+b x^2+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^2 + c*x^4)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(c*x**4+b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.338068, size = 121, normalized size = 0.92 \[ \frac{\frac{2 \left (-2 a^2 c+a b \left (b-3 c x^2\right )+b^3 x^2\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{2 b \left (b^2-6 a c\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\log \left (a+b x^2+c x^4\right )}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a + b*x^2 + c*x^4)^2,x]

[Out]

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

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Maple [B]  time = 0.021, size = 342, normalized size = 2.6 \[{\frac{1}{2\,c{x}^{4}+2\,b{x}^{2}+2\,a} \left ({\frac{b \left ( 3\,ac-{b}^{2} \right ){x}^{2}}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a \left ( 2\,ac-{b}^{2} \right ) }{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+{\frac{\ln \left ( c \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) \right ) }{4\,{c}^{2}}}-3\,{\frac{ab}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}\arctan \left ({\frac{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ){x}^{2}+c \left ( 4\,ac-{b}^{2} \right ) b}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}} \right ) }+{\frac{{b}^{3}}{2\,c}\arctan \left ({(2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ){x}^{2}+c \left ( 4\,ac-{b}^{2} \right ) b){\frac{1}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}}} \right ){\frac{1}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+b*x^2+a)^2,x)

[Out]

1/2*(b*(3*a*c-b^2)/c^2/(4*a*c-b^2)*x^2+a*(2*a*c-b^2)/(4*a*c-b^2)/c^2)/(c*x^4+b*x
^2+a)+1/4/c^2*ln(c*(4*a*c-b^2)*(c*x^4+b*x^2+a))-3/(64*a^3*c^5-48*a^2*b^2*c^4+12*
a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x^2+c*(4*a*c-b^2)*b)/(64*a^3*
c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*a*b+1/2/(64*a^3*c^5-48*a^2*b^2*c
^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x^2+c*(4*a*c-b^2)*b)/(6
4*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*b^3/c

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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(x^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276471, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (b^{3} c - 6 \, a b c^{2}\right )} x^{4} + a b^{3} - 6 \, a^{2} b c +{\left (b^{4} - 6 \, a b^{2} c\right )} x^{2}\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (2 \, a b^{2} - 4 \, a^{2} c + 2 \,{\left (b^{3} - 3 \, a b c\right )} x^{2} +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{4 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left ({\left (b^{3} c - 6 \, a b c^{2}\right )} x^{4} + a b^{3} - 6 \, a^{2} b c +{\left (b^{4} - 6 \, a b^{2} c\right )} x^{2}\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, a b^{2} - 4 \, a^{2} c + 2 \,{\left (b^{3} - 3 \, a b c\right )} x^{2} +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{4} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 12.2503, size = 745, normalized size = 5.64 \[ \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) \log{\left (x^{2} + \frac{- 32 a^{2} c^{3} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) + 8 a^{2} c + 16 a b^{2} c^{2} \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) - a b^{2} - 2 b^{4} c \left (- \frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) \log{\left (x^{2} + \frac{- 32 a^{2} c^{3} \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) + 8 a^{2} c + 16 a b^{2} c^{2} \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right ) - a b^{2} - 2 b^{4} c \left (\frac{b \sqrt{- \left (4 a c - b^{2}\right )^{3}} \left (6 a c - b^{2}\right )}{4 c^{2} \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )} + \frac{1}{4 c^{2}}\right )}{6 a b c - b^{3}} \right )} + \frac{2 a^{2} c - a b^{2} + x^{2} \left (3 a b c - b^{3}\right )}{8 a^{2} c^{3} - 2 a b^{2} c^{2} + x^{4} \left (8 a c^{4} - 2 b^{2} c^{3}\right ) + x^{2} \left (8 a b c^{3} - 2 b^{3} c^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(c*x**4+b*x**2+a)**2,x)

[Out]

(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6)) + 1/(4*c**2))*log(x**2 + (-32*a**2*c**3*(-b*sqrt(-(
4*a*c - b**2)**3)*(6*a*c - b**2)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*
a*b**4*c - b**6)) + 1/(4*c**2)) + 8*a**2*c + 16*a*b**2*c**2*(-b*sqrt(-(4*a*c - b
**2)**3)*(6*a*c - b**2)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c
- b**6)) + 1/(4*c**2)) - a*b**2 - 2*b**4*c*(-b*sqrt(-(4*a*c - b**2)**3)*(6*a*c -
b**2)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(4*c
**2)))/(6*a*b*c - b**3)) + (b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*c**2*(6
4*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(4*c**2))*log(x**2 +
(-32*a**2*c**3*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*c**2*(64*a**3*c**3
- 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(4*c**2)) + 8*a**2*c + 16*a*b**2*
c**2*(b*sqrt(-(4*a*c - b**2)**3)*(6*a*c - b**2)/(4*c**2*(64*a**3*c**3 - 48*a**2*
b**2*c**2 + 12*a*b**4*c - b**6)) + 1/(4*c**2)) - a*b**2 - 2*b**4*c*(b*sqrt(-(4*a
*c - b**2)**3)*(6*a*c - b**2)/(4*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b
**4*c - b**6)) + 1/(4*c**2)))/(6*a*b*c - b**3)) + (2*a**2*c - a*b**2 + x**2*(3*a
*b*c - b**3))/(8*a**2*c**3 - 2*a*b**2*c**2 + x**4*(8*a*c**4 - 2*b**2*c**3) + x**
2*(8*a*b*c**3 - 2*b**3*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^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError