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

Optimal. Leaf size=100 \[ \frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \sqrt{b^2-4 a c}}+\frac{\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}-\frac{b x^2}{2 c^2}+\frac{x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.282588, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{b \left (b^2-3 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \sqrt{b^2-4 a c}}+\frac{\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}-\frac{b x^2}{2 c^2}+\frac{x^4}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a + b*x^2 + c*x^4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.180034, size = 93, normalized size = 0.93 \[ \frac{-\frac{2 b \left (b^2-3 a c\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\left (b^2-a c\right ) \log \left (a+b x^2+c x^4\right )+c x^2 \left (c x^2-2 b\right )}{4 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 142, normalized size = 1.4 \[{\frac{{x}^{4}}{4\,c}}-{\frac{b{x}^{2}}{2\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) a}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}}{4\,{c}^{3}}}+{\frac{3\,ab}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}}{2\,{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(c*x^4+b*x^2+a),x)

[Out]

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

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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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270734, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b^{3} - 3 \, a b c\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (c^{2} x^{4} - 2 \, b c x^{2} +{\left (b^{2} - a c\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{4 \, \sqrt{b^{2} - 4 \, a c} c^{3}}, -\frac{2 \,{\left (b^{3} - 3 \, a b c\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (c^{2} x^{4} - 2 \, b c x^{2} +{\left (b^{2} - a c\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \, \sqrt{-b^{2} + 4 \, a c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.64268, size = 391, normalized size = 3.91 \[ - \frac{b x^{2}}{2 c^{2}} + \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{4 c^{3}}\right ) \log{\left (x^{2} + \frac{2 a^{2} c - a b^{2} + 8 a c^{3} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{4 c^{3}}\right ) - 2 b^{2} c^{2} \left (- \frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{4 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{4 c^{3}}\right ) \log{\left (x^{2} + \frac{2 a^{2} c - a b^{2} + 8 a c^{3} \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{4 c^{3}}\right ) - 2 b^{2} c^{2} \left (\frac{b \sqrt{- 4 a c + b^{2}} \left (3 a c - b^{2}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{a c - b^{2}}{4 c^{3}}\right )}{3 a b c - b^{3}} \right )} + \frac{x^{4}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*x**2/(2*c**2) + (-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*c**3*(4*a*c - b**2)
) - (a*c - b**2)/(4*c**3))*log(x**2 + (2*a**2*c - a*b**2 + 8*a*c**3*(-b*sqrt(-4*
a*c + b**2)*(3*a*c - b**2)/(4*c**3*(4*a*c - b**2)) - (a*c - b**2)/(4*c**3)) - 2*
b**2*c**2*(-b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*c**3*(4*a*c - b**2)) - (a*c
- b**2)/(4*c**3)))/(3*a*b*c - b**3)) + (b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*
c**3*(4*a*c - b**2)) - (a*c - b**2)/(4*c**3))*log(x**2 + (2*a**2*c - a*b**2 + 8*
a*c**3*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*c**3*(4*a*c - b**2)) - (a*c - b*
*2)/(4*c**3)) - 2*b**2*c**2*(b*sqrt(-4*a*c + b**2)*(3*a*c - b**2)/(4*c**3*(4*a*c
- b**2)) - (a*c - b**2)/(4*c**3)))/(3*a*b*c - b**3)) + x**4/(4*c)

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GIAC/XCAS [A]  time = 0.290605, size = 124, normalized size = 1.24 \[ \frac{c x^{4} - 2 \, b x^{2}}{4 \, c^{2}} + \frac{{\left (b^{2} - a c\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} - \frac{{\left (b^{3} - 3 \, a b c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(c*x^4 - 2*b*x^2)/c^2 + 1/4*(b^2 - a*c)*ln(c*x^4 + b*x^2 + a)/c^3 - 1/2*(b^3
- 3*a*b*c)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^3)