3.102 \(\int \frac{x^5 \left (A+B x^2\right )}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=133 \[ \frac{\left (-a B c-A b c+b^2 B\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}+\frac{\left (2 a A c^2-3 a b B c-A b^2 c+b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \sqrt{b^2-4 a c}}-\frac{x^2 (b B-A c)}{2 c^2}+\frac{B x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.432134, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\left (-a B c-A b c+b^2 B\right ) \log \left (a+b x^2+c x^4\right )}{4 c^3}+\frac{\left (2 a A c^2-3 a b B c-A b^2 c+b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \sqrt{b^2-4 a c}}-\frac{x^2 (b B-A c)}{2 c^2}+\frac{B x^4}{4 c} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4),x]

[Out]

-((b*B - A*c)*x^2)/(2*c^2) + (B*x^4)/(4*c) + ((b^3*B - A*b^2*c - 3*a*b*B*c + 2*a
*A*c^2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*Sqrt[b^2 - 4*a*c]) + ((
b^2*B - A*b*c - a*B*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. \[ \frac{B \int ^{x^{2}} x\, dx}{2 c} + \left (\frac{A c}{2} - \frac{B b}{2}\right ) \int ^{x^{2}} \frac{1}{c^{2}}\, dx + \frac{\left (- A b c - B a c + B b^{2}\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c^{3}} + \frac{\left (2 A a c^{2} - A b^{2} c - 3 B a b c + B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c^{3} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(B*x**2+A)/(c*x**4+b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.101535, size = 126, normalized size = 0.95 \[ \frac{\left (-a B c-A b c+b^2 B\right ) \log \left (a+b x^2+c x^4\right )+\frac{2 \left (-2 a A c^2+3 a b B c+A b^2 c+b^3 (-B)\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+2 c x^2 (A c-b B)+B c^2 x^4}{4 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(A + B*x^2))/(a + b*x^2 + c*x^4),x]

[Out]

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

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Maple [B]  time = 0.008, size = 261, normalized size = 2. \[{\frac{B{x}^{4}}{4\,c}}+{\frac{A{x}^{2}}{2\,c}}-{\frac{bB{x}^{2}}{2\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) Ab}{4\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) aB}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}B}{4\,{c}^{3}}}-{\frac{aA}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{3\,abB}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{A{b}^{2}}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}B}{2\,{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(B*x^2+A)/(c*x^4+b*x^2+a),x)

[Out]

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

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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((B*x^2 + A)*x^5/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.279729, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (B b^{3} + 2 \, A a c^{2} -{\left (3 \, B a b + A b^{2}\right )} c\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (B c^{2} x^{4} - 2 \,{\left (B b c - A c^{2}\right )} x^{2} +{\left (B b^{2} -{\left (B a + A b\right )} c\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{4 \, \sqrt{b^{2} - 4 \, a c} c^{3}}, -\frac{2 \,{\left (B b^{3} + 2 \, A a c^{2} -{\left (3 \, B a b + A b^{2}\right )} c\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (B c^{2} x^{4} - 2 \,{\left (B b c - A c^{2}\right )} x^{2} +{\left (B b^{2} -{\left (B a + A b\right )} c\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \, \sqrt{-b^{2} + 4 \, a c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^5/(c*x^4 + b*x^2 + a),x, algorithm="fricas")

[Out]

[1/4*((B*b^3 + 2*A*a*c^2 - (3*B*a*b + A*b^2)*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)) + (B*c^2*x^4 - 2*(B*b*c - A*c^2)*x^2 + (B*b^2 - (B*a + A*b)*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*b^3 +
2*A*a*c^2 - (3*B*a*b + A*b^2)*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2
- 4*a*c)) - (B*c^2*x^4 - 2*(B*b*c - A*c^2)*x^2 + (B*b^2 - (B*a + A*b)*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 = 24.859, size = 619, normalized size = 4.65 \[ \frac{B x^{4}}{4 c} + \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (- 2 A a c^{2} + A b^{2} c + 3 B a b c - B b^{3}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{A b c + B a c - B b^{2}}{4 c^{3}}\right ) \log{\left (x^{2} + \frac{A a b c + 2 B a^{2} c - B a b^{2} + 8 a c^{3} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (- 2 A a c^{2} + A b^{2} c + 3 B a b c - B b^{3}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{A b c + B a c - B b^{2}}{4 c^{3}}\right ) - 2 b^{2} c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (- 2 A a c^{2} + A b^{2} c + 3 B a b c - B b^{3}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{A b c + B a c - B b^{2}}{4 c^{3}}\right )}{- 2 A a c^{2} + A b^{2} c + 3 B a b c - B b^{3}} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (- 2 A a c^{2} + A b^{2} c + 3 B a b c - B b^{3}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{A b c + B a c - B b^{2}}{4 c^{3}}\right ) \log{\left (x^{2} + \frac{A a b c + 2 B a^{2} c - B a b^{2} + 8 a c^{3} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (- 2 A a c^{2} + A b^{2} c + 3 B a b c - B b^{3}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{A b c + B a c - B b^{2}}{4 c^{3}}\right ) - 2 b^{2} c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (- 2 A a c^{2} + A b^{2} c + 3 B a b c - B b^{3}\right )}{4 c^{3} \left (4 a c - b^{2}\right )} - \frac{A b c + B a c - B b^{2}}{4 c^{3}}\right )}{- 2 A a c^{2} + A b^{2} c + 3 B a b c - B b^{3}} \right )} - \frac{x^{2} \left (- A c + B b\right )}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(B*x**2+A)/(c*x**4+b*x**2+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.294702, size = 170, normalized size = 1.28 \[ \frac{B c x^{4} - 2 \, B b x^{2} + 2 \, A c x^{2}}{4 \, c^{2}} + \frac{{\left (B b^{2} - B a c - A b c\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} - \frac{{\left (B b^{3} - 3 \, B a b c - A b^{2} c + 2 \, A a c^{2}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^5/(c*x^4 + b*x^2 + a),x, algorithm="giac")

[Out]

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