3.103 \(\int \frac{x^3 \left (A+B x^2\right )}{a+b x^2+c x^4} \, dx\)

Optimal. Leaf size=97 \[ -\frac{\left (-2 a B c-A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}-\frac{(b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{B x^2}{2 c} \]

[Out]

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

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Rubi [A]  time = 0.259455, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ -\frac{\left (-2 a B c-A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}-\frac{(b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{B x^2}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.123189, size = 93, normalized size = 0.96 \[ \frac{\frac{2 \left (-2 a B c-A b c+b^2 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+(A c-b B) \log \left (a+b x^2+c x^4\right )+2 B c x^2}{4 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 175, normalized size = 1.8 \[{\frac{B{x}^{2}}{2\,c}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) A}{4\,c}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bB}{4\,{c}^{2}}}-{\frac{Ba}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{Ab}{2\,c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}B}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*B*x^2/c+1/4/c*ln(c*x^4+b*x^2+a)*A-1/4/c^2*ln(c*x^4+b*x^2+a)*b*B-1/c/(4*a*c-b
^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*B*a-1/2/c/(4*a*c-b^2)^(1/2)*arct
an((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*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))*b^2*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^3/(c*x^4 + b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.273215, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (2 \, B c x^{2} -{\left (B b - A c\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{4 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, \frac{2 \,{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, B c x^{2} -{\left (B b - A c\right )} \log \left (c x^{4} + b x^{2} + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{4 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 15.7278, size = 434, normalized size = 4.47 \[ \frac{B x^{2}}{2 c} + \left (- \frac{- A c + B b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{2 A a c - B a b - 8 a c^{2} \left (- \frac{- A c + B b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac{- A c + B b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right )}{A b c + 2 B a c - B b^{2}} \right )} + \left (- \frac{- A c + B b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{2 A a c - B a b - 8 a c^{2} \left (- \frac{- A c + B b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac{- A c + B b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right )}{A b c + 2 B a c - B b^{2}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.291799, size = 123, normalized size = 1.27 \[ \frac{B x^{2}}{2 \, c} - \frac{{\left (B b - A c\right )}{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac{{\left (B b^{2} - 2 \, B a c - A b c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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