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

Optimal. Leaf size=58 \[ \frac{\sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{x (b B-A c)}{c^2}+\frac{B x^3}{3 c} \]

[Out]

-(((b*B - A*c)*x)/c^2) + (B*x^3)/(3*c) + (Sqrt[b]*(b*B - A*c)*ArcTan[(Sqrt[c]*x)
/Sqrt[b]])/c^(5/2)

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Rubi [A]  time = 0.11399, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}-\frac{x (b B-A c)}{c^2}+\frac{B x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

-(((b*B - A*c)*x)/c^2) + (B*x^3)/(3*c) + (Sqrt[b]*(b*B - A*c)*ArcTan[(Sqrt[c]*x)
/Sqrt[b]])/c^(5/2)

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Rubi in Sympy [A]  time = 16.9807, size = 49, normalized size = 0.84 \[ \frac{B x^{3}}{3 c} - \frac{\sqrt{b} \left (A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{c^{\frac{5}{2}}} + \frac{x \left (A c - B b\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**3/(3*c) - sqrt(b)*(A*c - B*b)*atan(sqrt(c)*x/sqrt(b))/c**(5/2) + x*(A*c - B
*b)/c**2

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Mathematica [A]  time = 0.064945, size = 57, normalized size = 0.98 \[ \frac{\sqrt{b} (b B-A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{c^{5/2}}+\frac{x (A c-b B)}{c^2}+\frac{B x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

((-(b*B) + A*c)*x)/c^2 + (B*x^3)/(3*c) + (Sqrt[b]*(b*B - A*c)*ArcTan[(Sqrt[c]*x)
/Sqrt[b]])/c^(5/2)

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Maple [A]  time = 0.005, size = 68, normalized size = 1.2 \[{\frac{B{x}^{3}}{3\,c}}+{\frac{Ax}{c}}-{\frac{xBb}{{c}^{2}}}-{\frac{Ab}{c}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{{b}^{2}B}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217328, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, B c x^{3} - 3 \,{\left (B b - A c\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 6 \,{\left (B b - A c\right )} x}{6 \, c^{2}}, \frac{B c x^{3} + 3 \,{\left (B b - A c\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) - 3 \,{\left (B b - A c\right )} x}{3 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.87534, size = 90, normalized size = 1.55 \[ \frac{B x^{3}}{3 c} - \frac{\sqrt{- \frac{b}{c^{5}}} \left (- A c + B b\right ) \log{\left (- c^{2} \sqrt{- \frac{b}{c^{5}}} + x \right )}}{2} + \frac{\sqrt{- \frac{b}{c^{5}}} \left (- A c + B b\right ) \log{\left (c^{2} \sqrt{- \frac{b}{c^{5}}} + x \right )}}{2} - \frac{x \left (- A c + B b\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**3/(3*c) - sqrt(-b/c**5)*(-A*c + B*b)*log(-c**2*sqrt(-b/c**5) + x)/2 + sqrt(
-b/c**5)*(-A*c + B*b)*log(c**2*sqrt(-b/c**5) + x)/2 - x*(-A*c + B*b)/c**2

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GIAC/XCAS [A]  time = 0.208004, size = 77, normalized size = 1.33 \[ \frac{{\left (B b^{2} - A b c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{\sqrt{b c} c^{2}} + \frac{B c^{2} x^{3} - 3 \, B b c x + 3 \, A c^{2} x}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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