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

Optimal. Leaf size=54 \[ \frac{b (b B-A c) \log \left (b+c x^2\right )}{2 c^3}-\frac{x^2 (b B-A c)}{2 c^2}+\frac{B x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.141343, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{b (b B-A c) \log \left (b+c x^2\right )}{2 c^3}-\frac{x^2 (b B-A c)}{2 c^2}+\frac{B x^4}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B \int ^{x^{2}} x\, dx}{2 c} - \frac{b \left (A c - B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{3}} + \left (\frac{A c}{2} - \frac{B b}{2}\right ) \int ^{x^{2}} \frac{1}{c^{2}}\, dx \]

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),x)

[Out]

B*Integral(x, (x, x**2))/(2*c) - b*(A*c - B*b)*log(b + c*x**2)/(2*c**3) + (A*c/2
- B*b/2)*Integral(c**(-2), (x, x**2))

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Mathematica [A]  time = 0.0337198, size = 47, normalized size = 0.87 \[ \frac{c x^2 \left (2 A c-2 b B+B c x^2\right )+2 b (b B-A c) \log \left (b+c x^2\right )}{4 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 62, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,c}}+{\frac{A{x}^{2}}{2\,c}}-{\frac{Bb{x}^{2}}{2\,{c}^{2}}}-{\frac{b\ln \left ( c{x}^{2}+b \right ) A}{2\,{c}^{2}}}+{\frac{{b}^{2}\ln \left ( c{x}^{2}+b \right ) B}{2\,{c}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37736, size = 68, normalized size = 1.26 \[ \frac{B c x^{4} - 2 \,{\left (B b - A c\right )} x^{2}}{4 \, c^{2}} + \frac{{\left (B b^{2} - A b c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203181, size = 69, normalized size = 1.28 \[ \frac{B c^{2} x^{4} - 2 \,{\left (B b c - A c^{2}\right )} x^{2} + 2 \,{\left (B b^{2} - A b c\right )} \log \left (c x^{2} + b\right )}{4 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.815906, size = 44, normalized size = 0.81 \[ \frac{B x^{4}}{4 c} + \frac{b \left (- A c + B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{3}} - \frac{x^{2} \left (- A c + B b\right )}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.21203, size = 70, normalized size = 1.3 \[ \frac{B c x^{4} - 2 \, B b x^{2} + 2 \, A c x^{2}}{4 \, c^{2}} + \frac{{\left (B b^{2} - A b c\right )}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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