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

Optimal. Leaf size=70 \[ \frac{c (b B-A c) \log \left (b+c x^2\right )}{2 b^3}-\frac{c \log (x) (b B-A c)}{b^3}-\frac{b B-A c}{2 b^2 x^2}-\frac{A}{4 b x^4} \]

[Out]

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

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Rubi [A]  time = 0.160004, antiderivative size = 70, 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 \[ \frac{c (b B-A c) \log \left (b+c x^2\right )}{2 b^3}-\frac{c \log (x) (b B-A c)}{b^3}-\frac{b B-A c}{2 b^2 x^2}-\frac{A}{4 b x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.1106, size = 63, normalized size = 0.9 \[ - \frac{A}{4 b x^{4}} + \frac{A c - B b}{2 b^{2} x^{2}} + \frac{c \left (A c - B b\right ) \log{\left (x^{2} \right )}}{2 b^{3}} - \frac{c \left (A c - B b\right ) \log{\left (b + c x^{2} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0513998, size = 70, normalized size = 1. \[ \frac{4 c x^4 \log (x) (A c-b B)-b \left (A b-2 A c x^2+2 b B x^2\right )+2 c x^4 (b B-A c) \log \left (b+c x^2\right )}{4 b^3 x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 81, normalized size = 1.2 \[ -{\frac{A}{4\,b{x}^{4}}}+{\frac{Ac}{2\,{b}^{2}{x}^{2}}}-{\frac{B}{2\,b{x}^{2}}}+{\frac{A\ln \left ( x \right ){c}^{2}}{{b}^{3}}}-{\frac{Bc\ln \left ( x \right ) }{{b}^{2}}}-{\frac{{c}^{2}\ln \left ( c{x}^{2}+b \right ) A}{2\,{b}^{3}}}+{\frac{c\ln \left ( c{x}^{2}+b \right ) B}{2\,{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.37831, size = 95, normalized size = 1.36 \[ \frac{{\left (B b c - A c^{2}\right )} \log \left (c x^{2} + b\right )}{2 \, b^{3}} - \frac{{\left (B b c - A c^{2}\right )} \log \left (x^{2}\right )}{2 \, b^{3}} - \frac{2 \,{\left (B b - A c\right )} x^{2} + A b}{4 \, b^{2} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.218197, size = 99, normalized size = 1.41 \[ \frac{2 \,{\left (B b c - A c^{2}\right )} x^{4} \log \left (c x^{2} + b\right ) - 4 \,{\left (B b c - A c^{2}\right )} x^{4} \log \left (x\right ) - A b^{2} - 2 \,{\left (B b^{2} - A b c\right )} x^{2}}{4 \, b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.82504, size = 61, normalized size = 0.87 \[ - \frac{A b + x^{2} \left (- 2 A c + 2 B b\right )}{4 b^{2} x^{4}} - \frac{c \left (- A c + B b\right ) \log{\left (x \right )}}{b^{3}} + \frac{c \left (- A c + B b\right ) \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209852, size = 135, normalized size = 1.93 \[ -\frac{{\left (B b c - A c^{2}\right )}{\rm ln}\left (x^{2}\right )}{2 \, b^{3}} + \frac{{\left (B b c^{2} - A c^{3}\right )}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{3} c} + \frac{3 \, B b c x^{4} - 3 \, A c^{2} x^{4} - 2 \, B b^{2} x^{2} + 2 \, A b c x^{2} - A b^{2}}{4 \, b^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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