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

Optimal. Leaf size=97 \[ -\frac{b (3 A b-2 a B) \log \left (a+b x^2\right )}{2 a^4}+\frac{b \log (x) (3 A b-2 a B)}{a^4}+\frac{b (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{A}{4 a^2 x^4} \]

[Out]

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

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Rubi [A]  time = 0.228237, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{b (3 A b-2 a B) \log \left (a+b x^2\right )}{2 a^4}+\frac{b \log (x) (3 A b-2 a B)}{a^4}+\frac{b (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac{2 A b-a B}{2 a^3 x^2}-\frac{A}{4 a^2 x^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.7476, size = 94, normalized size = 0.97 \[ - \frac{A}{4 a^{2} x^{4}} + \frac{b \left (A b - B a\right )}{2 a^{3} \left (a + b x^{2}\right )} + \frac{2 A b - B a}{2 a^{3} x^{2}} + \frac{b \left (3 A b - 2 B a\right ) \log{\left (x^{2} \right )}}{2 a^{4}} - \frac{b \left (3 A b - 2 B a\right ) \log{\left (a + b x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.160471, size = 85, normalized size = 0.88 \[ -\frac{\frac{a^2 A}{x^4}+\frac{2 a b (a B-A b)}{a+b x^2}+\frac{2 a (a B-2 A b)}{x^2}+2 b (3 A b-2 a B) \log \left (a+b x^2\right )-4 b \log (x) (3 A b-2 a B)}{4 a^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 114, normalized size = 1.2 \[ -{\frac{A}{4\,{a}^{2}{x}^{4}}}+{\frac{Ab}{{a}^{3}{x}^{2}}}-{\frac{B}{2\,{a}^{2}{x}^{2}}}+3\,{\frac{A\ln \left ( x \right ){b}^{2}}{{a}^{4}}}-2\,{\frac{bB\ln \left ( x \right ) }{{a}^{3}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{4}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) B}{{a}^{3}}}+{\frac{{b}^{2}A}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{Bb}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36033, size = 143, normalized size = 1.47 \[ -\frac{2 \,{\left (2 \, B a b - 3 \, A b^{2}\right )} x^{4} + A a^{2} +{\left (2 \, B a^{2} - 3 \, A a b\right )} x^{2}}{4 \,{\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} + \frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac{{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x^{2}\right )}{2 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*(2*B*a*b - 3*A*b^2)*x^4 + A*a^2 + (2*B*a^2 - 3*A*a*b)*x^2)/(a^3*b*x^6 +
a^4*x^4) + 1/2*(2*B*a*b - 3*A*b^2)*log(b*x^2 + a)/a^4 - 1/2*(2*B*a*b - 3*A*b^2)*
log(x^2)/a^4

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Fricas [A]  time = 0.228635, size = 208, normalized size = 2.14 \[ -\frac{2 \,{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4} + A a^{3} +{\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x^{2} - 2 \,{\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{6} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{6} +{\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{4}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*(2*B*a^2*b - 3*A*a*b^2)*x^4 + A*a^3 + (2*B*a^3 - 3*A*a^2*b)*x^2 - 2*((2*
B*a*b^2 - 3*A*b^3)*x^6 + (2*B*a^2*b - 3*A*a*b^2)*x^4)*log(b*x^2 + a) + 4*((2*B*a
*b^2 - 3*A*b^3)*x^6 + (2*B*a^2*b - 3*A*a*b^2)*x^4)*log(x))/(a^4*b*x^6 + a^5*x^4)

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Sympy [A]  time = 4.97154, size = 100, normalized size = 1.03 \[ - \frac{A a^{2} + x^{4} \left (- 6 A b^{2} + 4 B a b\right ) + x^{2} \left (- 3 A a b + 2 B a^{2}\right )}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} - \frac{b \left (- 3 A b + 2 B a\right ) \log{\left (x \right )}}{a^{4}} + \frac{b \left (- 3 A b + 2 B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A*a**2 + x**4*(-6*A*b**2 + 4*B*a*b) + x**2*(-3*A*a*b + 2*B*a**2))/(4*a**4*x**4
+ 4*a**3*b*x**6) - b*(-3*A*b + 2*B*a)*log(x)/a**4 + b*(-3*A*b + 2*B*a)*log(a/b
+ x**2)/(2*a**4)

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GIAC/XCAS [A]  time = 0.234286, size = 203, normalized size = 2.09 \[ -\frac{{\left (2 \, B a b - 3 \, A b^{2}\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} + \frac{{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b} - \frac{2 \, B a b^{2} x^{2} - 3 \, A b^{3} x^{2} + 3 \, B a^{2} b - 4 \, A a b^{2}}{2 \,{\left (b x^{2} + a\right )} a^{4}} + \frac{6 \, B a b x^{4} - 9 \, A b^{2} x^{4} - 2 \, B a^{2} x^{2} + 4 \, A a b x^{2} - A a^{2}}{4 \, a^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*B*a*b - 3*A*b^2)*ln(x^2)/a^4 + 1/2*(2*B*a*b^2 - 3*A*b^3)*ln(abs(b*x^2 +
a))/(a^4*b) - 1/2*(2*B*a*b^2*x^2 - 3*A*b^3*x^2 + 3*B*a^2*b - 4*A*a*b^2)/((b*x^2
+ a)*a^4) + 1/4*(6*B*a*b*x^4 - 9*A*b^2*x^4 - 2*B*a^2*x^2 + 4*A*a*b*x^2 - A*a^2)/
(a^4*x^4)