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

Optimal. Leaf size=101 \[ \frac{(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac{A}{2 a^3 x^2}-\frac{A b-a B}{4 a^2 \left (a+b x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.245005, antiderivative size = 101, 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 \[ \frac{(3 A b-a B) \log \left (a+b x^2\right )}{2 a^4}-\frac{\log (x) (3 A b-a B)}{a^4}-\frac{2 A b-a B}{2 a^3 \left (a+b x^2\right )}-\frac{A}{2 a^3 x^2}-\frac{A b-a B}{4 a^2 \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.6393, size = 90, normalized size = 0.89 \[ - \frac{A}{2 a^{3} x^{2}} - \frac{A b - B a}{4 a^{2} \left (a + b x^{2}\right )^{2}} - \frac{2 A b - B a}{2 a^{3} \left (a + b x^{2}\right )} - \frac{\left (3 A b - B a\right ) \log{\left (x^{2} \right )}}{2 a^{4}} + \frac{\left (3 A b - B a\right ) \log{\left (a + b x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0955015, size = 87, normalized size = 0.86 \[ \frac{\frac{a^2 (a B-A b)}{\left (a+b x^2\right )^2}+\frac{2 a (a B-2 A b)}{a+b x^2}+2 (3 A b-a B) \log \left (a+b x^2\right )+4 \log (x) (a B-3 A b)-\frac{2 a A}{x^2}}{4 a^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.022, size = 118, normalized size = 1.2 \[ -{\frac{A}{2\,{a}^{3}{x}^{2}}}-3\,{\frac{A\ln \left ( x \right ) b}{{a}^{4}}}+{\frac{\ln \left ( x \right ) B}{{a}^{3}}}-{\frac{Ab}{4\,{a}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{B}{4\,a \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,b\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{4}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,{a}^{3}}}-{\frac{Ab}{{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{B}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34389, size = 147, normalized size = 1.46 \[ \frac{2 \,{\left (B a b - 3 \, A b^{2}\right )} x^{4} - 2 \, A a^{2} + 3 \,{\left (B a^{2} - 3 \, A a b\right )} x^{2}}{4 \,{\left (a^{3} b^{2} x^{6} + 2 \, a^{4} b x^{4} + a^{5} x^{2}\right )}} - \frac{{\left (B a - 3 \, A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220231, size = 266, normalized size = 2.63 \[ \frac{2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} - 2 \, A a^{3} + 3 \,{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2} - 2 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left ({\left (B a b^{2} - 3 \, A b^{3}\right )} x^{6} + 2 \,{\left (B a^{2} b - 3 \, A a b^{2}\right )} x^{4} +{\left (B a^{3} - 3 \, A a^{2} b\right )} x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{4} b^{2} x^{6} + 2 \, a^{5} b x^{4} + a^{6} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.1195, size = 107, normalized size = 1.06 \[ \frac{- 2 A a^{2} + x^{4} \left (- 6 A b^{2} + 2 B a b\right ) + x^{2} \left (- 9 A a b + 3 B a^{2}\right )}{4 a^{5} x^{2} + 8 a^{4} b x^{4} + 4 a^{3} b^{2} x^{6}} + \frac{\left (- 3 A b + B a\right ) \log{\left (x \right )}}{a^{4}} - \frac{\left (- 3 A b + B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*A*a**2 + x**4*(-6*A*b**2 + 2*B*a*b) + x**2*(-9*A*a*b + 3*B*a**2))/(4*a**5*x*
*2 + 8*a**4*b*x**4 + 4*a**3*b**2*x**6) + (-3*A*b + B*a)*log(x)/a**4 - (-3*A*b +
B*a)*log(a/b + x**2)/(2*a**4)

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GIAC/XCAS [A]  time = 0.224067, size = 186, normalized size = 1.84 \[ \frac{{\left (B a - 3 \, A b\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} - \frac{{\left (B a b - 3 \, A b^{2}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b} + \frac{3 \, B a b^{2} x^{4} - 9 \, A b^{3} x^{4} + 8 \, B a^{2} b x^{2} - 22 \, A a b^{2} x^{2} + 6 \, B a^{3} - 14 \, A a^{2} b}{4 \,{\left (b x^{2} + a\right )}^{2} a^{4}} - \frac{B a x^{2} - 3 \, A b x^{2} + A a}{2 \, a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*a - 3*A*b)*ln(x^2)/a^4 - 1/2*(B*a*b - 3*A*b^2)*ln(abs(b*x^2 + a))/(a^4*b)
+ 1/4*(3*B*a*b^2*x^4 - 9*A*b^3*x^4 + 8*B*a^2*b*x^2 - 22*A*a*b^2*x^2 + 6*B*a^3 -
14*A*a^2*b)/((b*x^2 + a)^2*a^4) - 1/2*(B*a*x^2 - 3*A*b*x^2 + A*a)/(a^4*x^2)