3.69 \(\int \frac{A+B x^2}{x^7 \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=93 \[ \frac{b^2 (A b-a B) \log \left (a+b x^2\right )}{2 a^4}-\frac{b^2 \log (x) (A b-a B)}{a^4}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{6 a x^6} \]

[Out]

-A/(6*a*x^6) + (A*b - a*B)/(4*a^2*x^4) - (b*(A*b - a*B))/(2*a^3*x^2) - (b^2*(A*b
- a*B)*Log[x])/a^4 + (b^2*(A*b - a*B)*Log[a + b*x^2])/(2*a^4)

_______________________________________________________________________________________

Rubi [A]  time = 0.191651, antiderivative size = 93, 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 \[ \frac{b^2 (A b-a B) \log \left (a+b x^2\right )}{2 a^4}-\frac{b^2 \log (x) (A b-a B)}{a^4}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x^2)/(x^7*(a + b*x^2)),x]

[Out]

-A/(6*a*x^6) + (A*b - a*B)/(4*a^2*x^4) - (b*(A*b - a*B))/(2*a^3*x^2) - (b^2*(A*b
- a*B)*Log[x])/a^4 + (b^2*(A*b - a*B)*Log[a + b*x^2])/(2*a^4)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 24.2299, size = 83, normalized size = 0.89 \[ - \frac{A}{6 a x^{6}} + \frac{A b - B a}{4 a^{2} x^{4}} - \frac{b \left (A b - B a\right )}{2 a^{3} x^{2}} - \frac{b^{2} \left (A b - B a\right ) \log{\left (x^{2} \right )}}{2 a^{4}} + \frac{b^{2} \left (A b - B a\right ) \log{\left (a + b x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x**2+A)/x**7/(b*x**2+a),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0645537, size = 96, normalized size = 1.03 \[ \frac{\left (A b^3-a b^2 B\right ) \log \left (a+b x^2\right )}{2 a^4}+\frac{\log (x) \left (a b^2 B-A b^3\right )}{a^4}+\frac{b (a B-A b)}{2 a^3 x^2}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x^2)/(x^7*(a + b*x^2)),x]

[Out]

-A/(6*a*x^6) + (A*b - a*B)/(4*a^2*x^4) + (b*(-(A*b) + a*B))/(2*a^3*x^2) + ((-(A*
b^3) + a*b^2*B)*Log[x])/a^4 + ((A*b^3 - a*b^2*B)*Log[a + b*x^2])/(2*a^4)

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 107, normalized size = 1.2 \[ -{\frac{A}{6\,a{x}^{6}}}+{\frac{Ab}{4\,{a}^{2}{x}^{4}}}-{\frac{B}{4\,a{x}^{4}}}-{\frac{{b}^{2}A}{2\,{a}^{3}{x}^{2}}}+{\frac{Bb}{2\,{a}^{2}{x}^{2}}}-{\frac{{b}^{3}\ln \left ( x \right ) A}{{a}^{4}}}+{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{3}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) A}{2\,{a}^{4}}}-{\frac{{b}^{2}\ln \left ( b{x}^{2}+a \right ) B}{2\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x^2+A)/x^7/(b*x^2+a),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34099, size = 130, normalized size = 1.4 \[ -\frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac{6 \,{\left (B a b - A b^{2}\right )} x^{4} - 2 \, A a^{2} - 3 \,{\left (B a^{2} - A a b\right )} x^{2}}{12 \, a^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)*x^7),x, algorithm="maxima")

[Out]

-1/2*(B*a*b^2 - A*b^3)*log(b*x^2 + a)/a^4 + 1/2*(B*a*b^2 - A*b^3)*log(x^2)/a^4 +
1/12*(6*(B*a*b - A*b^2)*x^4 - 2*A*a^2 - 3*(B*a^2 - A*a*b)*x^2)/(a^3*x^6)

_______________________________________________________________________________________

Fricas [A]  time = 0.237238, size = 132, normalized size = 1.42 \[ -\frac{6 \,{\left (B a b^{2} - A b^{3}\right )} x^{6} \log \left (b x^{2} + a\right ) - 12 \,{\left (B a b^{2} - A b^{3}\right )} x^{6} \log \left (x\right ) - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{4} + 2 \, A a^{3} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x^{2}}{12 \, a^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)*x^7),x, algorithm="fricas")

[Out]

-1/12*(6*(B*a*b^2 - A*b^3)*x^6*log(b*x^2 + a) - 12*(B*a*b^2 - A*b^3)*x^6*log(x)
- 6*(B*a^2*b - A*a*b^2)*x^4 + 2*A*a^3 + 3*(B*a^3 - A*a^2*b)*x^2)/(a^4*x^6)

_______________________________________________________________________________________

Sympy [A]  time = 4.39526, size = 88, normalized size = 0.95 \[ \frac{- 2 A a^{2} + x^{4} \left (- 6 A b^{2} + 6 B a b\right ) + x^{2} \left (3 A a b - 3 B a^{2}\right )}{12 a^{3} x^{6}} + \frac{b^{2} \left (- A b + B a\right ) \log{\left (x \right )}}{a^{4}} - \frac{b^{2} \left (- A b + B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x**2+A)/x**7/(b*x**2+a),x)

[Out]

(-2*A*a**2 + x**4*(-6*A*b**2 + 6*B*a*b) + x**2*(3*A*a*b - 3*B*a**2))/(12*a**3*x*
*6) + b**2*(-A*b + B*a)*log(x)/a**4 - b**2*(-A*b + B*a)*log(a/b + x**2)/(2*a**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.230746, size = 170, normalized size = 1.83 \[ \frac{{\left (B a b^{2} - A b^{3}\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} - \frac{{\left (B a b^{3} - A b^{4}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4} b} - \frac{11 \, B a b^{2} x^{6} - 11 \, A b^{3} x^{6} - 6 \, B a^{2} b x^{4} + 6 \, A a b^{2} x^{4} + 3 \, B a^{3} x^{2} - 3 \, A a^{2} b x^{2} + 2 \, A a^{3}}{12 \, a^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)/((b*x^2 + a)*x^7),x, algorithm="giac")

[Out]

1/2*(B*a*b^2 - A*b^3)*ln(x^2)/a^4 - 1/2*(B*a*b^3 - A*b^4)*ln(abs(b*x^2 + a))/(a^
4*b) - 1/12*(11*B*a*b^2*x^6 - 11*A*b^3*x^6 - 6*B*a^2*b*x^4 + 6*A*a*b^2*x^4 + 3*B
*a^3*x^2 - 3*A*a^2*b*x^2 + 2*A*a^3)/(a^4*x^6)