\(\int \frac {A+B x^2}{x^7 (a+b x^2)^2} \, dx\) [87]

Optimal result

Integrand size = 20, antiderivative size = 124 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx=-\frac {A}{6 a^2 x^6}+\frac {2 A b-a B}{4 a^3 x^4}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}-\frac {b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b^2 (4 A b-3 a B) \log (x)}{a^5}+\frac {b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {457, 78} \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx=\frac {b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5}-\frac {b^2 \log (x) (4 A b-3 a B)}{a^5}-\frac {b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}+\frac {2 A b-a B}{4 a^3 x^4}-\frac {A}{6 a^2 x^6} \]

[In]

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

[Out]

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

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 457

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {A+B x}{x^4 (a+b x)^2} \, dx,x,x^2\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a^2 x^4}+\frac {-2 A b+a B}{a^3 x^3}-\frac {b (-3 A b+2 a B)}{a^4 x^2}+\frac {b^2 (-4 A b+3 a B)}{a^5 x}-\frac {b^3 (-A b+a B)}{a^4 (a+b x)^2}-\frac {b^3 (-4 A b+3 a B)}{a^5 (a+b x)}\right ) \, dx,x,x^2\right ) \\ & = -\frac {A}{6 a^2 x^6}+\frac {2 A b-a B}{4 a^3 x^4}-\frac {b (3 A b-2 a B)}{2 a^4 x^2}-\frac {b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}-\frac {b^2 (4 A b-3 a B) \log (x)}{a^5}+\frac {b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.89 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx=\frac {-\frac {2 a^3 A}{x^6}-\frac {3 a^2 (-2 A b+a B)}{x^4}+\frac {6 a b (-3 A b+2 a B)}{x^2}+\frac {6 a b^2 (-A b+a B)}{a+b x^2}+12 b^2 (-4 A b+3 a B) \log (x)+6 b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{12 a^5} \]

[In]

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

[Out]

((-2*a^3*A)/x^6 - (3*a^2*(-2*A*b + a*B))/x^4 + (6*a*b*(-3*A*b + 2*a*B))/x^2 + (6*a*b^2*(-(A*b) + a*B))/(a + b*
x^2) + 12*b^2*(-4*A*b + 3*a*B)*Log[x] + 6*b^2*(4*A*b - 3*a*B)*Log[a + b*x^2])/(12*a^5)

Maple [A] (verified)

Time = 2.48 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94

method	result	size
default	\(-\frac {A}{6 a^{2} x^{6}}-\frac {-2 A b +B a}{4 x^{4} a^{3}}-\frac {b \left (3 A b -2 B a \right )}{2 a^{4} x^{2}}-\frac {b^{2} \left (4 A b -3 B a \right ) \ln \left (x \right )}{a^{5}}+\frac {b^{3} \left (\frac {\left (4 A b -3 B a \right ) \ln \left (b \,x^{2}+a \right )}{b}-\frac {a \left (A b -B a \right )}{b \left (b \,x^{2}+a \right )}\right )}{2 a^{5}}\)	\(117\)
norman	\(\frac {-\frac {A}{6 a}+\frac {\left (4 A b -3 B a \right ) x^{2}}{12 a^{2}}-\frac {b \left (4 A b -3 B a \right ) x^{4}}{4 a^{3}}+\frac {b \left (4 b^{3} A -3 a \,b^{2} B \right ) x^{8}}{2 a^{5}}}{x^{6} \left (b \,x^{2}+a \right )}-\frac {b^{2} \left (4 A b -3 B a \right ) \ln \left (x \right )}{a^{5}}+\frac {b^{2} \left (4 A b -3 B a \right ) \ln \left (b \,x^{2}+a \right )}{2 a^{5}}\)	\(124\)
risch	\(\frac {-\frac {b^{2} \left (4 A b -3 B a \right ) x^{6}}{2 a^{4}}-\frac {b \left (4 A b -3 B a \right ) x^{4}}{4 a^{3}}+\frac {\left (4 A b -3 B a \right ) x^{2}}{12 a^{2}}-\frac {A}{6 a}}{x^{6} \left (b \,x^{2}+a \right )}-\frac {4 b^{3} \ln \left (x \right ) A}{a^{5}}+\frac {3 b^{2} \ln \left (x \right ) B}{a^{4}}+\frac {2 b^{3} \ln \left (-b \,x^{2}-a \right ) A}{a^{5}}-\frac {3 b^{2} \ln \left (-b \,x^{2}-a \right ) B}{2 a^{4}}\)	\(139\)
parallelrisch	\(-\frac {48 A \ln \left (x \right ) x^{8} b^{4}-24 A \ln \left (b \,x^{2}+a \right ) x^{8} b^{4}-36 B \ln \left (x \right ) x^{8} a \,b^{3}+18 B \ln \left (b \,x^{2}+a \right ) x^{8} a \,b^{3}-24 A \,x^{8} b^{4}+18 B \,x^{8} a \,b^{3}+48 A \ln \left (x \right ) x^{6} a \,b^{3}-24 A \ln \left (b \,x^{2}+a \right ) x^{6} a \,b^{3}-36 B \ln \left (x \right ) x^{6} a^{2} b^{2}+18 B \ln \left (b \,x^{2}+a \right ) x^{6} a^{2} b^{2}+12 A \,x^{4} a^{2} b^{2}-9 B \,x^{4} a^{3} b -4 A \,x^{2} a^{3} b +3 B \,x^{2} a^{4}+2 A \,a^{4}}{12 a^{5} x^{6} \left (b \,x^{2}+a \right )}\)	\(207\)

[In]

int((B*x^2+A)/x^7/(b*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.48 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx=\frac {6 \, {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} - 2 \, A a^{4} + 3 \, {\left (3 \, B a^{3} b - 4 \, A a^{2} b^{2}\right )} x^{4} - {\left (3 \, B a^{4} - 4 \, A a^{3} b\right )} x^{2} - 6 \, {\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} + {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left ({\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} x^{8} + {\left (3 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6}\right )} \log \left (x\right )}{12 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}} \]

[In]

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

[Out]

1/12*(6*(3*B*a^2*b^2 - 4*A*a*b^3)*x^6 - 2*A*a^4 + 3*(3*B*a^3*b - 4*A*a^2*b^2)*x^4 - (3*B*a^4 - 4*A*a^3*b)*x^2
- 6*((3*B*a*b^3 - 4*A*b^4)*x^8 + (3*B*a^2*b^2 - 4*A*a*b^3)*x^6)*log(b*x^2 + a) + 12*((3*B*a*b^3 - 4*A*b^4)*x^8
+ (3*B*a^2*b^2 - 4*A*a*b^3)*x^6)*log(x))/(a^5*b*x^8 + a^6*x^6)

Sympy [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx=\frac {- 2 A a^{3} + x^{6} \left (- 24 A b^{3} + 18 B a b^{2}\right ) + x^{4} \left (- 12 A a b^{2} + 9 B a^{2} b\right ) + x^{2} \cdot \left (4 A a^{2} b - 3 B a^{3}\right )}{12 a^{5} x^{6} + 12 a^{4} b x^{8}} + \frac {b^{2} \left (- 4 A b + 3 B a\right ) \log {\left (x \right )}}{a^{5}} - \frac {b^{2} \left (- 4 A b + 3 B a\right ) \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{5}} \]

[In]

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

[Out]

(-2*A*a**3 + x**6*(-24*A*b**3 + 18*B*a*b**2) + x**4*(-12*A*a*b**2 + 9*B*a**2*b) + x**2*(4*A*a**2*b - 3*B*a**3)
)/(12*a**5*x**6 + 12*a**4*b*x**8) + b**2*(-4*A*b + 3*B*a)*log(x)/a**5 - b**2*(-4*A*b + 3*B*a)*log(a/b + x**2)/
(2*a**5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx=\frac {6 \, {\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} x^{6} + 3 \, {\left (3 \, B a^{2} b - 4 \, A a b^{2}\right )} x^{4} - 2 \, A a^{3} - {\left (3 \, B a^{3} - 4 \, A a^{2} b\right )} x^{2}}{12 \, {\left (a^{4} b x^{8} + a^{5} x^{6}\right )}} - \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{5}} + \frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} \]

[In]

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

[Out]

1/12*(6*(3*B*a*b^2 - 4*A*b^3)*x^6 + 3*(3*B*a^2*b - 4*A*a*b^2)*x^4 - 2*A*a^3 - (3*B*a^3 - 4*A*a^2*b)*x^2)/(a^4*
b*x^8 + a^5*x^6) - 1/2*(3*B*a*b^2 - 4*A*b^3)*log(b*x^2 + a)/a^5 + 1/2*(3*B*a*b^2 - 4*A*b^3)*log(x^2)/a^5

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.44 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx=\frac {{\left (3 \, B a b^{2} - 4 \, A b^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{5}} - \frac {{\left (3 \, B a b^{3} - 4 \, A b^{4}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{5} b} + \frac {3 \, B a b^{3} x^{2} - 4 \, A b^{4} x^{2} + 4 \, B a^{2} b^{2} - 5 \, A a b^{3}}{2 \, {\left (b x^{2} + a\right )} a^{5}} - \frac {33 \, B a b^{2} x^{6} - 44 \, A b^{3} x^{6} - 12 \, B a^{2} b x^{4} + 18 \, A a b^{2} x^{4} + 3 \, B a^{3} x^{2} - 6 \, A a^{2} b x^{2} + 2 \, A a^{3}}{12 \, a^{5} x^{6}} \]

[In]

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

[Out]

1/2*(3*B*a*b^2 - 4*A*b^3)*log(x^2)/a^5 - 1/2*(3*B*a*b^3 - 4*A*b^4)*log(abs(b*x^2 + a))/(a^5*b) + 1/2*(3*B*a*b^
3*x^2 - 4*A*b^4*x^2 + 4*B*a^2*b^2 - 5*A*a*b^3)/((b*x^2 + a)*a^5) - 1/12*(33*B*a*b^2*x^6 - 44*A*b^3*x^6 - 12*B*
a^2*b*x^4 + 18*A*a*b^2*x^4 + 3*B*a^3*x^2 - 6*A*a^2*b*x^2 + 2*A*a^3)/(a^5*x^6)

Mupad [B] (verification not implemented)

Time = 5.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.02 \[ \int \frac {A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx=\frac {\ln \left (b\,x^2+a\right )\,\left (4\,A\,b^3-3\,B\,a\,b^2\right )}{2\,a^5}-\frac {\frac {A}{6\,a}-\frac {x^2\,\left (4\,A\,b-3\,B\,a\right )}{12\,a^2}+\frac {b^2\,x^6\,\left (4\,A\,b-3\,B\,a\right )}{2\,a^4}+\frac {b\,x^4\,\left (4\,A\,b-3\,B\,a\right )}{4\,a^3}}{b\,x^8+a\,x^6}-\frac {\ln \left (x\right )\,\left (4\,A\,b^3-3\,B\,a\,b^2\right )}{a^5} \]

[In]

int((A + B*x^2)/(x^7*(a + b*x^2)^2),x)

[Out]

(log(a + b*x^2)*(4*A*b^3 - 3*B*a*b^2))/(2*a^5) - (A/(6*a) - (x^2*(4*A*b - 3*B*a))/(12*a^2) + (b^2*x^6*(4*A*b -
3*B*a))/(2*a^4) + (b*x^4*(4*A*b - 3*B*a))/(4*a^3))/(a*x^6 + b*x^8) - (log(x)*(4*A*b^3 - 3*B*a*b^2))/a^5