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

Optimal. Leaf size=124 \[ -\frac{b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}+\frac{b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}-\frac{b (3 A b-2 a B)}{2 a^4 x^2}+\frac{2 A b-a B}{4 a^3 x^4}-\frac{A}{6 a^2 x^6} \]

[Out]

-A/(6*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)

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Rubi [A]  time = 0.129539, antiderivative size = 124, 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, Rules used = {446, 77} \[ -\frac{b^2 (A b-a B)}{2 a^4 \left (a+b x^2\right )}+\frac{b^2 (4 A b-3 a B) \log \left (a+b x^2\right )}{2 a^5}-\frac{b^2 \log (x) (4 A b-3 a B)}{a^5}-\frac{b (3 A b-2 a B)}{2 a^4 x^2}+\frac{2 A b-a B}{4 a^3 x^4}-\frac{A}{6 a^2 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-A/(6*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 446

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]]

Rule 77

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]))))

Rubi steps

\begin{align*} \int \frac{A+B x^2}{x^7 \left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{A+B x}{x^4 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{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]  time = 0.0984883, size = 110, normalized size = 0.89 \[ \frac{-\frac{3 a^2 (a B-2 A b)}{x^4}-\frac{2 a^3 A}{x^6}+\frac{6 a b^2 (a B-A b)}{a+b x^2}+6 b^2 (4 A b-3 a B) \log \left (a+b x^2\right )+12 b^2 \log (x) (3 a B-4 A b)+\frac{6 a b (2 a B-3 A b)}{x^2}}{12 a^5} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.014, size = 143, normalized size = 1.2 \begin{align*} -{\frac{A}{6\,{a}^{2}{x}^{6}}}+{\frac{Ab}{2\,{a}^{3}{x}^{4}}}-{\frac{B}{4\,{a}^{2}{x}^{4}}}-{\frac{3\,A{b}^{2}}{2\,{a}^{4}{x}^{2}}}+{\frac{Bb}{{a}^{3}{x}^{2}}}-4\,{\frac{{b}^{3}\ln \left ( x \right ) A}{{a}^{5}}}+3\,{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{4}}}+2\,{\frac{{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}}}-{\frac{A{b}^{3}}{2\,{a}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{B{b}^{2}}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.980521, size = 184, normalized size = 1.48 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.41584, size = 385, normalized size = 3.1 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 1.58456, size = 129, normalized size = 1.04 \begin{align*} \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} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.62136, size = 240, normalized size = 1.94 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)