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

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)

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Rubi [A]  time = 0.0823365, 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, Rules used = {446, 77} \[ \frac{b^2 (A b-a B) \log \left (a+b x^2\right )}{2 a^4}-\frac{b^2 \log (x) (A b-a B)}{a^4}-\frac{b (A b-a B)}{2 a^3 x^2}+\frac{A b-a B}{4 a^2 x^4}-\frac{A}{6 a x^6} \]

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)

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

Mathematica [A]  time = 0.0359372, 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)

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Maple [A]  time = 0.007, size = 107, normalized size = 1.2 \begin{align*} -{\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}}} \end{align*}

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

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Maxima [A]  time = 1.0235, size = 130, normalized size = 1.4 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Fricas [A]  time = 1.22837, size = 211, normalized size = 2.27 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Sympy [A]  time = 1.16445, size = 88, normalized size = 0.95 \begin{align*} \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}} \end{align*}

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)

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Giac [A]  time = 1.35631, size = 170, normalized size = 1.83 \begin{align*} \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac{{\left (B a b^{3} - A b^{4}\right )} \log \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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