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

Optimal. Leaf size=33 \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.021635, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {446, 76} \[ -\frac{a B+A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*(A + B*x^2))/x^7,x]

[Out]

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

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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right ) \left (A+B x^2\right )}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x) (A+B x)}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a A}{x^4}+\frac{A b+a B}{x^3}+\frac{b B}{x^2}\right ) \, dx,x,x^2\right )\\ &=-\frac{a A}{6 x^6}-\frac{A b+a B}{4 x^4}-\frac{b B}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0090123, size = 35, normalized size = 1.06 \[ \frac{-a B-A b}{4 x^4}-\frac{a A}{6 x^6}-\frac{b B}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 28, normalized size = 0.9 \begin{align*} -{\frac{Ab+Ba}{4\,{x}^{4}}}-{\frac{Bb}{2\,{x}^{2}}}-{\frac{Aa}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47851, size = 39, normalized size = 1.18 \begin{align*} -\frac{6 \, B b x^{4} + 3 \,{\left (B a + A b\right )} x^{2} + 2 \, A a}{12 \, x^{6}} \end{align*}

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]

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

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Fricas [A]  time = 1.41149, size = 69, normalized size = 2.09 \begin{align*} -\frac{6 \, B b x^{4} + 3 \,{\left (B a + A b\right )} x^{2} + 2 \, A a}{12 \, x^{6}} \end{align*}

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

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Sympy [A]  time = 0.602801, size = 32, normalized size = 0.97 \begin{align*} - \frac{2 A a + 6 B b x^{4} + x^{2} \left (3 A b + 3 B a\right )}{12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10017, size = 42, normalized size = 1.27 \begin{align*} -\frac{6 \, B b x^{4} + 3 \, B a x^{2} + 3 \, A b x^{2} + 2 \, A a}{12 \, x^{6}} \end{align*}

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]

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