3.22 \(\int \frac{(a+b x^2)^2 (A+B x^2)}{x^9} \, dx\)

Optimal. Leaf size=48 \[ \frac{\left (a+b x^2\right )^3 (A b-4 a B)}{24 a^2 x^6}-\frac{A \left (a+b x^2\right )^3}{8 a x^8} \]

[Out]

-(A*(a + b*x^2)^3)/(8*a*x^8) + ((A*b - 4*a*B)*(a + b*x^2)^3)/(24*a^2*x^6)

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Rubi [A]  time = 0.0311508, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {446, 78, 37} \[ \frac{\left (a+b x^2\right )^3 (A b-4 a B)}{24 a^2 x^6}-\frac{A \left (a+b x^2\right )^3}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(A + B*x^2))/x^9,x]

[Out]

-(A*(a + b*x^2)^3)/(8*a*x^8) + ((A*b - 4*a*B)*(a + b*x^2)^3)/(24*a^2*x^6)

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (A+B x^2\right )}{x^9} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2 (A+B x)}{x^5} \, dx,x,x^2\right )\\ &=-\frac{A \left (a+b x^2\right )^3}{8 a x^8}+\frac{(-A b+4 a B) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x^4} \, dx,x,x^2\right )}{8 a}\\ &=-\frac{A \left (a+b x^2\right )^3}{8 a x^8}+\frac{(A b-4 a B) \left (a+b x^2\right )^3}{24 a^2 x^6}\\ \end{align*}

Mathematica [A]  time = 0.0161696, size = 55, normalized size = 1.15 \[ -\frac{a^2 \left (3 A+4 B x^2\right )+4 a b x^2 \left (2 A+3 B x^2\right )+6 b^2 x^4 \left (A+2 B x^2\right )}{24 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(A + B*x^2))/x^9,x]

[Out]

-(6*b^2*x^4*(A + 2*B*x^2) + 4*a*b*x^2*(2*A + 3*B*x^2) + a^2*(3*A + 4*B*x^2))/(24*x^8)

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Maple [A]  time = 0.005, size = 48, normalized size = 1. \begin{align*} -{\frac{b \left ( Ab+2\,Ba \right ) }{4\,{x}^{4}}}-{\frac{A{a}^{2}}{8\,{x}^{8}}}-{\frac{B{b}^{2}}{2\,{x}^{2}}}-{\frac{a \left ( 2\,Ab+Ba \right ) }{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(B*x^2+A)/x^9,x)

[Out]

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

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Maxima [A]  time = 0.99105, size = 72, normalized size = 1.5 \begin{align*} -\frac{12 \, B b^{2} x^{6} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^9,x, algorithm="maxima")

[Out]

-1/24*(12*B*b^2*x^6 + 6*(2*B*a*b + A*b^2)*x^4 + 3*A*a^2 + 4*(B*a^2 + 2*A*a*b)*x^2)/x^8

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Fricas [A]  time = 1.40432, size = 119, normalized size = 2.48 \begin{align*} -\frac{12 \, B b^{2} x^{6} + 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 3 \, A a^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^9,x, algorithm="fricas")

[Out]

-1/24*(12*B*b^2*x^6 + 6*(2*B*a*b + A*b^2)*x^4 + 3*A*a^2 + 4*(B*a^2 + 2*A*a*b)*x^2)/x^8

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Sympy [A]  time = 1.67552, size = 56, normalized size = 1.17 \begin{align*} - \frac{3 A a^{2} + 12 B b^{2} x^{6} + x^{4} \left (6 A b^{2} + 12 B a b\right ) + x^{2} \left (8 A a b + 4 B a^{2}\right )}{24 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(B*x**2+A)/x**9,x)

[Out]

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

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Giac [A]  time = 1.26124, size = 74, normalized size = 1.54 \begin{align*} -\frac{12 \, B b^{2} x^{6} + 12 \, B a b x^{4} + 6 \, A b^{2} x^{4} + 4 \, B a^{2} x^{2} + 8 \, A a b x^{2} + 3 \, A a^{2}}{24 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(B*x^2+A)/x^9,x, algorithm="giac")

[Out]

-1/24*(12*B*b^2*x^6 + 12*B*a*b*x^4 + 6*A*b^2*x^4 + 4*B*a^2*x^2 + 8*A*a*b*x^2 + 3*A*a^2)/x^8