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

Optimal. Leaf size=51 \[ -\frac{a^2 A}{6 x^6}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{2 x^2}+b^2 B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0367519, antiderivative size = 51, 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, 76} \[ -\frac{a^2 A}{6 x^6}-\frac{a (a B+2 A b)}{4 x^4}-\frac{b (2 a B+A b)}{2 x^2}+b^2 B \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0251778, size = 54, normalized size = 1.06 \[ b^2 B \log (x)-\frac{a^2 \left (2 A+3 B x^2\right )+6 a b x^2 \left (A+2 B x^2\right )+6 A b^2 x^4}{12 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 52, normalized size = 1. \begin{align*}{b}^{2}B\ln \left ( x \right ) -{\frac{abA}{2\,{x}^{4}}}-{\frac{{a}^{2}B}{4\,{x}^{4}}}-{\frac{{b}^{2}A}{2\,{x}^{2}}}-{\frac{abB}{{x}^{2}}}-{\frac{A{a}^{2}}{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^7,x)

[Out]

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

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Maxima [A]  time = 0.975843, size = 74, normalized size = 1.45 \begin{align*} \frac{1}{2} \, B b^{2} \log \left (x^{2}\right ) - \frac{6 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 2 \, A a^{2} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.44511, size = 127, normalized size = 2.49 \begin{align*} \frac{12 \, B b^{2} x^{6} \log \left (x\right ) - 6 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 2 \, A a^{2} - 3 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.11371, size = 53, normalized size = 1.04 \begin{align*} B b^{2} \log{\left (x \right )} - \frac{2 A a^{2} + x^{4} \left (6 A b^{2} + 12 B a b\right ) + x^{2} \left (6 A a b + 3 B a^{2}\right )}{12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17604, size = 89, normalized size = 1.75 \begin{align*} \frac{1}{2} \, B b^{2} \log \left (x^{2}\right ) - \frac{11 \, B b^{2} x^{6} + 12 \, B a b x^{4} + 6 \, A b^{2} x^{4} + 3 \, B a^{2} x^{2} + 6 \, A a b x^{2} + 2 \, A a^{2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*B*b^2*log(x^2) - 1/12*(11*B*b^2*x^6 + 12*B*a*b*x^4 + 6*A*b^2*x^4 + 3*B*a^2*x^2 + 6*A*a*b*x^2 + 2*A*a^2)/x^
6