3.23 \(\int \frac{(A+B x^2) (b x^2+c x^4)^2}{x^{11}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.044925, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1584, 446, 76} \[ -\frac{A b^2}{6 x^6}-\frac{b (2 A c+b B)}{4 x^4}-\frac{c (A c+2 b B)}{2 x^2}+B c^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^11,x]

[Out]

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

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

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 (b x^2+c x^4\right )^2}{x^{11}} \, dx &=\int \frac{\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x^7} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(A+B x) (b+c x)^2}{x^4} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{A b^2}{x^4}+\frac{b (b B+2 A c)}{x^3}+\frac{c (2 b B+A c)}{x^2}+\frac{B c^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{A b^2}{6 x^6}-\frac{b (b B+2 A c)}{4 x^4}-\frac{c (2 b B+A c)}{2 x^2}+B c^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0261184, size = 53, normalized size = 1.04 \[ B c^2 \log (x)-\frac{2 A \left (b^2+3 b c x^2+3 c^2 x^4\right )+3 b B x^2 \left (b+4 c x^2\right )}{12 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4)^2)/x^11,x]

[Out]

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

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Maple [A]  time = 0.005, size = 52, normalized size = 1. \begin{align*} B{c}^{2}\ln \left ( x \right ) -{\frac{Abc}{2\,{x}^{4}}}-{\frac{B{b}^{2}}{4\,{x}^{4}}}-{\frac{A{c}^{2}}{2\,{x}^{2}}}-{\frac{Bbc}{{x}^{2}}}-{\frac{A{b}^{2}}{6\,{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)*(c*x^4+b*x^2)^2/x^11,x)

[Out]

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

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Maxima [A]  time = 1.12125, size = 74, normalized size = 1.45 \begin{align*} \frac{1}{2} \, B c^{2} \log \left (x^{2}\right ) - \frac{6 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} + 2 \, A b^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^2/x^11,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.470435, size = 127, normalized size = 2.49 \begin{align*} \frac{12 \, B c^{2} x^{6} \log \left (x\right ) - 6 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} - 2 \, A b^{2} - 3 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^2/x^11,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.16495, size = 53, normalized size = 1.04 \begin{align*} B c^{2} \log{\left (x \right )} - \frac{2 A b^{2} + x^{4} \left (6 A c^{2} + 12 B b c\right ) + x^{2} \left (6 A b c + 3 B b^{2}\right )}{12 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x**2+A)*(c*x**4+b*x**2)**2/x**11,x)

[Out]

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

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Giac [A]  time = 1.25292, size = 89, normalized size = 1.75 \begin{align*} \frac{1}{2} \, B c^{2} \log \left (x^{2}\right ) - \frac{11 \, B c^{2} x^{6} + 12 \, B b c x^{4} + 6 \, A c^{2} x^{4} + 3 \, B b^{2} x^{2} + 6 \, A b c x^{2} + 2 \, A b^{2}}{12 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)*(c*x^4+b*x^2)^2/x^11,x, algorithm="giac")

[Out]

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