3.18 \(\int \frac{(A+B x^2) (b x^2+c x^4)^2}{x^6} \, dx\)

Optimal. Leaf size=48 \[ -\frac{A b^2}{x}+\frac{1}{3} c x^3 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{5} B c^2 x^5 \]

[Out]

-((A*b^2)/x) + b*(b*B + 2*A*c)*x + (c*(2*b*B + A*c)*x^3)/3 + (B*c^2*x^5)/5

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Rubi [A]  time = 0.037229, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1584, 448} \[ -\frac{A b^2}{x}+\frac{1}{3} c x^3 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{5} B c^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*b^2)/x) + b*(b*B + 2*A*c)*x + (c*(2*b*B + A*c)*x^3)/3 + (B*c^2*x^5)/5

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 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

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

Mathematica [A]  time = 0.0174977, size = 48, normalized size = 1. \[ -\frac{A b^2}{x}+\frac{1}{3} c x^3 (A c+2 b B)+b x (2 A c+b B)+\frac{1}{5} B c^2 x^5 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((A*b^2)/x) + b*(b*B + 2*A*c)*x + (c*(2*b*B + A*c)*x^3)/3 + (B*c^2*x^5)/5

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Maple [A]  time = 0.003, size = 49, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{5}}{5}}+{\frac{A{x}^{3}{c}^{2}}{3}}+{\frac{2\,B{x}^{3}bc}{3}}+2\,Abcx+B{b}^{2}x-{\frac{A{b}^{2}}{x}} \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^6,x)

[Out]

1/5*B*c^2*x^5+1/3*A*x^3*c^2+2/3*B*x^3*b*c+2*A*b*c*x+B*b^2*x-A*b^2/x

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Maxima [A]  time = 1.14033, size = 65, normalized size = 1.35 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{1}{3} \,{\left (2 \, B b c + A c^{2}\right )} x^{3} - \frac{A b^{2}}{x} +{\left (B b^{2} + 2 \, A b c\right )} x \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^6,x, algorithm="maxima")

[Out]

1/5*B*c^2*x^5 + 1/3*(2*B*b*c + A*c^2)*x^3 - A*b^2/x + (B*b^2 + 2*A*b*c)*x

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Fricas [A]  time = 0.442007, size = 116, normalized size = 2.42 \begin{align*} \frac{3 \, B c^{2} x^{6} + 5 \,{\left (2 \, B b c + A c^{2}\right )} x^{4} - 15 \, A b^{2} + 15 \,{\left (B b^{2} + 2 \, A b c\right )} x^{2}}{15 \, x} \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^6,x, algorithm="fricas")

[Out]

1/15*(3*B*c^2*x^6 + 5*(2*B*b*c + A*c^2)*x^4 - 15*A*b^2 + 15*(B*b^2 + 2*A*b*c)*x^2)/x

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Sympy [A]  time = 0.286375, size = 48, normalized size = 1. \begin{align*} - \frac{A b^{2}}{x} + \frac{B c^{2} x^{5}}{5} + x^{3} \left (\frac{A c^{2}}{3} + \frac{2 B b c}{3}\right ) + x \left (2 A b c + B b^{2}\right ) \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**6,x)

[Out]

-A*b**2/x + B*c**2*x**5/5 + x**3*(A*c**2/3 + 2*B*b*c/3) + x*(2*A*b*c + B*b**2)

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Giac [A]  time = 1.23109, size = 65, normalized size = 1.35 \begin{align*} \frac{1}{5} \, B c^{2} x^{5} + \frac{2}{3} \, B b c x^{3} + \frac{1}{3} \, A c^{2} x^{3} + B b^{2} x + 2 \, A b c x - \frac{A b^{2}}{x} \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^6,x, algorithm="giac")

[Out]

1/5*B*c^2*x^5 + 2/3*B*b*c*x^3 + 1/3*A*c^2*x^3 + B*b^2*x + 2*A*b*c*x - A*b^2/x