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

Optimal. Leaf size=55 \[ \frac{1}{3} A b^2 x^3+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{5} b x^5 (2 A c+b B)+\frac{1}{9} B c^2 x^9 \]

[Out]

(A*b^2*x^3)/3 + (b*(b*B + 2*A*c)*x^5)/5 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^9)/9

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Rubi [A]  time = 0.038727, antiderivative size = 55, 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} \[ \frac{1}{3} A b^2 x^3+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{5} b x^5 (2 A c+b B)+\frac{1}{9} B c^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^2*x^3)/3 + (b*(b*B + 2*A*c)*x^5)/5 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^9)/9

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^2} \, dx &=\int x^2 \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2 x^2+b (b B+2 A c) x^4+c (2 b B+A c) x^6+B c^2 x^8\right ) \, dx\\ &=\frac{1}{3} A b^2 x^3+\frac{1}{5} b (b B+2 A c) x^5+\frac{1}{7} c (2 b B+A c) x^7+\frac{1}{9} B c^2 x^9\\ \end{align*}

Mathematica [A]  time = 0.0089094, size = 55, normalized size = 1. \[ \frac{1}{3} A b^2 x^3+\frac{1}{7} c x^7 (A c+2 b B)+\frac{1}{5} b x^5 (2 A c+b B)+\frac{1}{9} B c^2 x^9 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*b^2*x^3)/3 + (b*(b*B + 2*A*c)*x^5)/5 + (c*(2*b*B + A*c)*x^7)/7 + (B*c^2*x^9)/9

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Maple [A]  time = 0.001, size = 52, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{9}}{9}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{7}}{7}}+{\frac{ \left ( 2\,Abc+B{b}^{2} \right ){x}^{5}}{5}}+{\frac{A{b}^{2}{x}^{3}}{3}} \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^2,x)

[Out]

1/9*B*c^2*x^9+1/7*(A*c^2+2*B*b*c)*x^7+1/5*(2*A*b*c+B*b^2)*x^5+1/3*A*b^2*x^3

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Maxima [A]  time = 1.0644, size = 69, normalized size = 1.25 \begin{align*} \frac{1}{9} \, B c^{2} x^{9} + \frac{1}{7} \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac{1}{3} \, A b^{2} x^{3} + \frac{1}{5} \,{\left (B b^{2} + 2 \, A b c\right )} x^{5} \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^2,x, algorithm="maxima")

[Out]

1/9*B*c^2*x^9 + 1/7*(2*B*b*c + A*c^2)*x^7 + 1/3*A*b^2*x^3 + 1/5*(B*b^2 + 2*A*b*c)*x^5

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Fricas [A]  time = 0.509525, size = 117, normalized size = 2.13 \begin{align*} \frac{1}{9} \, B c^{2} x^{9} + \frac{1}{7} \,{\left (2 \, B b c + A c^{2}\right )} x^{7} + \frac{1}{3} \, A b^{2} x^{3} + \frac{1}{5} \,{\left (B b^{2} + 2 \, A b c\right )} x^{5} \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^2,x, algorithm="fricas")

[Out]

1/9*B*c^2*x^9 + 1/7*(2*B*b*c + A*c^2)*x^7 + 1/3*A*b^2*x^3 + 1/5*(B*b^2 + 2*A*b*c)*x^5

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Sympy [A]  time = 0.068828, size = 56, normalized size = 1.02 \begin{align*} \frac{A b^{2} x^{3}}{3} + \frac{B c^{2} x^{9}}{9} + x^{7} \left (\frac{A c^{2}}{7} + \frac{2 B b c}{7}\right ) + x^{5} \left (\frac{2 A b c}{5} + \frac{B b^{2}}{5}\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**2,x)

[Out]

A*b**2*x**3/3 + B*c**2*x**9/9 + x**7*(A*c**2/7 + 2*B*b*c/7) + x**5*(2*A*b*c/5 + B*b**2/5)

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Giac [A]  time = 1.26311, size = 72, normalized size = 1.31 \begin{align*} \frac{1}{9} \, B c^{2} x^{9} + \frac{2}{7} \, B b c x^{7} + \frac{1}{7} \, A c^{2} x^{7} + \frac{1}{5} \, B b^{2} x^{5} + \frac{2}{5} \, A b c x^{5} + \frac{1}{3} \, A b^{2} x^{3} \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^2,x, algorithm="giac")

[Out]

1/9*B*c^2*x^9 + 2/7*B*b*c*x^7 + 1/7*A*c^2*x^7 + 1/5*B*b^2*x^5 + 2/5*A*b*c*x^5 + 1/3*A*b^2*x^3