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

Optimal. Leaf size=50 \[ A b^2 x+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

[Out]

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

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Rubi [A]  time = 0.0322901, antiderivative size = 50, 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, 373} \[ A b^2 x+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, 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^4} \, dx &=\int \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\int \left (A b^2+b (b B+2 A c) x^2+c (2 b B+A c) x^4+B c^2 x^6\right ) \, dx\\ &=A b^2 x+\frac{1}{3} b (b B+2 A c) x^3+\frac{1}{5} c (2 b B+A c) x^5+\frac{1}{7} B c^2 x^7\\ \end{align*}

Mathematica [A]  time = 0.0084935, size = 50, normalized size = 1. \[ A b^2 x+\frac{1}{5} c x^5 (A c+2 b B)+\frac{1}{3} b x^3 (2 A c+b B)+\frac{1}{7} B c^2 x^7 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0., size = 49, normalized size = 1. \begin{align*}{\frac{B{c}^{2}{x}^{7}}{7}}+{\frac{ \left ( A{c}^{2}+2\,Bbc \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,Abc+B{b}^{2} \right ){x}^{3}}{3}}+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^4,x)

[Out]

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

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Maxima [A]  time = 1.1253, size = 65, normalized size = 1.3 \begin{align*} \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} 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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.480369, size = 109, normalized size = 2.18 \begin{align*} \frac{1}{7} \, B c^{2} x^{7} + \frac{1}{5} \,{\left (2 \, B b c + A c^{2}\right )} x^{5} + A b^{2} x + \frac{1}{3} \,{\left (B b^{2} + 2 \, A b c\right )} 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^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.068293, size = 53, normalized size = 1.06 \begin{align*} A b^{2} x + \frac{B c^{2} x^{7}}{7} + x^{5} \left (\frac{A c^{2}}{5} + \frac{2 B b c}{5}\right ) + x^{3} \left (\frac{2 A b c}{3} + \frac{B b^{2}}{3}\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**4,x)

[Out]

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

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Giac [A]  time = 1.3346, size = 68, normalized size = 1.36 \begin{align*} \frac{1}{7} \, B c^{2} x^{7} + \frac{2}{5} \, B b c x^{5} + \frac{1}{5} \, A c^{2} x^{5} + \frac{1}{3} \, B b^{2} x^{3} + \frac{2}{3} \, A b c x^{3} + 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^4,x, algorithm="giac")

[Out]

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