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

Optimal. Leaf size=26 \[ x (A c+b B)-\frac{A b}{x}+\frac{1}{3} B c x^3 \]

[Out]

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

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Rubi [A]  time = 0.0234336, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1584, 448} \[ x (A c+b B)-\frac{A b}{x}+\frac{1}{3} B c x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0090501, size = 26, normalized size = 1. \[ x (A c+b B)-\frac{A b}{x}+\frac{1}{3} B c x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 24, normalized size = 0.9 \begin{align*}{\frac{Bc{x}^{3}}{3}}+Acx+Bbx-{\frac{Ab}{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)/x^4,x)

[Out]

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

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Maxima [A]  time = 1.41248, size = 32, normalized size = 1.23 \begin{align*} \frac{1}{3} \, B c x^{3} +{\left (B b + A c\right )} x - \frac{A b}{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)/x^4,x, algorithm="maxima")

[Out]

1/3*B*c*x^3 + (B*b + A*c)*x - A*b/x

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Fricas [A]  time = 0.477661, size = 61, normalized size = 2.35 \begin{align*} \frac{B c x^{4} + 3 \,{\left (B b + A c\right )} x^{2} - 3 \, A b}{3 \, 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)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.252091, size = 20, normalized size = 0.77 \begin{align*} - \frac{A b}{x} + \frac{B c x^{3}}{3} + x \left (A c + B b\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)/x**4,x)

[Out]

-A*b/x + B*c*x**3/3 + x*(A*c + B*b)

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Giac [A]  time = 1.20154, size = 31, normalized size = 1.19 \begin{align*} \frac{1}{3} \, B c x^{3} + B b x + A c x - \frac{A b}{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)/x^4,x, algorithm="giac")

[Out]

1/3*B*c*x^3 + B*b*x + A*c*x - A*b/x