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

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

[Out]

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

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Rubi [A]  time = 0.0220519, 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} \[ -\frac{A c+b B}{x}-\frac{A b}{3 x^3}+B c x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0111177, size = 27, normalized size = 1.04 \[ \frac{-A c-b B}{x}-\frac{A b}{3 x^3}+B c x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 25, normalized size = 1. \begin{align*} Bcx-{\frac{Ab}{3\,{x}^{3}}}-{\frac{Ac+Bb}{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^6,x)

[Out]

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

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Maxima [A]  time = 1.12318, size = 35, normalized size = 1.35 \begin{align*} B c x - \frac{3 \,{\left (B b + A c\right )} x^{2} + A b}{3 \, 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)/x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.448699, size = 63, normalized size = 2.42 \begin{align*} \frac{3 \, B c x^{4} - 3 \,{\left (B b + A c\right )} x^{2} - A b}{3 \, 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)/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.341664, size = 26, normalized size = 1. \begin{align*} B c x - \frac{A b + x^{2} \left (3 A c + 3 B b\right )}{3 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)/x**6,x)

[Out]

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

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Giac [A]  time = 1.20146, size = 38, normalized size = 1.46 \begin{align*} B c x - \frac{3 \, B b x^{2} + 3 \, A c x^{2} + A b}{3 \, 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)/x^6,x, algorithm="giac")

[Out]

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