3.8 \(\int \frac{(A+B x^2) (b x^2+c x^4)}{x^5} \, dx\)

Optimal. Leaf size=29 \[ \log (x) (A c+b B)-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2 \]

[Out]

-(A*b)/(2*x^2) + (B*c*x^2)/2 + (b*B + A*c)*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0275993, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1584, 446, 76} \[ \log (x) (A c+b B)-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2 \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x^2)*(b*x^2 + c*x^4))/x^5,x]

[Out]

-(A*b)/(2*x^2) + (B*c*x^2)/2 + (b*B + A*c)*Log[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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

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

Mathematica [A]  time = 0.0108132, size = 29, normalized size = 1. \[ \log (x) (A c+b B)-\frac{A b}{2 x^2}+\frac{1}{2} B c x^2 \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x^2)*(b*x^2 + c*x^4))/x^5,x]

[Out]

-(A*b)/(2*x^2) + (B*c*x^2)/2 + (b*B + A*c)*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 26, normalized size = 0.9 \begin{align*}{\frac{Bc{x}^{2}}{2}}+A\ln \left ( x \right ) c+B\ln \left ( x \right ) b-{\frac{Ab}{2\,{x}^{2}}} \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^5,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.16114, size = 38, normalized size = 1.31 \begin{align*} \frac{1}{2} \, B c x^{2} + \frac{1}{2} \,{\left (B b + A c\right )} \log \left (x^{2}\right ) - \frac{A b}{2 \, x^{2}} \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^5,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.52777, size = 70, normalized size = 2.41 \begin{align*} \frac{B c x^{4} + 2 \,{\left (B b + A c\right )} x^{2} \log \left (x\right ) - A b}{2 \, x^{2}} \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^5,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.326274, size = 26, normalized size = 0.9 \begin{align*} - \frac{A b}{2 x^{2}} + \frac{B c x^{2}}{2} + \left (A c + B b\right ) \log{\left (x \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**5,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17334, size = 57, normalized size = 1.97 \begin{align*} \frac{1}{2} \, B c x^{2} + \frac{1}{2} \,{\left (B b + A c\right )} \log \left (x^{2}\right ) - \frac{B b x^{2} + A c x^{2} + A b}{2 \, x^{2}} \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^5,x, algorithm="giac")

[Out]

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