3.148 \(\int \frac{(a+b x^2)^2 (c+d x^2)}{x^3} \, dx\)

Optimal. Leaf size=51 \[ -\frac{a^2 c}{2 x^2}+\frac{1}{2} b x^2 (2 a d+b c)+a \log (x) (a d+2 b c)+\frac{1}{4} b^2 d x^4 \]

[Out]

-(a^2*c)/(2*x^2) + (b*(b*c + 2*a*d)*x^2)/2 + (b^2*d*x^4)/4 + a*(2*b*c + a*d)*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0421537, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {446, 76} \[ -\frac{a^2 c}{2 x^2}+\frac{1}{2} b x^2 (2 a d+b c)+a \log (x) (a d+2 b c)+\frac{1}{4} b^2 d x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2))/x^3,x]

[Out]

-(a^2*c)/(2*x^2) + (b*(b*c + 2*a*d)*x^2)/2 + (b^2*d*x^4)/4 + a*(2*b*c + a*d)*Log[x]

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

Mathematica [A]  time = 0.0224635, size = 49, normalized size = 0.96 \[ \frac{1}{4} \left (-\frac{2 a^2 c}{x^2}+2 b x^2 (2 a d+b c)+4 a \log (x) (a d+2 b c)+b^2 d x^4\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2))/x^3,x]

[Out]

((-2*a^2*c)/x^2 + 2*b*(b*c + 2*a*d)*x^2 + b^2*d*x^4 + 4*a*(2*b*c + a*d)*Log[x])/4

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 50, normalized size = 1. \begin{align*}{\frac{{b}^{2}d{x}^{4}}{4}}+{x}^{2}abd+{\frac{{b}^{2}c{x}^{2}}{2}}+\ln \left ( x \right ){a}^{2}d+2\,\ln \left ( x \right ) abc-{\frac{{a}^{2}c}{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)/x^3,x)

[Out]

1/4*b^2*d*x^4+x^2*a*b*d+1/2*b^2*c*x^2+ln(x)*a^2*d+2*ln(x)*a*b*c-1/2*a^2*c/x^2

________________________________________________________________________________________

Maxima [A]  time = 0.981489, size = 70, normalized size = 1.37 \begin{align*} \frac{1}{4} \, b^{2} d x^{4} + \frac{1}{2} \,{\left (b^{2} c + 2 \, a b d\right )} x^{2} + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} \log \left (x^{2}\right ) - \frac{a^{2} c}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)/x^3,x, algorithm="maxima")

[Out]

1/4*b^2*d*x^4 + 1/2*(b^2*c + 2*a*b*d)*x^2 + 1/2*(2*a*b*c + a^2*d)*log(x^2) - 1/2*a^2*c/x^2

________________________________________________________________________________________

Fricas [A]  time = 1.2384, size = 122, normalized size = 2.39 \begin{align*} \frac{b^{2} d x^{6} + 2 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + 4 \,{\left (2 \, a b c + a^{2} d\right )} x^{2} \log \left (x\right ) - 2 \, a^{2} c}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)/x^3,x, algorithm="fricas")

[Out]

1/4*(b^2*d*x^6 + 2*(b^2*c + 2*a*b*d)*x^4 + 4*(2*a*b*c + a^2*d)*x^2*log(x) - 2*a^2*c)/x^2

________________________________________________________________________________________

Sympy [A]  time = 0.368969, size = 48, normalized size = 0.94 \begin{align*} - \frac{a^{2} c}{2 x^{2}} + a \left (a d + 2 b c\right ) \log{\left (x \right )} + \frac{b^{2} d x^{4}}{4} + x^{2} \left (a b d + \frac{b^{2} c}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)/x**3,x)

[Out]

-a**2*c/(2*x**2) + a*(a*d + 2*b*c)*log(x) + b**2*d*x**4/4 + x**2*(a*b*d + b**2*c/2)

________________________________________________________________________________________

Giac [A]  time = 1.16662, size = 95, normalized size = 1.86 \begin{align*} \frac{1}{4} \, b^{2} d x^{4} + \frac{1}{2} \, b^{2} c x^{2} + a b d x^{2} + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} \log \left (x^{2}\right ) - \frac{2 \, a b c x^{2} + a^{2} d x^{2} + a^{2} c}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2*(d*x^2+c)/x^3,x, algorithm="giac")

[Out]

1/4*b^2*d*x^4 + 1/2*b^2*c*x^2 + a*b*d*x^2 + 1/2*(2*a*b*c + a^2*d)*log(x^2) - 1/2*(2*a*b*c*x^2 + a^2*d*x^2 + a^
2*c)/x^2