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

Optimal. Leaf size=43 \[ a^2 c \log (x)+a b c x^2+\frac{d \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 c x^4 \]

[Out]

a*b*c*x^2 + (b^2*c*x^4)/4 + (d*(a + b*x^2)^3)/(6*b) + a^2*c*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0312346, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {446, 80, 43} \[ a^2 c \log (x)+a b c x^2+\frac{d \left (a+b x^2\right )^3}{6 b}+\frac{1}{4} b^2 c x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2))/x,x]

[Out]

a*b*c*x^2 + (b^2*c*x^4)/4 + (d*(a + b*x^2)^3)/(6*b) + a^2*c*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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a+b x^2\right )^2 \left (c+d x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2 (c+d x)}{x} \, dx,x,x^2\right )\\ &=\frac{d \left (a+b x^2\right )^3}{6 b}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{(a+b x)^2}{x} \, dx,x,x^2\right )\\ &=\frac{d \left (a+b x^2\right )^3}{6 b}+\frac{1}{2} c \operatorname{Subst}\left (\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx,x,x^2\right )\\ &=a b c x^2+\frac{1}{4} b^2 c x^4+\frac{d \left (a+b x^2\right )^3}{6 b}+a^2 c \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0144786, size = 51, normalized size = 1.19 \[ a^2 c \log (x)+\frac{1}{4} b x^4 (2 a d+b c)+\frac{1}{2} a x^2 (a d+2 b c)+\frac{1}{6} b^2 d x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(2*b*c + a*d)*x^2)/2 + (b*(b*c + 2*a*d)*x^4)/4 + (b^2*d*x^6)/6 + a^2*c*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 51, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}d{x}^{6}}{6}}+{\frac{{x}^{4}abd}{2}}+{\frac{{b}^{2}c{x}^{4}}{4}}+{\frac{{x}^{2}{a}^{2}d}{2}}+abc{x}^{2}+{a}^{2}c\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.00555, size = 70, normalized size = 1.63 \begin{align*} \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + \frac{1}{2} \, a^{2} c \log \left (x^{2}\right ) + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} 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,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.2487, size = 116, normalized size = 2.7 \begin{align*} \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + a^{2} c \log \left (x\right ) + \frac{1}{2} \,{\left (2 \, a b c + a^{2} d\right )} 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,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.282249, size = 49, normalized size = 1.14 \begin{align*} a^{2} c \log{\left (x \right )} + \frac{b^{2} d x^{6}}{6} + x^{4} \left (\frac{a b d}{2} + \frac{b^{2} c}{4}\right ) + x^{2} \left (\frac{a^{2} d}{2} + a b c\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,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19921, size = 72, normalized size = 1.67 \begin{align*} \frac{1}{6} \, b^{2} d x^{6} + \frac{1}{4} \, b^{2} c x^{4} + \frac{1}{2} \, a b d x^{4} + a b c x^{2} + \frac{1}{2} \, a^{2} d x^{2} + \frac{1}{2} \, a^{2} c \log \left (x^{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,x, algorithm="giac")

[Out]

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