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

Optimal. Leaf size=84 \[ \frac{1}{2} x^2 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{2 x^2}+\frac{1}{2} b d x^4 (a d+b c)+2 a c \log (x) (a d+b c)+\frac{1}{6} b^2 d^2 x^6 \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.078103, antiderivative size = 84, 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 = {446, 88} \[ \frac{1}{2} x^2 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{a^2 c^2}{2 x^2}+\frac{1}{2} b d x^4 (a d+b c)+2 a c \log (x) (a d+b c)+\frac{1}{6} b^2 d^2 x^6 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*c^2)/(2*x^2) + ((b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^2)/2 + (b*d*(b*c + a*d)*x^4)/2 + (b^2*d^2*x^6)/6 + 2*a
*c*(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 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.0413864, size = 83, normalized size = 0.99 \[ \frac{1}{6} \left (\frac{3 a^2 \left (d^2 x^4-c^2\right )}{x^2}+3 a b d x^2 \left (4 c+d x^2\right )+12 a c \log (x) (a d+b c)+b^2 x^2 \left (3 c^2+3 c d x^2+d^2 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*b*d*x^2*(4*c + d*x^2) + (3*a^2*(-c^2 + d^2*x^4))/x^2 + b^2*x^2*(3*c^2 + 3*c*d*x^2 + d^2*x^4) + 12*a*c*(b*
c + a*d)*Log[x])/6

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 93, normalized size = 1.1 \begin{align*}{\frac{{b}^{2}{d}^{2}{x}^{6}}{6}}+{\frac{{x}^{4}ab{d}^{2}}{2}}+{\frac{{x}^{4}{b}^{2}cd}{2}}+{\frac{{x}^{2}{a}^{2}{d}^{2}}{2}}+2\,{x}^{2}abcd+{\frac{{x}^{2}{b}^{2}{c}^{2}}{2}}+2\,\ln \left ( x \right ){a}^{2}cd+2\,\ln \left ( x \right ) ab{c}^{2}-{\frac{{a}^{2}{c}^{2}}{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)^2/x^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.995944, size = 115, normalized size = 1.37 \begin{align*} \frac{1}{6} \, b^{2} d^{2} x^{6} + \frac{1}{2} \,{\left (b^{2} c d + a b d^{2}\right )} x^{4} + \frac{1}{2} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{2} - \frac{a^{2} c^{2}}{2 \, x^{2}} +{\left (a b c^{2} + a^{2} c d\right )} \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)^2/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.26927, size = 188, normalized size = 2.24 \begin{align*} \frac{b^{2} d^{2} x^{8} + 3 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 3 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 3 \, a^{2} c^{2} + 12 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2} \log \left (x\right )}{6 \, 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)^2/x^3,x, algorithm="fricas")

[Out]

1/6*(b^2*d^2*x^8 + 3*(b^2*c*d + a*b*d^2)*x^6 + 3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 3*a^2*c^2 + 12*(a*b*c^2
+ a^2*c*d)*x^2*log(x))/x^2

________________________________________________________________________________________

Sympy [A]  time = 0.412152, size = 87, normalized size = 1.04 \begin{align*} - \frac{a^{2} c^{2}}{2 x^{2}} + 2 a c \left (a d + b c\right ) \log{\left (x \right )} + \frac{b^{2} d^{2} x^{6}}{6} + x^{4} \left (\frac{a b d^{2}}{2} + \frac{b^{2} c d}{2}\right ) + x^{2} \left (\frac{a^{2} d^{2}}{2} + 2 a b c d + \frac{b^{2} c^{2}}{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)**2/x**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.19398, size = 154, normalized size = 1.83 \begin{align*} \frac{1}{6} \, b^{2} d^{2} x^{6} + \frac{1}{2} \, b^{2} c d x^{4} + \frac{1}{2} \, a b d^{2} x^{4} + \frac{1}{2} \, b^{2} c^{2} x^{2} + 2 \, a b c d x^{2} + \frac{1}{2} \, a^{2} d^{2} x^{2} +{\left (a b c^{2} + a^{2} c d\right )} \log \left (x^{2}\right ) - \frac{2 \, a b c^{2} x^{2} + 2 \, a^{2} c d x^{2} + a^{2} c^{2}}{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)^2/x^3,x, algorithm="giac")

[Out]

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