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

Optimal. Leaf size=81 \[ -\frac{a^2}{2 c^2 x^2}-\frac{(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )}-\frac{a (b c-a d) \log \left (c+d x^2\right )}{c^3}+\frac{2 a \log (x) (b c-a d)}{c^3} \]

[Out]

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

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Rubi [A]  time = 0.0789378, antiderivative size = 81, 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} \[ -\frac{a^2}{2 c^2 x^2}-\frac{(b c-a d)^2}{2 c^2 d \left (c+d x^2\right )}-\frac{a (b c-a d) \log \left (c+d x^2\right )}{c^3}+\frac{2 a \log (x) (b c-a d)}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0927358, size = 72, normalized size = 0.89 \[ -\frac{\frac{a^2 c}{x^2}+\frac{c (b c-a d)^2}{d \left (c+d x^2\right )}-2 a (a d-b c) \log \left (c+d x^2\right )+4 a \log (x) (a d-b c)}{2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.014, size = 114, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}\ln \left ( d{x}^{2}+c \right ) d}{{c}^{3}}}-{\frac{a\ln \left ( d{x}^{2}+c \right ) b}{{c}^{2}}}-{\frac{{a}^{2}d}{2\,{c}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{ab}{c \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}}{2\,d \left ( d{x}^{2}+c \right ) }}-{\frac{{a}^{2}}{2\,{c}^{2}{x}^{2}}}-2\,{\frac{\ln \left ( x \right ){a}^{2}d}{{c}^{3}}}+2\,{\frac{a\ln \left ( x \right ) b}{{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*a^2*ln(d*x^2+c)*d-1/c^2*a*ln(d*x^2+c)*b-1/2/c^2/(d*x^2+c)*a^2*d+1/c/(d*x^2+c)*a*b-1/2/d/(d*x^2+c)*b^2-1/
2*a^2/c^2/x^2-2*a^2/c^3*ln(x)*d+2*a/c^2*ln(x)*b

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Maxima [A]  time = 0.981822, size = 135, normalized size = 1.67 \begin{align*} -\frac{a^{2} c d +{\left (b^{2} c^{2} - 2 \, a b c d + 2 \, a^{2} d^{2}\right )} x^{2}}{2 \,{\left (c^{2} d^{2} x^{4} + c^{3} d x^{2}\right )}} - \frac{{\left (a b c - a^{2} d\right )} \log \left (d x^{2} + c\right )}{c^{3}} + \frac{{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.46689, size = 316, normalized size = 3.9 \begin{align*} -\frac{a^{2} c^{2} d +{\left (b^{2} c^{3} - 2 \, a b c^{2} d + 2 \, a^{2} c d^{2}\right )} x^{2} + 2 \,{\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} +{\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \,{\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{4} +{\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (c^{3} d^{2} x^{4} + c^{4} d x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.52865, size = 92, normalized size = 1.14 \begin{align*} - \frac{2 a \left (a d - b c\right ) \log{\left (x \right )}}{c^{3}} + \frac{a \left (a d - b c\right ) \log{\left (\frac{c}{d} + x^{2} \right )}}{c^{3}} - \frac{a^{2} c d + x^{2} \left (2 a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{2 c^{3} d x^{2} + 2 c^{2} d^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a*(a*d - b*c)*log(x)/c**3 + a*(a*d - b*c)*log(c/d + x**2)/c**3 - (a**2*c*d + x**2*(2*a**2*d**2 - 2*a*b*c*d
+ b**2*c**2))/(2*c**3*d*x**2 + 2*c**2*d**2*x**4)

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Giac [A]  time = 1.17575, size = 147, normalized size = 1.81 \begin{align*} \frac{{\left (a b c - a^{2} d\right )} \log \left (x^{2}\right )}{c^{3}} - \frac{{\left (a b c d - a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{c^{3} d} - \frac{b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + 2 \, a^{2} d^{2} x^{2} + a^{2} c d}{2 \,{\left (d x^{4} + c x^{2}\right )} c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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