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

Optimal. Leaf size=67 \[ -\frac{1}{2} \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \log \left (c+d x^2\right )+\frac{a^2 \log (x)}{c^2}+\frac{(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0635326, antiderivative size = 67, 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} \[ -\frac{1}{2} \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \log \left (c+d x^2\right )+\frac{a^2 \log (x)}{c^2}+\frac{(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0438614, size = 70, normalized size = 1.04 \[ \frac{2 a^2 \log (x)+\frac{(b c-a d) \left (\left (c+d x^2\right ) (a d+b c) \log \left (c+d x^2\right )+c (b c-a d)\right )}{d^2 \left (c+d x^2\right )}}{2 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 94, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,{d}^{2}}}+{\frac{{a}^{2}}{2\,c \left ( d{x}^{2}+c \right ) }}-{\frac{ab}{d \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}c}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{2}\ln \left ( x \right ) }{{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01021, size = 116, normalized size = 1.73 \begin{align*} \frac{a^{2} \log \left (x^{2}\right )}{2 \, c^{2}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \,{\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4596, size = 228, normalized size = 3.4 \begin{align*} \frac{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{3} - a^{2} c d^{2} +{\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \,{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \log \left (x\right )}{2 \,{\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.33387, size = 80, normalized size = 1.19 \begin{align*} \frac{a^{2} \log{\left (x \right )}}{c^{2}} + \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 c^{2} d^{2} + 2 c d^{3} x^{2}} - \frac{\left (a d - b c\right ) \left (a d + b c\right ) \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c^{2} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12126, size = 134, normalized size = 2. \begin{align*} \frac{a^{2} \log \left (x^{2}\right )}{2 \, c^{2}} + \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{2} d^{2}} - \frac{b^{2} c^{2} x^{2} - a^{2} d^{2} x^{2} + 2 \, a b c^{2} - 2 \, a^{2} c d}{2 \,{\left (d x^{2} + c\right )} c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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