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

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

[Out]

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

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Rubi [A]  time = 0.0646807, 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} \[ -\frac{1}{2} \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \log \left (a+b x^2\right )+\frac{c^2 \log (x)}{a^2}+\frac{(b c-a d)^2}{2 a b^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0428798, size = 70, normalized size = 1.04 \[ \frac{\frac{(a d-b c) \left (\left (a+b x^2\right ) (a d+b c) \log \left (a+b x^2\right )+a (a d-b c)\right )}{b^2 \left (a+b x^2\right )}+2 c^2 \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 94, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}\ln \left ( x \right ) }{{a}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){d}^{2}}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,{a}^{2}}}+{\frac{a{d}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{cd}{b \left ( b{x}^{2}+a \right ) }}+{\frac{{c}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04862, size = 116, normalized size = 1.73 \begin{align*} \frac{c^{2} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} - \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.55434, size = 228, normalized size = 3.4 \begin{align*} \frac{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} -{\left (a b^{2} c^{2} - a^{3} d^{2} +{\left (b^{3} c^{2} - a^{2} b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (b^{3} c^{2} x^{2} + a b^{2} c^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.37925, size = 80, normalized size = 1.19 \begin{align*} \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac{c^{2} \log{\left (x \right )}}{a^{2}} + \frac{\left (a d - b c\right ) \left (a d + b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15932, size = 134, normalized size = 2. \begin{align*} \frac{c^{2} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{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 (b x^{2} + a\right )} a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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