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

Optimal. Leaf size=80 \[ -\frac{(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}+\frac{c (b c-a d) \log \left (a+b x^2\right )}{a^3}-\frac{2 c \log (x) (b c-a d)}{a^3}-\frac{c^2}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0814483, antiderivative size = 80, 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} \[ -\frac{(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}+\frac{c (b c-a d) \log \left (a+b x^2\right )}{a^3}-\frac{2 c \log (x) (b c-a d)}{a^3}-\frac{c^2}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0972196, size = 72, normalized size = 0.9 \[ -\frac{\frac{a (b c-a d)^2}{b \left (a+b x^2\right )}-2 c (b c-a d) \log \left (a+b x^2\right )+4 c \log (x) (b c-a d)+\frac{a c^2}{x^2}}{2 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

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

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Maxima [A]  time = 1.17934, size = 135, normalized size = 1.69 \begin{align*} -\frac{a b c^{2} +{\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{2 \,{\left (a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )}} + \frac{{\left (b c^{2} - a c d\right )} \log \left (b x^{2} + a\right )}{a^{3}} - \frac{{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.52795, size = 316, normalized size = 3.95 \begin{align*} -\frac{a^{2} b c^{2} +{\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 2 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} +{\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} +{\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.61149, size = 92, normalized size = 1.15 \begin{align*} - \frac{a b c^{2} + x^{2} \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{4}} + \frac{2 c \left (a d - b c\right ) \log{\left (x \right )}}{a^{3}} - \frac{c \left (a d - b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12221, size = 147, normalized size = 1.84 \begin{align*} -\frac{{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{3}} + \frac{{\left (b^{2} c^{2} - a b c d\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{3} b} - \frac{2 \, b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + a^{2} d^{2} x^{2} + a b c^{2}}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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