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

Optimal. Leaf size=86 \[ \frac{\frac{a^2}{c^2}-\frac{b^2}{d^2}}{2 \left (c+d x^2\right )}-\frac{a^2 \log \left (c+d x^2\right )}{2 c^3}+\frac{a^2 \log (x)}{c^3}+\frac{(b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0829324, antiderivative size = 86, 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} \[ \frac{\frac{a^2}{c^2}-\frac{b^2}{d^2}}{2 \left (c+d x^2\right )}-\frac{a^2 \log \left (c+d x^2\right )}{2 c^3}+\frac{a^2 \log (x)}{c^3}+\frac{(b c-a d)^2}{4 c d^2 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0469384, size = 103, normalized size = 1.2 \[ \frac{a^2 d^2-b^2 c^2}{2 c^2 d^2 \left (c+d x^2\right )}+\frac{a^2 d^2-2 a b c d+b^2 c^2}{4 c d^2 \left (c+d x^2\right )^2}-\frac{a^2 \log \left (c+d x^2\right )}{2 c^3}+\frac{a^2 \log (x)}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.011, size = 112, normalized size = 1.3 \begin{align*} -{\frac{{a}^{2}\ln \left ( d{x}^{2}+c \right ) }{2\,{c}^{3}}}+{\frac{{a}^{2}}{4\,c \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{ab}{2\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}c}{4\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{a}^{2}}{2\,{c}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{b}^{2}}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{2}\ln \left ( x \right ) }{{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00211, size = 147, normalized size = 1.71 \begin{align*} -\frac{b^{2} c^{3} + 2 \, a b c^{2} d - 3 \, a^{2} c d^{2} + 2 \,{\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}}{4 \,{\left (c^{2} d^{4} x^{4} + 2 \, c^{3} d^{3} x^{2} + c^{4} d^{2}\right )}} - \frac{a^{2} \log \left (d x^{2} + c\right )}{2 \, c^{3}} + \frac{a^{2} \log \left (x^{2}\right )}{2 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.4595, size = 324, normalized size = 3.77 \begin{align*} -\frac{b^{2} c^{4} + 2 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + 2 \,{\left (b^{2} c^{3} d - a^{2} c d^{3}\right )} x^{2} + 2 \,{\left (a^{2} d^{4} x^{4} + 2 \, a^{2} c d^{3} x^{2} + a^{2} c^{2} d^{2}\right )} \log \left (d x^{2} + c\right ) - 4 \,{\left (a^{2} d^{4} x^{4} + 2 \, a^{2} c d^{3} x^{2} + a^{2} c^{2} d^{2}\right )} \log \left (x\right )}{4 \,{\left (c^{3} d^{4} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{5} 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)^3,x, algorithm="fricas")

[Out]

-1/4*(b^2*c^4 + 2*a*b*c^3*d - 3*a^2*c^2*d^2 + 2*(b^2*c^3*d - a^2*c*d^3)*x^2 + 2*(a^2*d^4*x^4 + 2*a^2*c*d^3*x^2
+ a^2*c^2*d^2)*log(d*x^2 + c) - 4*(a^2*d^4*x^4 + 2*a^2*c*d^3*x^2 + a^2*c^2*d^2)*log(x))/(c^3*d^4*x^4 + 2*c^4*
d^3*x^2 + c^5*d^2)

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Sympy [A]  time = 1.48476, size = 107, normalized size = 1.24 \begin{align*} \frac{a^{2} \log{\left (x \right )}}{c^{3}} - \frac{a^{2} \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c^{3}} + \frac{3 a^{2} c d^{2} - 2 a b c^{2} d - b^{2} c^{3} + x^{2} \left (2 a^{2} d^{3} - 2 b^{2} c^{2} d\right )}{4 c^{4} d^{2} + 8 c^{3} d^{3} x^{2} + 4 c^{2} d^{4} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.14995, size = 149, normalized size = 1.73 \begin{align*} \frac{a^{2} \log \left (x^{2}\right )}{2 \, c^{3}} - \frac{a^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{3}} + \frac{3 \, a^{2} d^{4} x^{4} - 2 \, b^{2} c^{3} d x^{2} + 8 \, a^{2} c d^{3} x^{2} - b^{2} c^{4} - 2 \, a b c^{3} d + 6 \, a^{2} c^{2} d^{2}}{4 \,{\left (d x^{2} + c\right )}^{2} c^{3} 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)^3,x, algorithm="giac")

[Out]

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