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

Optimal. Leaf size=51 \[ \frac{a^2 \log (x)}{c}-\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c d^2}+\frac{b^2 x^2}{2 d} \]

[Out]

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

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Rubi [A]  time = 0.0463171, antiderivative size = 51, 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, 72} \[ \frac{a^2 \log (x)}{c}-\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c d^2}+\frac{b^2 x^2}{2 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/(x*(c + d*x^2)),x]

[Out]

(b^2*x^2)/(2*d) + (a^2*Log[x])/c - ((b*c - a*d)^2*Log[c + d*x^2])/(2*c*d^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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 0.0215287, size = 50, normalized size = 0.98 \[ \frac{2 a^2 d^2 \log (x)-(b c-a d)^2 \log \left (c+d x^2\right )+b^2 c d x^2}{2 c d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 69, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2\,d}}-{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,c}}+{\frac{\ln \left ( d{x}^{2}+c \right ) ab}{d}}-{\frac{c\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,{d}^{2}}}+{\frac{{a}^{2}\ln \left ( x \right ) }{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.996459, size = 82, normalized size = 1.61 \begin{align*} \frac{b^{2} x^{2}}{2 \, d} + \frac{a^{2} \log \left (x^{2}\right )}{2 \, c} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.33602, size = 128, normalized size = 2.51 \begin{align*} \frac{b^{2} c d x^{2} + 2 \, a^{2} d^{2} \log \left (x\right ) -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.29811, size = 41, normalized size = 0.8 \begin{align*} \frac{a^{2} \log{\left (x \right )}}{c} + \frac{b^{2} x^{2}}{2 d} - \frac{\left (a d - b c\right )^{2} \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c 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),x)

[Out]

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

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Giac [A]  time = 1.15437, size = 84, normalized size = 1.65 \begin{align*} \frac{b^{2} x^{2}}{2 \, d} + \frac{a^{2} \log \left (x^{2}\right )}{2 \, c} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, c 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),x, algorithm="giac")

[Out]

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