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

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

[Out]

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

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Rubi [A]  time = 0.0482164, 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} \[ -\frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 a b^2}+\frac{c^2 \log (x)}{a}+\frac{d^2 x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x^2)^2/(x*(a + b*x^2)),x]

[Out]

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

Mathematica [A]  time = 0.0220223, size = 50, normalized size = 0.98 \[ \frac{-(b c-a d)^2 \log \left (a+b x^2\right )+a b d^2 x^2+2 b^2 c^2 \log (x)}{2 a b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03126, size = 82, normalized size = 1.61 \begin{align*} \frac{d^{2} x^{2}}{2 \, b} + \frac{c^{2} \log \left (x^{2}\right )}{2 \, a} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a 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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49935, size = 128, normalized size = 2.51 \begin{align*} \frac{a b d^{2} x^{2} + 2 \, b^{2} c^{2} \log \left (x\right ) -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a 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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.34536, size = 41, normalized size = 0.8 \begin{align*} \frac{d^{2} x^{2}}{2 b} + \frac{c^{2} \log{\left (x \right )}}{a} - \frac{\left (a d - b c\right )^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a 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),x)

[Out]

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

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Giac [A]  time = 1.1355, size = 84, normalized size = 1.65 \begin{align*} \frac{d^{2} x^{2}}{2 \, b} + \frac{c^{2} \log \left (x^{2}\right )}{2 \, a} - \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a 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),x, algorithm="giac")

[Out]

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