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]

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

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Rubi [A]  time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, 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 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]

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]]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.47, size = 59, normalized size = 1.16 \[ \frac {a b d^{2} x^{2} + 2 \, b^{2} c^{2} \log \relax (x) - {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}} \]

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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giac [A]  time = 0.33, size = 62, normalized size = 1.22 \[ \frac {d^{2} x^{2}}{2 \, b} + \frac {c^{2} \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a b^{2}} \]

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)

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maple [A]  time = 0.00, size = 69, normalized size = 1.35 \[ \frac {d^{2} x^{2}}{2 b}-\frac {a \,d^{2} \ln \left (b \,x^{2}+a \right )}{2 b^{2}}+\frac {c^{2} \ln \relax (x )}{a}-\frac {c^{2} \ln \left (b \,x^{2}+a \right )}{2 a}+\frac {c d \ln \left (b \,x^{2}+a \right )}{b} \]

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

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maxima [A]  time = 1.11, size = 61, normalized size = 1.20 \[ \frac {d^{2} x^{2}}{2 \, b} + \frac {c^{2} \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a b^{2}} \]

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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mupad [B]  time = 0.17, size = 58, normalized size = 1.14 \[ \frac {d^2\,x^2}{2\,b}+\frac {c^2\,\ln \relax (x)}{a}-\frac {\ln \left (b\,x^2+a\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{2\,a\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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sympy [A]  time = 1.23, size = 41, normalized size = 0.80 \[ \frac {d^{2} x^{2}}{2 b} + \frac {c^{2} \log {\relax (x )}}{a} - \frac {\left (a d - b c\right )^{2} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a b^{2}} \]

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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