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

Optimal. Leaf size=67 \[ -\frac{1}{2} \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \log \left (a+b x^2\right )+\frac{c^2 \log (x)}{a^2}+\frac{(b c-a d)^2}{2 a b^2 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.159148, antiderivative size = 67, 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 \[ -\frac{1}{2} \left (\frac{c^2}{a^2}-\frac{d^2}{b^2}\right ) \log \left (a+b x^2\right )+\frac{c^2 \log (x)}{a^2}+\frac{(b c-a d)^2}{2 a b^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.6669, size = 60, normalized size = 0.9 \[ - \left (- \frac{d^{2}}{2 b^{2}} + \frac{c^{2}}{2 a^{2}}\right ) \log{\left (a + b x^{2} \right )} + \frac{\left (a d - b c\right )^{2}}{2 a b^{2} \left (a + b x^{2}\right )} + \frac{c^{2} \log{\left (x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**2/x/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0711076, size = 70, normalized size = 1.04 \[ \frac{\frac{(a d-b c) \left (\left (a+b x^2\right ) (a d+b c) \log \left (a+b x^2\right )+a (a d-b c)\right )}{b^2 \left (a+b x^2\right )}+2 c^2 \log (x)}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 94, normalized size = 1.4 \[{\frac{{c}^{2}\ln \left ( x \right ) }{{a}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){d}^{2}}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,{a}^{2}}}+{\frac{a{d}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{cd}{b \left ( b{x}^{2}+a \right ) }}+{\frac{{c}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33789, size = 116, normalized size = 1.73 \[ \frac{c^{2} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \,{\left (a b^{3} x^{2} + a^{2} b^{2}\right )}} - \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.22979, size = 158, normalized size = 2.36 \[ \frac{a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2} -{\left (a b^{2} c^{2} - a^{3} d^{2} +{\left (b^{3} c^{2} - a^{2} b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (b^{3} c^{2} x^{2} + a b^{2} c^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{3} x^{2} + a^{3} b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.35883, size = 80, normalized size = 1.19 \[ \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 a^{2} b^{2} + 2 a b^{3} x^{2}} + \frac{c^{2} \log{\left (x \right )}}{a^{2}} + \frac{\left (a d - b c\right ) \left (a d + b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**2/x/(b*x**2+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.259983, size = 134, normalized size = 2. \[ \frac{c^{2}{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b^{2}} + \frac{b^{2} c^{2} x^{2} - a^{2} d^{2} x^{2} + 2 \, a b c^{2} - 2 \, a^{2} c d}{2 \,{\left (b x^{2} + a\right )} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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