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

Optimal. Leaf size=80 \[ \frac{c (b c-a d) \log \left (a+b x^2\right )}{a^3}-\frac{2 c \log (x) (b c-a d)}{a^3}-\frac{(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac{c^2}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.200642, antiderivative size = 80, 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 \[ \frac{c (b c-a d) \log \left (a+b x^2\right )}{a^3}-\frac{2 c \log (x) (b c-a d)}{a^3}-\frac{(b c-a d)^2}{2 a^2 b \left (a+b x^2\right )}-\frac{c^2}{2 a^2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.9634, size = 70, normalized size = 0.88 \[ - \frac{c^{2}}{2 a^{2} x^{2}} - \frac{\left (a d - b c\right )^{2}}{2 a^{2} b \left (a + b x^{2}\right )} + \frac{c \left (a d - b c\right ) \log{\left (x^{2} \right )}}{a^{3}} - \frac{c \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**2/x**3/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.173657, size = 72, normalized size = 0.9 \[ -\frac{\frac{a (b c-a d)^2}{b \left (a+b x^2\right )}-2 c (b c-a d) \log \left (a+b x^2\right )+4 c \log (x) (b c-a d)+\frac{a c^2}{x^2}}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 114, normalized size = 1.4 \[ -{\frac{{c}^{2}}{2\,{a}^{2}{x}^{2}}}+2\,{\frac{c\ln \left ( x \right ) d}{{a}^{2}}}-2\,{\frac{{c}^{2}\ln \left ( x \right ) b}{{a}^{3}}}-{\frac{c\ln \left ( b{x}^{2}+a \right ) d}{{a}^{2}}}+{\frac{{c}^{2}\ln \left ( b{x}^{2}+a \right ) b}{{a}^{3}}}-{\frac{{d}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{cd}{a \left ( b{x}^{2}+a \right ) }}-{\frac{b{c}^{2}}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33887, size = 135, normalized size = 1.69 \[ -\frac{a b c^{2} +{\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2}}{2 \,{\left (a^{2} b^{2} x^{4} + a^{3} b x^{2}\right )}} + \frac{{\left (b c^{2} - a c d\right )} \log \left (b x^{2} + a\right )}{a^{3}} - \frac{{\left (b c^{2} - a c d\right )} \log \left (x^{2}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.232108, size = 215, normalized size = 2.69 \[ -\frac{a^{2} b c^{2} +{\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x^{2} - 2 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} +{\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{4} +{\left (a b^{2} c^{2} - a^{2} b c d\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^2/((b*x^2 + a)^2*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.19215, size = 92, normalized size = 1.15 \[ - \frac{a b c^{2} + x^{2} \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{4}} + \frac{2 c \left (a d - b c\right ) \log{\left (x \right )}}{a^{3}} - \frac{c \left (a d - b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.237616, size = 147, normalized size = 1.84 \[ -\frac{{\left (b c^{2} - a c d\right )}{\rm ln}\left (x^{2}\right )}{a^{3}} + \frac{{\left (b^{2} c^{2} - a b c d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{3} b} - \frac{2 \, b^{2} c^{2} x^{2} - 2 \, a b c d x^{2} + a^{2} d^{2} x^{2} + a b c^{2}}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^2/((b*x^2 + a)^2*x^3),x, algorithm="giac")

[Out]

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