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

Optimal. Leaf size=58 \[ \frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b}-\frac{c \log (x) (b c-2 a d)}{a^2}-\frac{c^2}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.148687, antiderivative size = 58, 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 \[ \frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b}-\frac{c \log (x) (b c-2 a d)}{a^2}-\frac{c^2}{2 a x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.9183, size = 53, normalized size = 0.91 \[ - \frac{c^{2}}{2 a x^{2}} + \frac{c \left (2 a d - b c\right ) \log{\left (x^{2} \right )}}{2 a^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.046068, size = 60, normalized size = 1.03 \[ \frac{-a b c^2-2 b c x^2 \log (x) (b c-2 a d)+x^2 (b c-a d)^2 \log \left (a+b x^2\right )}{2 a^2 b x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 81, normalized size = 1.4 \[ -{\frac{{c}^{2}}{2\,a{x}^{2}}}+2\,{\frac{c\ln \left ( x \right ) d}{a}}-{\frac{{c}^{2}\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){d}^{2}}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ) cd}{a}}+{\frac{b\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35515, size = 93, normalized size = 1.6 \[ -\frac{{\left (b c^{2} - 2 \, a c d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{c^{2}}{2 \, a x^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.237571, size = 99, normalized size = 1.71 \[ -\frac{a b c^{2} -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (b x^{2} + a\right ) + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d\right )} x^{2} \log \left (x\right )}{2 \, a^{2} b x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.81095, size = 49, normalized size = 0.84 \[ - \frac{c^{2}}{2 a x^{2}} + \frac{c \left (2 a d - b c\right ) \log{\left (x \right )}}{a^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226419, size = 122, normalized size = 2.1 \[ -\frac{{\left (b c^{2} - 2 \, a c d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac{b c^{2} x^{2} - 2 \, a c d x^{2} - a c^{2}}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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