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

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

[Out]

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

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Rubi [A]  time = 0.149173, 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 \[ -\frac{a^2}{2 c x^2}+\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d}+\frac{a \log (x) (2 b c-a d)}{c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.9969, size = 53, normalized size = 0.91 \[ - \frac{a^{2}}{2 c x^{2}} - \frac{a \left (a d - 2 b c\right ) \log{\left (x^{2} \right )}}{2 c^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x^{2} \right )}}{2 c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0473834, size = 60, normalized size = 1.03 \[ \frac{a^2 (-c) d-2 a d x^2 \log (x) (a d-2 b c)+x^2 (b c-a d)^2 \log \left (c+d x^2\right )}{2 c^2 d x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.36137, size = 95, normalized size = 1.64 \[ \frac{{\left (2 \, a b c - a^{2} d\right )} \log \left (x^{2}\right )}{2 \, c^{2}} - \frac{a^{2}}{2 \, c x^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239451, size = 100, normalized size = 1.72 \[ -\frac{a^{2} c d -{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x^{2} \log \left (d x^{2} + c\right ) - 2 \,{\left (2 \, a b c d - a^{2} d^{2}\right )} x^{2} \log \left (x\right )}{2 \, c^{2} d x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.63598, size = 49, normalized size = 0.84 \[ - \frac{a^{2}}{2 c x^{2}} - \frac{a \left (a d - 2 b c\right ) \log{\left (x \right )}}{c^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.228465, size = 123, normalized size = 2.12 \[ \frac{{\left (2 \, a b c - a^{2} d\right )}{\rm ln}\left (x^{2}\right )}{2 \, c^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{2} d} - \frac{2 \, a b c x^{2} - a^{2} d x^{2} + a^{2} c}{2 \, c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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