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

Optimal. Leaf size=67 \[ \frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3} \]

[Out]

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

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Rubi [A]  time = 0.151324, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.5464, size = 56, normalized size = 0.84 \[ \frac{b^{2} \log{\left (c + d x^{2} \right )}}{2 d^{3}} - \frac{b \left (a d - b c\right )}{d^{3} \left (c + d x^{2}\right )} - \frac{\left (a d - b c\right )^{2}}{4 d^{3} \left (c + d x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0503628, size = 75, normalized size = 1.12 \[ \frac{-a^2 d^2-2 a b d \left (c+2 d x^2\right )+b^2 c \left (3 c+4 d x^2\right )+2 b^2 \left (c+d x^2\right )^2 \log \left (c+d x^2\right )}{4 d^3 \left (c+d x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 105, normalized size = 1.6 \[ -{\frac{ab}{{d}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}c}{{d}^{3} \left ( d{x}^{2}+c \right ) }}-{\frac{{a}^{2}}{4\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{abc}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}}{4\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{{b}^{2}\ln \left ( d{x}^{2}+c \right ) }{2\,{d}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33405, size = 117, normalized size = 1.75 \[ \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2}}{4 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} + \frac{b^{2} \log \left (d x^{2} + c\right )}{2 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(3*b^2*c^2 - 2*a*b*c*d - a^2*d^2 + 4*(b^2*c*d - a*b*d^2)*x^2)/(d^5*x^4 + 2*c
*d^4*x^2 + c^2*d^3) + 1/2*b^2*log(d*x^2 + c)/d^3

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Fricas [A]  time = 0.233684, size = 146, normalized size = 2.18 \[ \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} d^{2} x^{4} + 2 \, b^{2} c d x^{2} + b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(3*b^2*c^2 - 2*a*b*c*d - a^2*d^2 + 4*(b^2*c*d - a*b*d^2)*x^2 + 2*(b^2*d^2*x^
4 + 2*b^2*c*d*x^2 + b^2*c^2)*log(d*x^2 + c))/(d^5*x^4 + 2*c*d^4*x^2 + c^2*d^3)

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Sympy [A]  time = 5.96983, size = 87, normalized size = 1.3 \[ \frac{b^{2} \log{\left (c + d x^{2} \right )}}{2 d^{3}} - \frac{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2} + x^{2} \left (4 a b d^{2} - 4 b^{2} c d\right )}{4 c^{2} d^{3} + 8 c d^{4} x^{2} + 4 d^{5} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**2*log(c + d*x**2)/(2*d**3) - (a**2*d**2 + 2*a*b*c*d - 3*b**2*c**2 + x**2*(4*a
*b*d**2 - 4*b**2*c*d))/(4*c**2*d**3 + 8*c*d**4*x**2 + 4*d**5*x**4)

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GIAC/XCAS [A]  time = 0.231473, size = 103, normalized size = 1.54 \[ \frac{b^{2}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{3}} + \frac{4 \,{\left (b^{2} c - a b d\right )} x^{2} + \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}}{d}}{4 \,{\left (d x^{2} + c\right )}^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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