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

Optimal. Leaf size=61 \[ -\frac{(b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac{d (b c-a d) \log \left (a+b x^2\right )}{b^3}+\frac{d^2 x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.147973, antiderivative size = 61, 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 \[ -\frac{(b c-a d)^2}{2 b^3 \left (a+b x^2\right )}+\frac{d (b c-a d) \log \left (a+b x^2\right )}{b^3}+\frac{d^2 x^2}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d^{2} \int ^{x^{2}} \frac{1}{b^{2}}\, dx}{2} - \frac{d \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{b^{3}} - \frac{\left (a d - b c\right )^{2}}{2 b^{3} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0759352, size = 56, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{a+b x^2}+2 d (b c-a d) \log \left (a+b x^2\right )+b d^2 x^2}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 97, normalized size = 1.6 \[{\frac{{d}^{2}{x}^{2}}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ){d}^{2}a}{{b}^{3}}}+{\frac{\ln \left ( b{x}^{2}+a \right ) dc}{{b}^{2}}}-{\frac{{a}^{2}{d}^{2}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{acd}{{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{c}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.3282, size = 99, normalized size = 1.62 \[ \frac{d^{2} x^{2}}{2 \, b^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} + \frac{{\left (b c d - a d^{2}\right )} \log \left (b x^{2} + a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.219055, size = 136, normalized size = 2.23 \[ \frac{b^{2} d^{2} x^{4} + a b d^{2} x^{2} - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} + 2 \,{\left (a b c d - a^{2} d^{2} +{\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.60889, size = 68, normalized size = 1.11 \[ - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 a b^{3} + 2 b^{4} x^{2}} + \frac{d^{2} x^{2}}{2 b^{2}} - \frac{d \left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.24152, size = 150, normalized size = 2.46 \[ \frac{{\left (b x^{2} + a\right )} d^{2}}{2 \, b^{3}} - \frac{{\left (b c d - a d^{2}\right )}{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} - \frac{\frac{b^{3} c^{2}}{b x^{2} + a} - \frac{2 \, a b^{2} c d}{b x^{2} + a} + \frac{a^{2} b d^{2}}{b x^{2} + a}}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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