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

Optimal. Leaf size=60 \[ \frac{a (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac{(b c-2 a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2}{2 b^2} \]

[Out]

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

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Rubi [A]  time = 0.150472, antiderivative size = 60, 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 \[ \frac{a (b c-a d)}{2 b^3 \left (a+b x^2\right )}+\frac{(b c-2 a d) \log \left (a+b x^2\right )}{2 b^3}+\frac{d x^2}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0582615, size = 50, normalized size = 0.83 \[ \frac{\frac{a (b c-a d)}{a+b x^2}+(b c-2 a d) \log \left (a+b x^2\right )+b d x^2}{2 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 74, normalized size = 1.2 \[{\frac{d{x}^{2}}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) ad}{{b}^{3}}}+{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}-{\frac{{a}^{2}d}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{ac}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.33961, size = 80, normalized size = 1.33 \[ \frac{d x^{2}}{2 \, b^{2}} + \frac{a b c - a^{2} d}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} + \frac{{\left (b c - 2 \, a d\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216687, size = 105, normalized size = 1.75 \[ \frac{b^{2} d x^{4} + a b d x^{2} + a b c - a^{2} d +{\left (a b c - 2 \, a^{2} d +{\left (b^{2} c - 2 \, a b d\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.76607, size = 56, normalized size = 0.93 \[ - \frac{a^{2} d - a b c}{2 a b^{3} + 2 b^{4} x^{2}} + \frac{d x^{2}}{2 b^{2}} - \frac{\left (2 a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23864, size = 122, normalized size = 2.03 \[ \frac{\frac{{\left (b x^{2} + a\right )} d}{b^{2}} - \frac{{\left (b c - 2 \, a d\right )}{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b^{2}} + \frac{\frac{a b^{2} c}{b x^{2} + a} - \frac{a^{2} b d}{b x^{2} + a}}{b^{3}}}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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