3.96 \(\int \frac{x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=134 \[ \frac{(A b-3 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x (A b-3 a C)}{2 a b^2}-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-2 a D) \log \left (a+b x^2\right )}{2 b^3}+\frac{D x^2}{2 b^2} \]

[Out]

-((A*b - 3*a*C)*x)/(2*a*b^2) + (D*x^2)/(2*b^2) - (x^2*(a*(B - (a*D)/b) - (A*b -
a*C)*x))/(2*a*b*(a + b*x^2)) + ((A*b - 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sq
rt[a]*b^(5/2)) + ((b*B - 2*a*D)*Log[a + b*x^2])/(2*b^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.478134, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{(A b-3 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}-\frac{x (A b-3 a C)}{2 a b^2}-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-2 a D) \log \left (a+b x^2\right )}{2 b^3}+\frac{D x^2}{2 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]

[Out]

-((A*b - 3*a*C)*x)/(2*a*b^2) + (D*x^2)/(2*b^2) - (x^2*(a*(B - (a*D)/b) - (A*b -
a*C)*x))/(2*a*b*(a + b*x^2)) + ((A*b - 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*Sq
rt[a]*b^(5/2)) + ((b*B - 2*a*D)*Log[a + b*x^2])/(2*b^3)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{C x}{b^{2}} + \frac{D \int x\, dx}{b^{2}} - \frac{x \left (A b - C a + x \left (B b - D a\right )\right )}{2 b^{2} \left (a + b x^{2}\right )} + \frac{\left (B b - 2 D a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \frac{\left (A b - 3 C a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 \sqrt{a} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)

[Out]

C*x/b**2 + D*Integral(x, x)/b**2 - x*(A*b - C*a + x*(B*b - D*a))/(2*b**2*(a + b*
x**2)) + (B*b - 2*D*a)*log(a + b*x**2)/(2*b**3) + (A*b - 3*C*a)*atan(sqrt(b)*x/s
qrt(a))/(2*sqrt(a)*b**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.147992, size = 100, normalized size = 0.75 \[ \frac{\frac{a^2 (-D)+a b (B+C x)-A b^2 x}{a+b x^2}+\frac{\sqrt{b} (A b-3 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}+(b B-2 a D) \log \left (a+b x^2\right )+2 b C x+b D x^2}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]

[Out]

(2*b*C*x + b*D*x^2 + (-(a^2*D) - A*b^2*x + a*b*(B + C*x))/(a + b*x^2) + (Sqrt[b]
*(A*b - 3*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[a] + (b*B - 2*a*D)*Log[a + b*x^
2])/(2*b^3)

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 154, normalized size = 1.2 \[{\frac{D{x}^{2}}{2\,{b}^{2}}}+{\frac{Cx}{{b}^{2}}}-{\frac{Ax}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{Cxa}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{Ba}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}D}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{B\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) aD}{{b}^{3}}}+{\frac{A}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,aC}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x)

[Out]

1/2*D*x^2/b^2+1/b^2*C*x-1/2/b/(b*x^2+a)*A*x+1/2/b^2/(b*x^2+a)*C*x*a+1/2/b^2/(b*x
^2+a)*B*a-1/2/b^3/(b*x^2+a)*a^2*D+1/2*B/b^2*ln(b*x^2+a)-1/b^3*ln(b*x^2+a)*a*D+1/
2/b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*A-3/2/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(
1/2))*a*C

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*x^2/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.239986, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (3 \, C a^{2} b - A a b^{2} +{\left (3 \, C a b^{2} - A b^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (D b^{2} x^{4} + 2 \, C b^{2} x^{3} + D a b x^{2} - D a^{2} + B a b +{\left (3 \, C a b - A b^{2}\right )} x -{\left (2 \, D a^{2} - B a b +{\left (2 \, D a b - B b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{-a b}}{4 \,{\left (b^{4} x^{2} + a b^{3}\right )} \sqrt{-a b}}, -\frac{{\left (3 \, C a^{2} b - A a b^{2} +{\left (3 \, C a b^{2} - A b^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (D b^{2} x^{4} + 2 \, C b^{2} x^{3} + D a b x^{2} - D a^{2} + B a b +{\left (3 \, C a b - A b^{2}\right )} x -{\left (2 \, D a^{2} - B a b +{\left (2 \, D a b - B b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{a b}}{2 \,{\left (b^{4} x^{2} + a b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*x^2/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[-1/4*((3*C*a^2*b - A*a*b^2 + (3*C*a*b^2 - A*b^3)*x^2)*log((2*a*b*x + (b*x^2 - a
)*sqrt(-a*b))/(b*x^2 + a)) - 2*(D*b^2*x^4 + 2*C*b^2*x^3 + D*a*b*x^2 - D*a^2 + B*
a*b + (3*C*a*b - A*b^2)*x - (2*D*a^2 - B*a*b + (2*D*a*b - B*b^2)*x^2)*log(b*x^2
+ a))*sqrt(-a*b))/((b^4*x^2 + a*b^3)*sqrt(-a*b)), -1/2*((3*C*a^2*b - A*a*b^2 + (
3*C*a*b^2 - A*b^3)*x^2)*arctan(sqrt(a*b)*x/a) - (D*b^2*x^4 + 2*C*b^2*x^3 + D*a*b
*x^2 - D*a^2 + B*a*b + (3*C*a*b - A*b^2)*x - (2*D*a^2 - B*a*b + (2*D*a*b - B*b^2
)*x^2)*log(b*x^2 + a))*sqrt(a*b))/((b^4*x^2 + a*b^3)*sqrt(a*b))]

_______________________________________________________________________________________

Sympy [A]  time = 6.5623, size = 284, normalized size = 2.12 \[ \frac{C x}{b^{2}} + \frac{D x^{2}}{2 b^{2}} + \left (- \frac{- B b + 2 D a}{2 b^{3}} - \frac{\sqrt{- a b^{7}} \left (- A b + 3 C a\right )}{4 a b^{6}}\right ) \log{\left (x + \frac{2 B a b - 4 D a^{2} - 4 a b^{3} \left (- \frac{- B b + 2 D a}{2 b^{3}} - \frac{\sqrt{- a b^{7}} \left (- A b + 3 C a\right )}{4 a b^{6}}\right )}{- A b^{2} + 3 C a b} \right )} + \left (- \frac{- B b + 2 D a}{2 b^{3}} + \frac{\sqrt{- a b^{7}} \left (- A b + 3 C a\right )}{4 a b^{6}}\right ) \log{\left (x + \frac{2 B a b - 4 D a^{2} - 4 a b^{3} \left (- \frac{- B b + 2 D a}{2 b^{3}} + \frac{\sqrt{- a b^{7}} \left (- A b + 3 C a\right )}{4 a b^{6}}\right )}{- A b^{2} + 3 C a b} \right )} + \frac{B a b - D a^{2} + x \left (- A b^{2} + C a b\right )}{2 a b^{3} + 2 b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)

[Out]

C*x/b**2 + D*x**2/(2*b**2) + (-(-B*b + 2*D*a)/(2*b**3) - sqrt(-a*b**7)*(-A*b + 3
*C*a)/(4*a*b**6))*log(x + (2*B*a*b - 4*D*a**2 - 4*a*b**3*(-(-B*b + 2*D*a)/(2*b**
3) - sqrt(-a*b**7)*(-A*b + 3*C*a)/(4*a*b**6)))/(-A*b**2 + 3*C*a*b)) + (-(-B*b +
2*D*a)/(2*b**3) + sqrt(-a*b**7)*(-A*b + 3*C*a)/(4*a*b**6))*log(x + (2*B*a*b - 4*
D*a**2 - 4*a*b**3*(-(-B*b + 2*D*a)/(2*b**3) + sqrt(-a*b**7)*(-A*b + 3*C*a)/(4*a*
b**6)))/(-A*b**2 + 3*C*a*b)) + (B*a*b - D*a**2 + x*(-A*b**2 + C*a*b))/(2*a*b**3
+ 2*b**4*x**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223688, size = 150, normalized size = 1.12 \[ -\frac{{\left (3 \, C a - A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} - \frac{{\left (2 \, D a - B b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{3}} + \frac{D b^{2} x^{2} + 2 \, C b^{2} x}{2 \, b^{4}} - \frac{D a^{2} - B a b -{\left (C a b - A b^{2}\right )} x}{2 \,{\left (b x^{2} + a\right )} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*x^2/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

-1/2*(3*C*a - A*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^2) - 1/2*(2*D*a - B*b)*ln(
b*x^2 + a)/b^3 + 1/2*(D*b^2*x^2 + 2*C*b^2*x)/b^4 - 1/2*(D*a^2 - B*a*b - (C*a*b -
A*b^2)*x)/((b*x^2 + a)*b^3)