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

Optimal. Leaf size=154 \[ \frac{(A b-2 a C) \log \left (a+b x^2\right )}{2 b^3}-\frac{x^2 (A b-2 a C)}{2 a b^2}-\frac{x^3 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}-\frac{\sqrt{a} (3 b B-5 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{x (3 b B-5 a D)}{2 b^3}+\frac{D x^3}{3 b^2} \]

[Out]

((3*b*B - 5*a*D)*x)/(2*b^3) - ((A*b - 2*a*C)*x^2)/(2*a*b^2) + (D*x^3)/(3*b^2) -
(x^3*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(2*a*b*(a + b*x^2)) - (Sqrt[a]*(3*b*B -
5*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2)) + ((A*b - 2*a*C)*Log[a + b*x^2])
/(2*b^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.50113, antiderivative size = 154, 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 \[ \frac{(A b-2 a C) \log \left (a+b x^2\right )}{2 b^3}-\frac{x^2 (A b-2 a C)}{2 a b^2}-\frac{x^3 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}-\frac{\sqrt{a} (3 b B-5 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{x (3 b B-5 a D)}{2 b^3}+\frac{D x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((3*b*B - 5*a*D)*x)/(2*b^3) - ((A*b - 2*a*C)*x^2)/(2*a*b^2) + (D*x^3)/(3*b^2) -
(x^3*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(2*a*b*(a + b*x^2)) - (Sqrt[a]*(3*b*B -
5*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2)) + ((A*b - 2*a*C)*Log[a + b*x^2])
/(2*b^3)

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{C \int x\, dx}{b^{2}} + \frac{D x^{3}}{3 b^{2}} - \frac{\sqrt{a} \left (3 B b - 5 D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{7}{2}}} + \frac{x \left (B b - 2 D a\right )}{b^{3}} + \frac{x \left (a \left (B b - D a\right ) - b x \left (A b - C a\right )\right )}{2 b^{3} \left (a + b x^{2}\right )} + \frac{\left (A b - 2 C a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

C*Integral(x, x)/b**2 + D*x**3/(3*b**2) - sqrt(a)*(3*B*b - 5*D*a)*atan(sqrt(b)*x
/sqrt(a))/(2*b**(7/2)) + x*(B*b - 2*D*a)/b**3 + x*(a*(B*b - D*a) - b*x*(A*b - C*
a))/(2*b**3*(a + b*x**2)) + (A*b - 2*C*a)*log(a + b*x**2)/(2*b**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.143791, size = 128, normalized size = 0.83 \[ \frac{a (-a (C+D x)+A b+b B x)}{2 b^3 \left (a+b x^2\right )}+\frac{(A b-2 a C) \log \left (a+b x^2\right )}{2 b^3}+\frac{\sqrt{a} (5 a D-3 b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{x (b B-2 a D)}{b^3}+\frac{C x^2}{2 b^2}+\frac{D x^3}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

((b*B - 2*a*D)*x)/b^3 + (C*x^2)/(2*b^2) + (D*x^3)/(3*b^2) + (a*(A*b + b*B*x - a*
(C + D*x)))/(2*b^3*(a + b*x^2)) + (Sqrt[a]*(-3*b*B + 5*a*D)*ArcTan[(Sqrt[b]*x)/S
qrt[a]])/(2*b^(7/2)) + ((A*b - 2*a*C)*Log[a + b*x^2])/(2*b^3)

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 177, normalized size = 1.2 \[{\frac{D{x}^{3}}{3\,{b}^{2}}}+{\frac{C{x}^{2}}{2\,{b}^{2}}}+{\frac{Bx}{{b}^{2}}}-2\,{\frac{Dxa}{{b}^{3}}}+{\frac{Bxa}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{Dx{a}^{2}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{aA}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}C}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) A}{2\,{b}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) aC}{{b}^{3}}}-{\frac{3\,Ba}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{a}^{2}D}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*D*x^3/b^2+1/2/b^2*C*x^2+B*x/b^2-2/b^3*D*x*a+1/2/b^2/(b*x^2+a)*B*x*a-1/2/b^3/
(b*x^2+a)*D*x*a^2+1/2/b^2*a/(b*x^2+a)*A-1/2/b^3/(b*x^2+a)*a^2*C+1/2/b^2*ln(b*x^2
+a)*A-1/b^3*ln(b*x^2+a)*a*C-3/2/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*a*B+5/2/
b^3/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*a^2*D

_______________________________________________________________________________________

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^3/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.238777, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, D b^{2} x^{5} + 6 \, C b^{2} x^{4} + 6 \, C a b x^{2} - 4 \,{\left (5 \, D a b - 3 \, B b^{2}\right )} x^{3} - 6 \, C a^{2} + 6 \, A a b - 3 \,{\left (5 \, D a^{2} - 3 \, B a b +{\left (5 \, D a b - 3 \, B b^{2}\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 6 \,{\left (5 \, D a^{2} - 3 \, B a b\right )} x - 6 \,{\left (2 \, C a^{2} - A a b +{\left (2 \, C a b - A b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{4} x^{2} + a b^{3}\right )}}, \frac{2 \, D b^{2} x^{5} + 3 \, C b^{2} x^{4} + 3 \, C a b x^{2} - 2 \,{\left (5 \, D a b - 3 \, B b^{2}\right )} x^{3} - 3 \, C a^{2} + 3 \, A a b + 3 \,{\left (5 \, D a^{2} - 3 \, B a b +{\left (5 \, D a b - 3 \, B b^{2}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 3 \,{\left (5 \, D a^{2} - 3 \, B a b\right )} x - 3 \,{\left (2 \, C a^{2} - A a b +{\left (2 \, C a b - A b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{6 \,{\left (b^{4} x^{2} + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/12*(4*D*b^2*x^5 + 6*C*b^2*x^4 + 6*C*a*b*x^2 - 4*(5*D*a*b - 3*B*b^2)*x^3 - 6*C
*a^2 + 6*A*a*b - 3*(5*D*a^2 - 3*B*a*b + (5*D*a*b - 3*B*b^2)*x^2)*sqrt(-a/b)*log(
(b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a)) - 6*(5*D*a^2 - 3*B*a*b)*x - 6*(2*C*a
^2 - A*a*b + (2*C*a*b - A*b^2)*x^2)*log(b*x^2 + a))/(b^4*x^2 + a*b^3), 1/6*(2*D*
b^2*x^5 + 3*C*b^2*x^4 + 3*C*a*b*x^2 - 2*(5*D*a*b - 3*B*b^2)*x^3 - 3*C*a^2 + 3*A*
a*b + 3*(5*D*a^2 - 3*B*a*b + (5*D*a*b - 3*B*b^2)*x^2)*sqrt(a/b)*arctan(x/sqrt(a/
b)) - 3*(5*D*a^2 - 3*B*a*b)*x - 3*(2*C*a^2 - A*a*b + (2*C*a*b - A*b^2)*x^2)*log(
b*x^2 + a))/(b^4*x^2 + a*b^3)]

_______________________________________________________________________________________

Sympy [A]  time = 6.90362, size = 287, normalized size = 1.86 \[ \frac{C x^{2}}{2 b^{2}} + \frac{D x^{3}}{3 b^{2}} + \left (- \frac{- A b + 2 C a}{2 b^{3}} - \frac{\sqrt{- a b^{7}} \left (- 3 B b + 5 D a\right )}{4 b^{7}}\right ) \log{\left (x + \frac{- 2 A b + 4 C a + 4 b^{3} \left (- \frac{- A b + 2 C a}{2 b^{3}} - \frac{\sqrt{- a b^{7}} \left (- 3 B b + 5 D a\right )}{4 b^{7}}\right )}{- 3 B b + 5 D a} \right )} + \left (- \frac{- A b + 2 C a}{2 b^{3}} + \frac{\sqrt{- a b^{7}} \left (- 3 B b + 5 D a\right )}{4 b^{7}}\right ) \log{\left (x + \frac{- 2 A b + 4 C a + 4 b^{3} \left (- \frac{- A b + 2 C a}{2 b^{3}} + \frac{\sqrt{- a b^{7}} \left (- 3 B b + 5 D a\right )}{4 b^{7}}\right )}{- 3 B b + 5 D a} \right )} - \frac{- A a b + C a^{2} + x \left (- B a b + D a^{2}\right )}{2 a b^{3} + 2 b^{4} x^{2}} - \frac{x \left (- B b + 2 D a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

C*x**2/(2*b**2) + D*x**3/(3*b**2) + (-(-A*b + 2*C*a)/(2*b**3) - sqrt(-a*b**7)*(-
3*B*b + 5*D*a)/(4*b**7))*log(x + (-2*A*b + 4*C*a + 4*b**3*(-(-A*b + 2*C*a)/(2*b*
*3) - sqrt(-a*b**7)*(-3*B*b + 5*D*a)/(4*b**7)))/(-3*B*b + 5*D*a)) + (-(-A*b + 2*
C*a)/(2*b**3) + sqrt(-a*b**7)*(-3*B*b + 5*D*a)/(4*b**7))*log(x + (-2*A*b + 4*C*a
+ 4*b**3*(-(-A*b + 2*C*a)/(2*b**3) + sqrt(-a*b**7)*(-3*B*b + 5*D*a)/(4*b**7)))/
(-3*B*b + 5*D*a)) - (-A*a*b + C*a**2 + x*(-B*a*b + D*a**2))/(2*a*b**3 + 2*b**4*x
**2) - x*(-B*b + 2*D*a)/b**3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223106, size = 177, normalized size = 1.15 \[ -\frac{{\left (2 \, C a - A b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{3}} + \frac{{\left (5 \, D a^{2} - 3 \, B a b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} - \frac{C a^{2} - A a b +{\left (D a^{2} - B a b\right )} x}{2 \,{\left (b x^{2} + a\right )} b^{3}} + \frac{2 \, D b^{4} x^{3} + 3 \, C b^{4} x^{2} - 12 \, D a b^{3} x + 6 \, B b^{4} x}{6 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*C*a - A*b)*ln(b*x^2 + a)/b^3 + 1/2*(5*D*a^2 - 3*B*a*b)*arctan(b*x/sqrt(a
*b))/(sqrt(a*b)*b^3) - 1/2*(C*a^2 - A*a*b + (D*a^2 - B*a*b)*x)/((b*x^2 + a)*b^3)
+ 1/6*(2*D*b^4*x^3 + 3*C*b^4*x^2 - 12*D*a*b^3*x + 6*B*b^4*x)/b^6