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

Optimal. Leaf size=119 \[ \frac{(3 a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}-\frac{2 (a C+A b)-x (b B-5 a D)}{8 a b^2 \left (a+b x^2\right )}-\frac{x \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2} \]

[Out]

-(x*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(4*a*b*(a + b*x^2)^2) - (2*(A*b + a*C) -
(b*B - 5*a*D)*x)/(8*a*b^2*(a + b*x^2)) + ((b*B + 3*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[
a]])/(8*a^(3/2)*b^(5/2))

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Rubi [A]  time = 0.249769, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(3 a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}-\frac{2 (a C+A b)-x (b B-5 a D)}{8 a b^2 \left (a+b x^2\right )}-\frac{x \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(4*a*b*(a + b*x^2)^2) - (2*(A*b + a*C) -
(b*B - 5*a*D)*x)/(8*a*b^2*(a + b*x^2)) + ((b*B + 3*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[
a]])/(8*a^(3/2)*b^(5/2))

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Rubi in Sympy [A]  time = 31.224, size = 88, normalized size = 0.74 \[ - \frac{A + B x + C x^{2} + D x^{3}}{4 b \left (a + b x^{2}\right )^{2}} - \frac{2 C a - x \left (B b - 3 D a\right )}{8 a b^{2} \left (a + b x^{2}\right )} + \frac{\left (B b + 3 D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 a^{\frac{3}{2}} b^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(A + B*x + C*x**2 + D*x**3)/(4*b*(a + b*x**2)**2) - (2*C*a - x*(B*b - 3*D*a))/(
8*a*b**2*(a + b*x**2)) + (B*b + 3*D*a)*atan(sqrt(b)*x/sqrt(a))/(8*a**(3/2)*b**(5
/2))

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Mathematica [A]  time = 0.204017, size = 99, normalized size = 0.83 \[ \frac{\frac{(3 a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\sqrt{b} \left (-a^2 (2 C+3 D x)-a b \left (2 A+x \left (B+4 C x+5 D x^2\right )\right )+b^2 B x^3\right )}{a \left (a+b x^2\right )^2}}{8 b^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((Sqrt[b]*(b^2*B*x^3 - a^2*(2*C + 3*D*x) - a*b*(2*A + x*(B + 4*C*x + 5*D*x^2))))
/(a*(a + b*x^2)^2) + ((b*B + 3*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/a^(3/2))/(8*b^(
5/2))

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Maple [A]  time = 0.013, size = 110, normalized size = 0.9 \[{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( Bb-5\,aD \right ){x}^{3}}{8\,ab}}-{\frac{C{x}^{2}}{2\,b}}-{\frac{ \left ( Bb+3\,aD \right ) x}{8\,{b}^{2}}}-{\frac{Ab+aC}{4\,{b}^{2}}} \right ) }+{\frac{B}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,D}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/8*(B*b-5*D*a)/a/b*x^3-1/2*C*x^2/b-1/8*(B*b+3*D*a)/b^2*x-1/4*(A*b+C*a)/b^2)/(b
*x^2+a)^2+1/8/b/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*B+3/8/b^2/(a*b)^(1/2)*arct
an(x*b/(a*b)^(1/2))*D

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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/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238685, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, D a b^{2} + B b^{3}\right )} x^{4} + 3 \, D a^{3} + B a^{2} b + 2 \,{\left (3 \, D a^{2} b + B a b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (4 \, C a b x^{2} +{\left (5 \, D a b - B b^{2}\right )} x^{3} + 2 \, C a^{2} + 2 \, A a b +{\left (3 \, D a^{2} + B a b\right )} x\right )} \sqrt{-a b}}{16 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \sqrt{-a b}}, \frac{{\left ({\left (3 \, D a b^{2} + B b^{3}\right )} x^{4} + 3 \, D a^{3} + B a^{2} b + 2 \,{\left (3 \, D a^{2} b + B a b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (4 \, C a b x^{2} +{\left (5 \, D a b - B b^{2}\right )} x^{3} + 2 \, C a^{2} + 2 \, A a b +{\left (3 \, D a^{2} + B a b\right )} x\right )} \sqrt{a b}}{8 \,{\left (a b^{4} x^{4} + 2 \, a^{2} b^{3} x^{2} + a^{3} b^{2}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(((3*D*a*b^2 + B*b^3)*x^4 + 3*D*a^3 + B*a^2*b + 2*(3*D*a^2*b + B*a*b^2)*x^
2)*log((2*a*b*x + (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) - 2*(4*C*a*b*x^2 + (5*D*a
*b - B*b^2)*x^3 + 2*C*a^2 + 2*A*a*b + (3*D*a^2 + B*a*b)*x)*sqrt(-a*b))/((a*b^4*x
^4 + 2*a^2*b^3*x^2 + a^3*b^2)*sqrt(-a*b)), 1/8*(((3*D*a*b^2 + B*b^3)*x^4 + 3*D*a
^3 + B*a^2*b + 2*(3*D*a^2*b + B*a*b^2)*x^2)*arctan(sqrt(a*b)*x/a) - (4*C*a*b*x^2
+ (5*D*a*b - B*b^2)*x^3 + 2*C*a^2 + 2*A*a*b + (3*D*a^2 + B*a*b)*x)*sqrt(a*b))/(
(a*b^4*x^4 + 2*a^2*b^3*x^2 + a^3*b^2)*sqrt(a*b))]

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Sympy [A]  time = 25.8248, size = 177, normalized size = 1.49 \[ - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (B b + 3 D a\right ) \log{\left (- a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (B b + 3 D a\right ) \log{\left (a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{16} - \frac{2 A a b + 2 C a^{2} + 4 C a b x^{2} + x^{3} \left (- B b^{2} + 5 D a b\right ) + x \left (B a b + 3 D a^{2}\right )}{8 a^{3} b^{2} + 16 a^{2} b^{3} x^{2} + 8 a b^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**3*b**5))*(B*b + 3*D*a)*log(-a**2*b**2*sqrt(-1/(a**3*b**5)) + x)/16
+ sqrt(-1/(a**3*b**5))*(B*b + 3*D*a)*log(a**2*b**2*sqrt(-1/(a**3*b**5)) + x)/16
- (2*A*a*b + 2*C*a**2 + 4*C*a*b*x**2 + x**3*(-B*b**2 + 5*D*a*b) + x*(B*a*b + 3*D
*a**2))/(8*a**3*b**2 + 16*a**2*b**3*x**2 + 8*a*b**4*x**4)

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GIAC/XCAS [A]  time = 0.223047, size = 131, normalized size = 1.1 \[ \frac{{\left (3 \, D a + B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{2}} - \frac{5 \, D a b x^{3} - B b^{2} x^{3} + 4 \, C a b x^{2} + 3 \, D a^{2} x + B a b x + 2 \, C a^{2} + 2 \, A a b}{8 \,{\left (b x^{2} + a\right )}^{2} a b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*D*a + B*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^2) - 1/8*(5*D*a*b*x^3 - B
*b^2*x^3 + 4*C*a*b*x^2 + 3*D*a^2*x + B*a*b*x + 2*C*a^2 + 2*A*a*b)/((b*x^2 + a)^2
*a*b^2)