3.94 \(\int \frac{x^4 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=176 \[ -\frac{\sqrt{a} (3 A b-5 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{x (3 A b-5 a C)}{2 b^3}-\frac{x^3 (3 A b-5 a C)}{6 a b^2}-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}-\frac{a (2 b B-3 a D) \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2 (2 b B-3 a D)}{2 b^3}+\frac{D x^4}{4 b^2} \]

[Out]

((3*A*b - 5*a*C)*x)/(2*b^3) + ((2*b*B - 3*a*D)*x^2)/(2*b^3) - ((3*A*b - 5*a*C)*x
^3)/(6*a*b^2) + (D*x^4)/(4*b^2) - (x^4*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(2*a*b
*(a + b*x^2)) - (Sqrt[a]*(3*A*b - 5*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2)
) - (a*(2*b*B - 3*a*D)*Log[a + b*x^2])/(2*b^4)

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Rubi [A]  time = 0.552634, antiderivative size = 176, 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 \[ -\frac{\sqrt{a} (3 A b-5 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 b^{7/2}}+\frac{x (3 A b-5 a C)}{2 b^3}-\frac{x^3 (3 A b-5 a C)}{6 a b^2}-\frac{x^4 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}-\frac{a (2 b B-3 a D) \log \left (a+b x^2\right )}{2 b^4}+\frac{x^2 (2 b B-3 a D)}{2 b^3}+\frac{D x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]

[Out]

((3*A*b - 5*a*C)*x)/(2*b^3) + ((2*b*B - 3*a*D)*x^2)/(2*b^3) - ((3*A*b - 5*a*C)*x
^3)/(6*a*b^2) + (D*x^4)/(4*b^2) - (x^4*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(2*a*b
*(a + b*x^2)) - (Sqrt[a]*(3*A*b - 5*a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(7/2)
) - (a*(2*b*B - 3*a*D)*Log[a + b*x^2])/(2*b^4)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{C x^{3}}{3 b^{2}} + \frac{D x^{4}}{4 b^{2}} - \frac{\sqrt{a} \left (3 A b - 5 C a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 b^{\frac{7}{2}}} + \frac{a x \left (A b - C a + x \left (B b - D a\right )\right )}{2 b^{3} \left (a + b x^{2}\right )} - \frac{a \left (2 B b - 3 D a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{4}} + \frac{x \left (A b - 2 C a\right )}{b^{3}} + \frac{\left (B b - 2 D a\right ) \int x\, dx}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.240112, size = 139, normalized size = 0.79 \[ \frac{\frac{6 a \left (a^2 D-a b (B+C x)+A b^2 x\right )}{a+b x^2}+12 b x (A b-2 a C)+6 \sqrt{a} \sqrt{b} (5 a C-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+6 b x^2 (b B-2 a D)+6 a (3 a D-2 b B) \log \left (a+b x^2\right )+4 b^2 C x^3+3 b^2 D x^4}{12 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(A + B*x + C*x^2 + D*x^3))/(a + b*x^2)^2,x]

[Out]

(12*b*(A*b - 2*a*C)*x + 6*b*(b*B - 2*a*D)*x^2 + 4*b^2*C*x^3 + 3*b^2*D*x^4 + (6*a
*(a^2*D + A*b^2*x - a*b*(B + C*x)))/(a + b*x^2) + 6*Sqrt[a]*Sqrt[b]*(-3*A*b + 5*
a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] + 6*a*(-2*b*B + 3*a*D)*Log[a + b*x^2])/(12*b^4)

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Maple [A]  time = 0.013, size = 201, normalized size = 1.1 \[{\frac{D{x}^{4}}{4\,{b}^{2}}}+{\frac{C{x}^{3}}{3\,{b}^{2}}}+{\frac{B{x}^{2}}{2\,{b}^{2}}}-{\frac{D{x}^{2}a}{{b}^{3}}}+{\frac{Ax}{{b}^{2}}}-2\,{\frac{Cxa}{{b}^{3}}}+{\frac{aAx}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{Cx{a}^{2}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}B}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{3}D}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{\ln \left ( b{x}^{2}+a \right ) Ba}{{b}^{3}}}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) D}{2\,{b}^{4}}}-{\frac{3\,Aa}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{a}^{2}C}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(D*x^3+C*x^2+B*x+A)/(b*x^2+a)^2,x)

[Out]

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

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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^4/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233795, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, D b^{3} x^{6} + 4 \, C b^{3} x^{5} - 3 \,{\left (3 \, D a b^{2} - 2 \, B b^{3}\right )} x^{4} + 6 \, D a^{3} - 6 \, B a^{2} b - 4 \,{\left (5 \, C a b^{2} - 3 \, A b^{3}\right )} x^{3} - 6 \,{\left (2 \, D a^{2} b - B a b^{2}\right )} x^{2} - 3 \,{\left (5 \, C a^{2} b - 3 \, A a b^{2} +{\left (5 \, C a b^{2} - 3 \, A b^{3}\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 6 \,{\left (5 \, C a^{2} b - 3 \, A a b^{2}\right )} x + 6 \,{\left (3 \, D a^{3} - 2 \, B a^{2} b +{\left (3 \, D a^{2} b - 2 \, B a b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{5} x^{2} + a b^{4}\right )}}, \frac{3 \, D b^{3} x^{6} + 4 \, C b^{3} x^{5} - 3 \,{\left (3 \, D a b^{2} - 2 \, B b^{3}\right )} x^{4} + 6 \, D a^{3} - 6 \, B a^{2} b - 4 \,{\left (5 \, C a b^{2} - 3 \, A b^{3}\right )} x^{3} - 6 \,{\left (2 \, D a^{2} b - B a b^{2}\right )} x^{2} + 6 \,{\left (5 \, C a^{2} b - 3 \, A a b^{2} +{\left (5 \, C a b^{2} - 3 \, A b^{3}\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 6 \,{\left (5 \, C a^{2} b - 3 \, A a b^{2}\right )} x + 6 \,{\left (3 \, D a^{3} - 2 \, B a^{2} b +{\left (3 \, D a^{2} b - 2 \, B a b^{2}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{12 \,{\left (b^{5} x^{2} + a b^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*x^4/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.29329, size = 333, normalized size = 1.89 \[ \frac{C x^{3}}{3 b^{2}} + \frac{D x^{4}}{4 b^{2}} + \left (\frac{a \left (- 2 B b + 3 D a\right )}{2 b^{4}} - \frac{\sqrt{- a b^{9}} \left (- 3 A b + 5 C a\right )}{4 b^{8}}\right ) \log{\left (x + \frac{4 B a b - 6 D a^{2} + 4 b^{4} \left (\frac{a \left (- 2 B b + 3 D a\right )}{2 b^{4}} - \frac{\sqrt{- a b^{9}} \left (- 3 A b + 5 C a\right )}{4 b^{8}}\right )}{- 3 A b^{2} + 5 C a b} \right )} + \left (\frac{a \left (- 2 B b + 3 D a\right )}{2 b^{4}} + \frac{\sqrt{- a b^{9}} \left (- 3 A b + 5 C a\right )}{4 b^{8}}\right ) \log{\left (x + \frac{4 B a b - 6 D a^{2} + 4 b^{4} \left (\frac{a \left (- 2 B b + 3 D a\right )}{2 b^{4}} + \frac{\sqrt{- a b^{9}} \left (- 3 A b + 5 C a\right )}{4 b^{8}}\right )}{- 3 A b^{2} + 5 C a b} \right )} - \frac{B a^{2} b - D a^{3} + x \left (- A a b^{2} + C a^{2} b\right )}{2 a b^{4} + 2 b^{5} x^{2}} - \frac{x^{2} \left (- B b + 2 D a\right )}{2 b^{3}} - \frac{x \left (- A b + 2 C a\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(D*x**3+C*x**2+B*x+A)/(b*x**2+a)**2,x)

[Out]

C*x**3/(3*b**2) + D*x**4/(4*b**2) + (a*(-2*B*b + 3*D*a)/(2*b**4) - sqrt(-a*b**9)
*(-3*A*b + 5*C*a)/(4*b**8))*log(x + (4*B*a*b - 6*D*a**2 + 4*b**4*(a*(-2*B*b + 3*
D*a)/(2*b**4) - sqrt(-a*b**9)*(-3*A*b + 5*C*a)/(4*b**8)))/(-3*A*b**2 + 5*C*a*b))
+ (a*(-2*B*b + 3*D*a)/(2*b**4) + sqrt(-a*b**9)*(-3*A*b + 5*C*a)/(4*b**8))*log(x
+ (4*B*a*b - 6*D*a**2 + 4*b**4*(a*(-2*B*b + 3*D*a)/(2*b**4) + sqrt(-a*b**9)*(-3
*A*b + 5*C*a)/(4*b**8)))/(-3*A*b**2 + 5*C*a*b)) - (B*a**2*b - D*a**3 + x*(-A*a*b
**2 + C*a**2*b))/(2*a*b**4 + 2*b**5*x**2) - x**2*(-B*b + 2*D*a)/(2*b**3) - x*(-A
*b + 2*C*a)/b**3

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GIAC/XCAS [A]  time = 0.224767, size = 215, normalized size = 1.22 \[ \frac{{\left (5 \, C a^{2} - 3 \, A a b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{3}} + \frac{{\left (3 \, D a^{2} - 2 \, B a b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{4}} + \frac{D a^{3} - B a^{2} b -{\left (C a^{2} b - A a b^{2}\right )} x}{2 \,{\left (b x^{2} + a\right )} b^{4}} + \frac{3 \, D b^{6} x^{4} + 4 \, C b^{6} x^{3} - 12 \, D a b^{5} x^{2} + 6 \, B b^{6} x^{2} - 24 \, C a b^{5} x + 12 \, A b^{6} x}{12 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*x^4/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

1/2*(5*C*a^2 - 3*A*a*b)*arctan(b*x/sqrt(a*b))/(sqrt(a*b)*b^3) + 1/2*(3*D*a^2 - 2
*B*a*b)*ln(b*x^2 + a)/b^4 + 1/2*(D*a^3 - B*a^2*b - (C*a^2*b - A*a*b^2)*x)/((b*x^
2 + a)*b^4) + 1/12*(3*D*b^6*x^4 + 4*C*b^6*x^3 - 12*D*a*b^5*x^2 + 6*B*b^6*x^2 - 2
4*C*a*b^5*x + 12*A*b^6*x)/b^8