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

Optimal. Leaf size=111 \[ -\frac{\sqrt{a} (A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{x (A b-a C)}{b^2}-\frac{a (b B-a D) \log \left (a+b x^2\right )}{2 b^3}+\frac{x^2 (b B-a D)}{2 b^2}+\frac{C x^3}{3 b}+\frac{D x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.248523, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\sqrt{a} (A b-a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{5/2}}+\frac{x (A b-a C)}{b^2}-\frac{a (b B-a D) \log \left (a+b x^2\right )}{2 b^3}+\frac{x^2 (b B-a D)}{2 b^2}+\frac{C x^3}{3 b}+\frac{D x^4}{4 b} \]

Antiderivative was successfully verified.

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

[Out]

((A*b - a*C)*x)/b^2 + ((b*B - a*D)*x^2)/(2*b^2) + (C*x^3)/(3*b) + (D*x^4)/(4*b)
- (Sqrt[a]*(A*b - a*C)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/b^(5/2) - (a*(b*B - 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. \[ \frac{C x^{3}}{3 b} + \frac{D x^{4}}{4 b} - \frac{\sqrt{a} \left (A b - C a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{b^{\frac{5}{2}}} - \frac{a \left (B b - D a\right ) \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \left (A b - C a\right ) \int \frac{1}{b^{2}}\, dx + \frac{\left (B b - D a\right ) \int x\, dx}{b^{2}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.101558, size = 95, normalized size = 0.86 \[ \frac{b x \left (-6 a (2 C+D x)+12 A b+b x \left (6 B+4 C x+3 D x^2\right )\right )+12 \sqrt{a} \sqrt{b} (a C-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+6 a (a D-b B) \log \left (a+b x^2\right )}{12 b^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 128, normalized size = 1.2 \[{\frac{D{x}^{4}}{4\,b}}+{\frac{C{x}^{3}}{3\,b}}+{\frac{B{x}^{2}}{2\,b}}-{\frac{D{x}^{2}a}{2\,{b}^{2}}}+{\frac{Ax}{b}}-{\frac{Cxa}{{b}^{2}}}-{\frac{a\ln \left ( b{x}^{2}+a \right ) B}{2\,{b}^{2}}}+{\frac{{a}^{2}\ln \left ( b{x}^{2}+a \right ) D}{2\,{b}^{3}}}-{\frac{aA}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{a}^{2}C}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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),x)

[Out]

1/4*D*x^4/b+1/3*C*x^3/b+1/2*B/b*x^2-1/2/b^2*D*x^2*a+1/b*A*x-1/b^2*C*x*a-1/2*a/b^
2*ln(b*x^2+a)*B+1/2*a^2/b^3*ln(b*x^2+a)*D-a/b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2)
)*A+a^2/b^2/(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^2/(b*x^2 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.271887, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, D b^{2} x^{4} + 4 \, C b^{2} x^{3} - 6 \,{\left (D a b - B b^{2}\right )} x^{2} - 6 \,{\left (C a b - A b^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 12 \,{\left (C a b - A b^{2}\right )} x + 6 \,{\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{3}}, \frac{3 \, D b^{2} x^{4} + 4 \, C b^{2} x^{3} - 6 \,{\left (D a b - B b^{2}\right )} x^{2} + 12 \,{\left (C a b - A b^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 12 \,{\left (C a b - A b^{2}\right )} x + 6 \,{\left (D a^{2} - B a b\right )} \log \left (b x^{2} + a\right )}{12 \, b^{3}}\right ] \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.90719, size = 243, normalized size = 2.19 \[ \frac{C x^{3}}{3 b} + \frac{D x^{4}}{4 b} + \left (\frac{a \left (- B b + D a\right )}{2 b^{3}} - \frac{\sqrt{- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right ) \log{\left (x + \frac{B a b - D a^{2} + 2 b^{3} \left (\frac{a \left (- B b + D a\right )}{2 b^{3}} - \frac{\sqrt{- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right )}{- A b^{2} + C a b} \right )} + \left (\frac{a \left (- B b + D a\right )}{2 b^{3}} + \frac{\sqrt{- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right ) \log{\left (x + \frac{B a b - D a^{2} + 2 b^{3} \left (\frac{a \left (- B b + D a\right )}{2 b^{3}} + \frac{\sqrt{- a b^{7}} \left (- A b + C a\right )}{2 b^{6}}\right )}{- A b^{2} + C a b} \right )} - \frac{x^{2} \left (- B b + D a\right )}{2 b^{2}} - \frac{x \left (- A b + C a\right )}{b^{2}} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220411, size = 151, normalized size = 1.36 \[ \frac{{\left (C a^{2} - A a b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{{\left (D a^{2} - B a b\right )}{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{3}} + \frac{3 \, D b^{3} x^{4} + 4 \, C b^{3} x^{3} - 6 \, D a b^{2} x^{2} + 6 \, B b^{3} x^{2} - 12 \, C a b^{2} x + 12 \, A b^{3} x}{12 \, b^{4}} \]

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),x, algorithm="giac")

[Out]

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