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

Optimal. Leaf size=155 \[ -\frac{x^3 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2}+\frac{3 (b B-5 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{7/2}}-\frac{3 x (b B-5 a D)}{8 a b^3}+\frac{C \log \left (a+b x^2\right )}{2 b^3}-\frac{x^2 (4 a C-x (3 b B-7 a D))}{8 a b^2 \left (a+b x^2\right )} \]

[Out]

(-3*(b*B - 5*a*D)*x)/(8*a*b^3) - (x^3*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(4*a*b*
(a + b*x^2)^2) - (x^2*(4*a*C - (3*b*B - 7*a*D)*x))/(8*a*b^2*(a + b*x^2)) + (3*(b
*B - 5*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(7/2)) + (C*Log[a + b*x^2]
)/(2*b^3)

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Rubi [A]  time = 0.494439, antiderivative size = 155, 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 \[ -\frac{x^3 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2}+\frac{3 (b B-5 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{7/2}}-\frac{3 x (b B-5 a D)}{8 a b^3}+\frac{C \log \left (a+b x^2\right )}{2 b^3}-\frac{x^2 (4 a C-x (3 b B-7 a D))}{8 a b^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(b*B - 5*a*D)*x)/(8*a*b^3) - (x^3*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(4*a*b*
(a + b*x^2)^2) - (x^2*(4*a*C - (3*b*B - 7*a*D)*x))/(8*a*b^2*(a + b*x^2)) + (3*(b
*B - 5*a*D)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(8*Sqrt[a]*b^(7/2)) + (C*Log[a + b*x^2]
)/(2*b^3)

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Rubi in Sympy [A]  time = 133.624, size = 131, normalized size = 0.85 \[ \frac{C \log{\left (a + b x^{2} \right )}}{2 b^{3}} + \frac{D x}{b^{3}} + \frac{x \left (a \left (B b - D a\right ) - b x \left (A b - C a\right )\right )}{4 b^{3} \left (a + b x^{2}\right )^{2}} - \frac{x \left (a \left (5 B b - 9 D a\right ) - 2 b x \left (A b - 3 C a\right )\right )}{8 a b^{3} \left (a + b x^{2}\right )} + \frac{3 \left (B b - 5 D a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{8 \sqrt{a} b^{\frac{7}{2}}} \]

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)**3,x)

[Out]

C*log(a + b*x**2)/(2*b**3) + D*x/b**3 + x*(a*(B*b - D*a) - b*x*(A*b - C*a))/(4*b
**3*(a + b*x**2)**2) - x*(a*(5*B*b - 9*D*a) - 2*b*x*(A*b - 3*C*a))/(8*a*b**3*(a
+ b*x**2)) + 3*(B*b - 5*D*a)*atan(sqrt(b)*x/sqrt(a))/(8*sqrt(a)*b**(7/2))

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Mathematica [A]  time = 0.150641, size = 126, normalized size = 0.81 \[ \frac{a (-a (C+D x)+A b+b B x)}{4 b^3 \left (a+b x^2\right )^2}+\frac{8 a C+9 a D x-4 A b-5 b B x}{8 b^3 \left (a+b x^2\right )}+\frac{3 (b B-5 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{7/2}}+\frac{C \log \left (a+b x^2\right )}{2 b^3}+\frac{D x}{b^3} \]

Antiderivative was successfully verified.

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

[Out]

(D*x)/b^3 + (-4*A*b + 8*a*C - 5*b*B*x + 9*a*D*x)/(8*b^3*(a + b*x^2)) + (a*(A*b +
b*B*x - a*(C + D*x)))/(4*b^3*(a + b*x^2)^2) + (3*(b*B - 5*a*D)*ArcTan[(Sqrt[b]*
x)/Sqrt[a]])/(8*Sqrt[a]*b^(7/2)) + (C*Log[a + b*x^2])/(2*b^3)

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Maple [A]  time = 0.015, size = 206, normalized size = 1.3 \[{\frac{Dx}{{b}^{3}}}-{\frac{5\,B{x}^{3}}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,D{x}^{3}a}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{A{x}^{2}}{2\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{C{x}^{2}a}{{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,Bxa}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,Dx{a}^{2}}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{Aa}{4\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{3\,{a}^{2}C}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{C\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}}+{\frac{3\,B}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,aD}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

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)^3,x)

[Out]

D/b^3*x-5/8/b/(b*x^2+a)^2*B*x^3+9/8/b^2/(b*x^2+a)^2*D*x^3*a-1/2/b/(b*x^2+a)^2*A*
x^2+1/b^2/(b*x^2+a)^2*C*x^2*a-3/8/b^2/(b*x^2+a)^2*B*x*a+7/8/b^3/(b*x^2+a)^2*D*x*
a^2-1/4*a/b^2/(b*x^2+a)^2*A+3/4/b^3/(b*x^2+a)^2*a^2*C+1/2*C*ln(b*x^2+a)/b^3+3/8/
b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*B-15/8/b^3/(a*b)^(1/2)*arctan(x*b/(a*b)^
(1/2))*a*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^3/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242437, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (5 \, D a b^{2} - B b^{3}\right )} x^{4} + 5 \, D a^{3} - B a^{2} b + 2 \,{\left (5 \, D a^{2} b - B a b^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (8 \, D b^{2} x^{5} + 5 \,{\left (5 \, D a b - B b^{2}\right )} x^{3} + 6 \, C a^{2} - 2 \, A a b + 4 \,{\left (2 \, C a b - A b^{2}\right )} x^{2} + 3 \,{\left (5 \, D a^{2} - B a b\right )} x + 4 \,{\left (C b^{2} x^{4} + 2 \, C a b x^{2} + C a^{2}\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{-a b}}{16 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{-a b}}, -\frac{3 \,{\left ({\left (5 \, D a b^{2} - B b^{3}\right )} x^{4} + 5 \, D a^{3} - B a^{2} b + 2 \,{\left (5 \, D a^{2} b - B a b^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (8 \, D b^{2} x^{5} + 5 \,{\left (5 \, D a b - B b^{2}\right )} x^{3} + 6 \, C a^{2} - 2 \, A a b + 4 \,{\left (2 \, C a b - A b^{2}\right )} x^{2} + 3 \,{\left (5 \, D a^{2} - B a b\right )} x + 4 \,{\left (C b^{2} x^{4} + 2 \, C a b x^{2} + C a^{2}\right )} \log \left (b x^{2} + a\right )\right )} \sqrt{a b}}{8 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/16*(3*((5*D*a*b^2 - B*b^3)*x^4 + 5*D*a^3 - B*a^2*b + 2*(5*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*(8*D*b^2*x^5 + 5*(
5*D*a*b - B*b^2)*x^3 + 6*C*a^2 - 2*A*a*b + 4*(2*C*a*b - A*b^2)*x^2 + 3*(5*D*a^2
- B*a*b)*x + 4*(C*b^2*x^4 + 2*C*a*b*x^2 + C*a^2)*log(b*x^2 + a))*sqrt(-a*b))/((b
^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*sqrt(-a*b)), -1/8*(3*((5*D*a*b^2 - B*b^3)*x^4 +
5*D*a^3 - B*a^2*b + 2*(5*D*a^2*b - B*a*b^2)*x^2)*arctan(sqrt(a*b)*x/a) - (8*D*b^
2*x^5 + 5*(5*D*a*b - B*b^2)*x^3 + 6*C*a^2 - 2*A*a*b + 4*(2*C*a*b - A*b^2)*x^2 +
3*(5*D*a^2 - B*a*b)*x + 4*(C*b^2*x^4 + 2*C*a*b*x^2 + C*a^2)*log(b*x^2 + a))*sqrt
(a*b))/((b^5*x^4 + 2*a*b^4*x^2 + a^2*b^3)*sqrt(a*b))]

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Sympy [A]  time = 38.2683, size = 282, normalized size = 1.82 \[ \frac{D x}{b^{3}} + \left (\frac{C}{2 b^{3}} - \frac{3 \sqrt{- a b^{7}} \left (- B b + 5 D a\right )}{16 a b^{7}}\right ) \log{\left (x + \frac{8 C a - 16 a b^{3} \left (\frac{C}{2 b^{3}} - \frac{3 \sqrt{- a b^{7}} \left (- B b + 5 D a\right )}{16 a b^{7}}\right )}{- 3 B b + 15 D a} \right )} + \left (\frac{C}{2 b^{3}} + \frac{3 \sqrt{- a b^{7}} \left (- B b + 5 D a\right )}{16 a b^{7}}\right ) \log{\left (x + \frac{8 C a - 16 a b^{3} \left (\frac{C}{2 b^{3}} + \frac{3 \sqrt{- a b^{7}} \left (- B b + 5 D a\right )}{16 a b^{7}}\right )}{- 3 B b + 15 D a} \right )} + \frac{- 2 A a b + 6 C a^{2} + x^{3} \left (- 5 B b^{2} + 9 D a b\right ) + x^{2} \left (- 4 A b^{2} + 8 C a b\right ) + x \left (- 3 B a b + 7 D a^{2}\right )}{8 a^{2} b^{3} + 16 a b^{4} x^{2} + 8 b^{5} x^{4}} \]

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)**3,x)

[Out]

D*x/b**3 + (C/(2*b**3) - 3*sqrt(-a*b**7)*(-B*b + 5*D*a)/(16*a*b**7))*log(x + (8*
C*a - 16*a*b**3*(C/(2*b**3) - 3*sqrt(-a*b**7)*(-B*b + 5*D*a)/(16*a*b**7)))/(-3*B
*b + 15*D*a)) + (C/(2*b**3) + 3*sqrt(-a*b**7)*(-B*b + 5*D*a)/(16*a*b**7))*log(x
+ (8*C*a - 16*a*b**3*(C/(2*b**3) + 3*sqrt(-a*b**7)*(-B*b + 5*D*a)/(16*a*b**7)))/
(-3*B*b + 15*D*a)) + (-2*A*a*b + 6*C*a**2 + x**3*(-5*B*b**2 + 9*D*a*b) + x**2*(-
4*A*b**2 + 8*C*a*b) + x*(-3*B*a*b + 7*D*a**2))/(8*a**2*b**3 + 16*a*b**4*x**2 + 8
*b**5*x**4)

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GIAC/XCAS [A]  time = 0.227834, size = 165, normalized size = 1.06 \[ \frac{D x}{b^{3}} + \frac{C{\rm ln}\left (b x^{2} + a\right )}{2 \, b^{3}} - \frac{3 \,{\left (5 \, D a - B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{3}} + \frac{{\left (9 \, D a b - 5 \, B b^{2}\right )} x^{3} + 6 \, C a^{2} - 2 \, A a b + 4 \,{\left (2 \, C a b - A b^{2}\right )} x^{2} +{\left (7 \, D a^{2} - 3 \, B a b\right )} x}{8 \,{\left (b x^{2} + a\right )}^{2} b^{3}} \]

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)^3,x, algorithm="giac")

[Out]

D*x/b^3 + 1/2*C*ln(b*x^2 + a)/b^3 - 3/8*(5*D*a - B*b)*arctan(b*x/sqrt(a*b))/(sqr
t(a*b)*b^3) + 1/8*((9*D*a*b - 5*B*b^2)*x^3 + 6*C*a^2 - 2*A*a*b + 4*(2*C*a*b - A*
b^2)*x^2 + (7*D*a^2 - 3*B*a*b)*x)/((b*x^2 + a)^2*b^3)