3.28 \(\int \frac{\left (c+d x^2\right )^4}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=142 \[ \frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{9/2}}+\frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{d^4 x^5}{5 b^2} \]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b
^3) + (d^4*x^5)/(5*b^2) + ((b*c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)
^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(9/2))

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Rubi [A]  time = 0.260008, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{9/2}}+\frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{d^4 x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^4/(a + b*x^2)^2,x]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b
^3) + (d^4*x^5)/(5*b^2) + ((b*c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)
^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(9/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ d^{2} \left (3 a^{2} d^{2} - 8 a b c d + 6 b^{2} c^{2}\right ) \int \frac{1}{b^{4}}\, dx + \frac{d^{4} x^{5}}{5 b^{2}} - \frac{2 d^{3} x^{3} \left (a d - 2 b c\right )}{3 b^{3}} + \frac{x \left (a d - b c\right )^{4}}{2 a b^{4} \left (a + b x^{2}\right )} - \frac{\left (a d - b c\right )^{3} \left (7 a d + b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} b^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**4/(b*x**2+a)**2,x)

[Out]

d**2*(3*a**2*d**2 - 8*a*b*c*d + 6*b**2*c**2)*Integral(b**(-4), x) + d**4*x**5/(5
*b**2) - 2*d**3*x**3*(a*d - 2*b*c)/(3*b**3) + x*(a*d - b*c)**4/(2*a*b**4*(a + b*
x**2)) - (a*d - b*c)**3*(7*a*d + b*c)*atan(sqrt(b)*x/sqrt(a))/(2*a**(3/2)*b**(9/
2))

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Mathematica [A]  time = 0.143361, size = 142, normalized size = 1. \[ \frac{(b c-a d)^3 (7 a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{9/2}}+\frac{d^2 x \left (3 a^2 d^2-8 a b c d+6 b^2 c^2\right )}{b^4}+\frac{x (b c-a d)^4}{2 a b^4 \left (a+b x^2\right )}+\frac{2 d^3 x^3 (2 b c-a d)}{3 b^3}+\frac{d^4 x^5}{5 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^4/(a + b*x^2)^2,x]

[Out]

(d^2*(6*b^2*c^2 - 8*a*b*c*d + 3*a^2*d^2)*x)/b^4 + (2*d^3*(2*b*c - a*d)*x^3)/(3*b
^3) + (d^4*x^5)/(5*b^2) + ((b*c - a*d)^4*x)/(2*a*b^4*(a + b*x^2)) + ((b*c - a*d)
^3*(b*c + 7*a*d)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*a^(3/2)*b^(9/2))

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Maple [B]  time = 0.016, size = 296, normalized size = 2.1 \[{\frac{{d}^{4}{x}^{5}}{5\,{b}^{2}}}-{\frac{2\,{d}^{4}{x}^{3}a}{3\,{b}^{3}}}+{\frac{4\,{d}^{3}{x}^{3}c}{3\,{b}^{2}}}+3\,{\frac{{a}^{2}{d}^{4}x}{{b}^{4}}}-8\,{\frac{ac{d}^{3}x}{{b}^{3}}}+6\,{\frac{{c}^{2}{d}^{2}x}{{b}^{2}}}+{\frac{x{a}^{3}{d}^{4}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{{a}^{2}cx{d}^{3}}{{b}^{3} \left ( b{x}^{2}+a \right ) }}+3\,{\frac{a{c}^{2}x{d}^{2}}{{b}^{2} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{x{c}^{3}d}{b \left ( b{x}^{2}+a \right ) }}+{\frac{x{c}^{4}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{a}^{3}{d}^{4}}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+10\,{\frac{{a}^{2}c{d}^{3}}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-9\,{\frac{a{c}^{2}{d}^{2}}{{b}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+2\,{\frac{{c}^{3}d}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{c}^{4}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^4/(b*x^2+a)^2,x)

[Out]

1/5*d^4*x^5/b^2-2/3*d^4/b^3*x^3*a+4/3*d^3/b^2*x^3*c+3*d^4/b^4*a^2*x-8*d^3/b^3*a*
c*x+6*d^2/b^2*c^2*x+1/2/b^4*x*a^3/(b*x^2+a)*d^4-2/b^3*x*a^2/(b*x^2+a)*c*d^3+3/b^
2*x*a/(b*x^2+a)*c^2*d^2-2/b*x/(b*x^2+a)*c^3*d+1/2*x/a/(b*x^2+a)*c^4-7/2/b^4*a^3/
(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d^4+10/b^3*a^2/(a*b)^(1/2)*arctan(x*b/(a*b)^
(1/2))*c*d^3-9/b^2*a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^2*d^2+2/b/(a*b)^(1/2)
*arctan(x*b/(a*b)^(1/2))*c^3*d+1/2/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^4

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.213542, size = 1, normalized size = 0.01 \[ \left [-\frac{15 \,{\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 20 \, a^{4} b c d^{3} - 7 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 18 \, a^{2} b^{3} c^{2} d^{2} + 20 \, a^{3} b^{2} c d^{3} - 7 \, a^{4} b d^{4}\right )} x^{2}\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (6 \, a b^{3} d^{4} x^{7} + 2 \,{\left (20 \, a b^{3} c d^{3} - 7 \, a^{2} b^{2} d^{4}\right )} x^{5} + 10 \,{\left (18 \, a b^{3} c^{2} d^{2} - 20 \, a^{2} b^{2} c d^{3} + 7 \, a^{3} b d^{4}\right )} x^{3} + 15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 20 \, a^{3} b c d^{3} + 7 \, a^{4} d^{4}\right )} x\right )} \sqrt{-a b}}{60 \,{\left (a b^{5} x^{2} + a^{2} b^{4}\right )} \sqrt{-a b}}, \frac{15 \,{\left (a b^{4} c^{4} + 4 \, a^{2} b^{3} c^{3} d - 18 \, a^{3} b^{2} c^{2} d^{2} + 20 \, a^{4} b c d^{3} - 7 \, a^{5} d^{4} +{\left (b^{5} c^{4} + 4 \, a b^{4} c^{3} d - 18 \, a^{2} b^{3} c^{2} d^{2} + 20 \, a^{3} b^{2} c d^{3} - 7 \, a^{4} b d^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (6 \, a b^{3} d^{4} x^{7} + 2 \,{\left (20 \, a b^{3} c d^{3} - 7 \, a^{2} b^{2} d^{4}\right )} x^{5} + 10 \,{\left (18 \, a b^{3} c^{2} d^{2} - 20 \, a^{2} b^{2} c d^{3} + 7 \, a^{3} b d^{4}\right )} x^{3} + 15 \,{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 18 \, a^{2} b^{2} c^{2} d^{2} - 20 \, a^{3} b c d^{3} + 7 \, a^{4} d^{4}\right )} x\right )} \sqrt{a b}}{30 \,{\left (a b^{5} x^{2} + a^{2} b^{4}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^4/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[-1/60*(15*(a*b^4*c^4 + 4*a^2*b^3*c^3*d - 18*a^3*b^2*c^2*d^2 + 20*a^4*b*c*d^3 -
7*a^5*d^4 + (b^5*c^4 + 4*a*b^4*c^3*d - 18*a^2*b^3*c^2*d^2 + 20*a^3*b^2*c*d^3 - 7
*a^4*b*d^4)*x^2)*log(-(2*a*b*x - (b*x^2 - a)*sqrt(-a*b))/(b*x^2 + a)) - 2*(6*a*b
^3*d^4*x^7 + 2*(20*a*b^3*c*d^3 - 7*a^2*b^2*d^4)*x^5 + 10*(18*a*b^3*c^2*d^2 - 20*
a^2*b^2*c*d^3 + 7*a^3*b*d^4)*x^3 + 15*(b^4*c^4 - 4*a*b^3*c^3*d + 18*a^2*b^2*c^2*
d^2 - 20*a^3*b*c*d^3 + 7*a^4*d^4)*x)*sqrt(-a*b))/((a*b^5*x^2 + a^2*b^4)*sqrt(-a*
b)), 1/30*(15*(a*b^4*c^4 + 4*a^2*b^3*c^3*d - 18*a^3*b^2*c^2*d^2 + 20*a^4*b*c*d^3
- 7*a^5*d^4 + (b^5*c^4 + 4*a*b^4*c^3*d - 18*a^2*b^3*c^2*d^2 + 20*a^3*b^2*c*d^3
- 7*a^4*b*d^4)*x^2)*arctan(sqrt(a*b)*x/a) + (6*a*b^3*d^4*x^7 + 2*(20*a*b^3*c*d^3
- 7*a^2*b^2*d^4)*x^5 + 10*(18*a*b^3*c^2*d^2 - 20*a^2*b^2*c*d^3 + 7*a^3*b*d^4)*x
^3 + 15*(b^4*c^4 - 4*a*b^3*c^3*d + 18*a^2*b^2*c^2*d^2 - 20*a^3*b*c*d^3 + 7*a^4*d
^4)*x)*sqrt(a*b))/((a*b^5*x^2 + a^2*b^4)*sqrt(a*b))]

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Sympy [A]  time = 6.65158, size = 398, normalized size = 2.8 \[ \frac{x \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{2 a^{2} b^{4} + 2 a b^{5} x^{2}} + \frac{\sqrt{- \frac{1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right ) \log{\left (- \frac{a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right )}{7 a^{4} d^{4} - 20 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right ) \log{\left (\frac{a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} \left (a d - b c\right )^{3} \left (7 a d + b c\right )}{7 a^{4} d^{4} - 20 a^{3} b c d^{3} + 18 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d - b^{4} c^{4}} + x \right )}}{4} + \frac{d^{4} x^{5}}{5 b^{2}} - \frac{x^{3} \left (2 a d^{4} - 4 b c d^{3}\right )}{3 b^{3}} + \frac{x \left (3 a^{2} d^{4} - 8 a b c d^{3} + 6 b^{2} c^{2} d^{2}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**4/(b*x**2+a)**2,x)

[Out]

x*(a**4*d**4 - 4*a**3*b*c*d**3 + 6*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d + b**4*
c**4)/(2*a**2*b**4 + 2*a*b**5*x**2) + sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d
+ b*c)*log(-a**2*b**4*sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b*c)/(7*a**4
*d**4 - 20*a**3*b*c*d**3 + 18*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d - b**4*c**4)
+ x)/4 - sqrt(-1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b*c)*log(a**2*b**4*sqrt(-
1/(a**3*b**9))*(a*d - b*c)**3*(7*a*d + b*c)/(7*a**4*d**4 - 20*a**3*b*c*d**3 + 18
*a**2*b**2*c**2*d**2 - 4*a*b**3*c**3*d - b**4*c**4) + x)/4 + d**4*x**5/(5*b**2)
- x**3*(2*a*d**4 - 4*b*c*d**3)/(3*b**3) + x*(3*a**2*d**4 - 8*a*b*c*d**3 + 6*b**2
*c**2*d**2)/b**4

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GIAC/XCAS [A]  time = 0.244342, size = 297, normalized size = 2.09 \[ \frac{{\left (b^{4} c^{4} + 4 \, a b^{3} c^{3} d - 18 \, a^{2} b^{2} c^{2} d^{2} + 20 \, a^{3} b c d^{3} - 7 \, a^{4} d^{4}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b^{4}} + \frac{b^{4} c^{4} x - 4 \, a b^{3} c^{3} d x + 6 \, a^{2} b^{2} c^{2} d^{2} x - 4 \, a^{3} b c d^{3} x + a^{4} d^{4} x}{2 \,{\left (b x^{2} + a\right )} a b^{4}} + \frac{3 \, b^{8} d^{4} x^{5} + 20 \, b^{8} c d^{3} x^{3} - 10 \, a b^{7} d^{4} x^{3} + 90 \, b^{8} c^{2} d^{2} x - 120 \, a b^{7} c d^{3} x + 45 \, a^{2} b^{6} d^{4} x}{15 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^4/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

1/2*(b^4*c^4 + 4*a*b^3*c^3*d - 18*a^2*b^2*c^2*d^2 + 20*a^3*b*c*d^3 - 7*a^4*d^4)*
arctan(b*x/sqrt(a*b))/(sqrt(a*b)*a*b^4) + 1/2*(b^4*c^4*x - 4*a*b^3*c^3*d*x + 6*a
^2*b^2*c^2*d^2*x - 4*a^3*b*c*d^3*x + a^4*d^4*x)/((b*x^2 + a)*a*b^4) + 1/15*(3*b^
8*d^4*x^5 + 20*b^8*c*d^3*x^3 - 10*a*b^7*d^4*x^3 + 90*b^8*c^2*d^2*x - 120*a*b^7*c
*d^3*x + 45*a^2*b^6*d^4*x)/b^10