3.29 \(\int \frac{\left (c+d x^2\right )^3}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=106 \[ \frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.204606, antiderivative size = 106, 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 \[ \frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^3/(a + b*x^2)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.100365, size = 106, normalized size = 1. \[ \frac{(5 a d+b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{3/2} b^{7/2}}+\frac{d^2 x (3 b c-2 a d)}{b^3}+\frac{x (b c-a d)^3}{2 a b^3 \left (a+b x^2\right )}+\frac{d^3 x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^3/(a + b*x^2)^2,x]

[Out]

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

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Maple [B]  time = 0.014, size = 205, normalized size = 1.9 \[{\frac{{d}^{3}{x}^{3}}{3\,{b}^{2}}}-2\,{\frac{a{d}^{3}x}{{b}^{3}}}+3\,{\frac{{d}^{2}xc}{{b}^{2}}}-{\frac{{a}^{2}x{d}^{3}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,acx{d}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,x{c}^{2}d}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{x{c}^{3}}{2\,a \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{a}^{2}{d}^{3}}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{9\,ac{d}^{2}}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{2}d}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{c}^{3}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^3/(b*x^2+a)^2,x)

[Out]

1/3*d^3*x^3/b^2-2*d^3/b^3*a*x+3*d^2/b^2*x*c-1/2/b^3*x*a^2/(b*x^2+a)*d^3+3/2/b^2*
x*a/(b*x^2+a)*c*d^2-3/2/b*x/(b*x^2+a)*c^2*d+1/2*x/a/(b*x^2+a)*c^3+5/2/b^3*a^2/(a
*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d^3-9/2/b^2*a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/
2))*c*d^2+3/2/b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^2*d+1/2/a/(a*b)^(1/2)*arct
an(x*b/(a*b)^(1/2))*c^3

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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)^3/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.210617, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} +{\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 9 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (2 \, a b^{2} d^{3} x^{5} + 2 \,{\left (9 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x^{3} + 3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} x\right )} \sqrt{-a b}}{12 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{-a b}}, \frac{3 \,{\left (a b^{3} c^{3} + 3 \, a^{2} b^{2} c^{2} d - 9 \, a^{3} b c d^{2} + 5 \, a^{4} d^{3} +{\left (b^{4} c^{3} + 3 \, a b^{3} c^{2} d - 9 \, a^{2} b^{2} c d^{2} + 5 \, a^{3} b d^{3}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (2 \, a b^{2} d^{3} x^{5} + 2 \,{\left (9 \, a b^{2} c d^{2} - 5 \, a^{2} b d^{3}\right )} x^{3} + 3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} - 5 \, a^{3} d^{3}\right )} x\right )} \sqrt{a b}}{6 \,{\left (a b^{4} x^{2} + a^{2} b^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[1/12*(3*(a*b^3*c^3 + 3*a^2*b^2*c^2*d - 9*a^3*b*c*d^2 + 5*a^4*d^3 + (b^4*c^3 + 3
*a*b^3*c^2*d - 9*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^2)*log((2*a*b*x + (b*x^2 - a)*sq
rt(-a*b))/(b*x^2 + a)) + 2*(2*a*b^2*d^3*x^5 + 2*(9*a*b^2*c*d^2 - 5*a^2*b*d^3)*x^
3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 5*a^3*d^3)*x)*sqrt(-a*b))/((a*b
^4*x^2 + a^2*b^3)*sqrt(-a*b)), 1/6*(3*(a*b^3*c^3 + 3*a^2*b^2*c^2*d - 9*a^3*b*c*d
^2 + 5*a^4*d^3 + (b^4*c^3 + 3*a*b^3*c^2*d - 9*a^2*b^2*c*d^2 + 5*a^3*b*d^3)*x^2)*
arctan(sqrt(a*b)*x/a) + (2*a*b^2*d^3*x^5 + 2*(9*a*b^2*c*d^2 - 5*a^2*b*d^3)*x^3 +
3*(b^3*c^3 - 3*a*b^2*c^2*d + 9*a^2*b*c*d^2 - 5*a^3*d^3)*x)*sqrt(a*b))/((a*b^4*x
^2 + a^2*b^3)*sqrt(a*b))]

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Sympy [A]  time = 5.17971, size = 313, normalized size = 2.95 \[ - \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 a^{2} b^{3} + 2 a b^{4} x^{2}} - \frac{\sqrt{- \frac{1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right ) \log{\left (- \frac{a^{2} b^{3} \sqrt{- \frac{1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right )}{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right ) \log{\left (\frac{a^{2} b^{3} \sqrt{- \frac{1}{a^{3} b^{7}}} \left (a d - b c\right )^{2} \left (5 a d + b c\right )}{5 a^{3} d^{3} - 9 a^{2} b c d^{2} + 3 a b^{2} c^{2} d + b^{3} c^{3}} + x \right )}}{4} + \frac{d^{3} x^{3}}{3 b^{2}} - \frac{x \left (2 a d^{3} - 3 b c d^{2}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**3/(b*x**2+a)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.235221, size = 205, normalized size = 1.93 \[ \frac{{\left (b^{3} c^{3} + 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 5 \, a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a b^{3}} + \frac{b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \,{\left (b x^{2} + a\right )} a b^{3}} + \frac{b^{4} d^{3} x^{3} + 9 \, b^{4} c d^{2} x - 6 \, a b^{3} d^{3} x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

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