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

Optimal. Leaf size=131 \[ -\frac{3 (b c-a d)^2 (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{5/2}}-\frac{c^2 (3 b c-a d)}{2 a^2 b x}-\frac{d^2 x (b c-3 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.334847, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{3 (b c-a d)^2 (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{5/2}}-\frac{c^2 (3 b c-a d)}{2 a^2 b x}-\frac{d^2 x (b c-3 a d)}{2 a b^2}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d**2*(3*a*d - b*c)*Integral(1/b, x)/(2*a*b) - (c + d*x**2)**2*(a*d - b*c)/(2*a*b
*x*(a + b*x**2)) + c**2*(a*d - 3*b*c)/(2*a**2*b*x) - 3*(a*d - b*c)**2*(a*d + b*c
)*atan(sqrt(b)*x/sqrt(a))/(2*a**(5/2)*b**(5/2))

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Mathematica [A]  time = 0.0973548, size = 94, normalized size = 0.72 \[ -\frac{3 (a d-b c)^2 (a d+b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} b^{5/2}}+\frac{x (a d-b c)^3}{2 a^2 b^2 \left (a+b x^2\right )}-\frac{c^3}{a^2 x}+\frac{d^3 x}{b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.017, size = 189, normalized size = 1.4 \[{\frac{{d}^{3}x}{{b}^{2}}}-{\frac{{c}^{3}}{{a}^{2}x}}+{\frac{ax{d}^{3}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,cx{d}^{2}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{3\,x{c}^{2}d}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bx{c}^{3}}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,a{d}^{3}}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,c{d}^{2}}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,{c}^{2}d}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,b{c}^{3}}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d^3*x/b^2-c^3/x/a^2+1/2*a/b^2*x/(b*x^2+a)*d^3-3/2/b*x/(b*x^2+a)*c*d^2+3/2/a*x/(b
*x^2+a)*c^2*d-1/2/a^2*b*x/(b*x^2+a)*c^3-3/2*a/b^2/(a*b)^(1/2)*arctan(x*b/(a*b)^(
1/2))*d^3+3/2/b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c*d^2+3/2/a/(a*b)^(1/2)*arct
an(x*b/(a*b)^(1/2))*c^2*d-3/2/a^2*b/(a*b)^(1/2)*arctan(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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240846, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (b^{4} c^{3} - a b^{3} c^{2} d - a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{3} +{\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + a^{4} d^{3}\right )} x\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (2 \, a^{2} b d^{3} x^{4} - 2 \, a b^{2} c^{3} - 3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b^{3} x^{3} + a^{3} b^{2} x\right )} \sqrt{-a b}}, -\frac{3 \,{\left ({\left (b^{4} c^{3} - a b^{3} c^{2} d - a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{3} +{\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + a^{4} d^{3}\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, a^{2} b d^{3} x^{4} - 2 \, a b^{2} c^{3} - 3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d + a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{2}\right )} \sqrt{a b}}{2 \,{\left (a^{2} b^{3} x^{3} + a^{3} b^{2} x\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*(3*((b^4*c^3 - a*b^3*c^2*d - a^2*b^2*c*d^2 + a^3*b*d^3)*x^3 + (a*b^3*c^3 -
a^2*b^2*c^2*d - a^3*b*c*d^2 + a^4*d^3)*x)*log(-(2*a*b*x - (b*x^2 - a)*sqrt(-a*b)
)/(b*x^2 + a)) + 2*(2*a^2*b*d^3*x^4 - 2*a*b^2*c^3 - 3*(b^3*c^3 - a*b^2*c^2*d + a
^2*b*c*d^2 - a^3*d^3)*x^2)*sqrt(-a*b))/((a^2*b^3*x^3 + a^3*b^2*x)*sqrt(-a*b)), -
1/2*(3*((b^4*c^3 - a*b^3*c^2*d - a^2*b^2*c*d^2 + a^3*b*d^3)*x^3 + (a*b^3*c^3 - a
^2*b^2*c^2*d - a^3*b*c*d^2 + a^4*d^3)*x)*arctan(sqrt(a*b)*x/a) - (2*a^2*b*d^3*x^
4 - 2*a*b^2*c^3 - 3*(b^3*c^3 - a*b^2*c^2*d + a^2*b*c*d^2 - a^3*d^3)*x^2)*sqrt(a*
b))/((a^2*b^3*x^3 + a^3*b^2*x)*sqrt(a*b))]

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Sympy [A]  time = 7.05431, size = 309, normalized size = 2.36 \[ \frac{3 \sqrt{- \frac{1}{a^{5} b^{5}}} \left (a d - b c\right )^{2} \left (a d + b c\right ) \log{\left (- \frac{3 a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} \left (a d - b c\right )^{2} \left (a d + b c\right )}{3 a^{3} d^{3} - 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + 3 b^{3} c^{3}} + x \right )}}{4} - \frac{3 \sqrt{- \frac{1}{a^{5} b^{5}}} \left (a d - b c\right )^{2} \left (a d + b c\right ) \log{\left (\frac{3 a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} \left (a d - b c\right )^{2} \left (a d + b c\right )}{3 a^{3} d^{3} - 3 a^{2} b c d^{2} - 3 a b^{2} c^{2} d + 3 b^{3} c^{3}} + x \right )}}{4} + \frac{- 2 a b^{2} c^{3} + x^{2} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 3 b^{3} c^{3}\right )}{2 a^{3} b^{2} x + 2 a^{2} b^{3} x^{3}} + \frac{d^{3} x}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.247841, size = 193, normalized size = 1.47 \[ \frac{d^{3} x}{b^{2}} - \frac{3 \,{\left (b^{3} c^{3} - a b^{2} c^{2} d - a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2} b^{2}} - \frac{3 \, b^{3} c^{3} x^{2} - 3 \, a b^{2} c^{2} d x^{2} + 3 \, a^{2} b c d^{2} x^{2} - a^{3} d^{3} x^{2} + 2 \, a b^{2} c^{3}}{2 \,{\left (b x^{3} + a x\right )} a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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