3.288 \(\int \frac{\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=147 \[ \frac{(b c-a d)^2 (a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} b^{3/2}}-\frac{c^2 (5 b c-3 a d)}{6 a^2 b x^3}+\frac{c \left (2 a^2 d^2-9 a b c d+5 b^2 c^2\right )}{2 a^3 b x}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^3 \left (a+b x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.351205, antiderivative size = 147, 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 \[ \frac{(b c-a d)^2 (a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} b^{3/2}}-\frac{c^2 (5 b c-3 a d)}{6 a^2 b x^3}+\frac{c \left (2 a^2 d^2-9 a b c d+5 b^2 c^2\right )}{2 a^3 b x}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{2 a b x^3 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x^2)^3/(x^4*(a + b*x^2)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 56.6265, size = 131, normalized size = 0.89 \[ - \frac{\left (c + d x^{2}\right )^{2} \left (a d - b c\right )}{2 a b x^{3} \left (a + b x^{2}\right )} + \frac{c^{2} \left (3 a d - 5 b c\right )}{6 a^{2} b x^{3}} + \frac{c \left (2 a^{2} d^{2} - 9 a b c d + 5 b^{2} c^{2}\right )}{2 a^{3} b x} + \frac{\left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}} b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**3/x**4/(b*x**2+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.102152, size = 109, normalized size = 0.74 \[ \frac{(a d-b c)^2 (a d+5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} b^{3/2}}-\frac{c^2 (3 a d-2 b c)}{a^3 x}-\frac{x (a d-b c)^3}{2 a^3 b \left (a+b x^2\right )}-\frac{c^3}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x^2)^3/(x^4*(a + b*x^2)^2),x]

[Out]

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

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Maple [A]  time = 0.019, size = 209, normalized size = 1.4 \[ -{\frac{{c}^{3}}{3\,{a}^{2}{x}^{3}}}-3\,{\frac{{c}^{2}d}{{a}^{2}x}}+2\,{\frac{{c}^{3}b}{{a}^{3}x}}-{\frac{x{d}^{3}}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{3\,cx{d}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{3\,bx{c}^{2}d}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{x{b}^{2}{c}^{3}}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{d}^{3}}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,c{d}^{2}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{9\,bd{c}^{2}}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}{c}^{3}}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^3/x^4/(b*x^2+a)^2,x)

[Out]

-1/3*c^3/a^2/x^3-3*c^2/a^2/x*d+2*c^3/a^3/x*b-1/2/b*x/(b*x^2+a)*d^3+3/2/a*x/(b*x^
2+a)*c*d^2-3/2/a^2*b*x/(b*x^2+a)*c^2*d+1/2/a^3*b^2*x/(b*x^2+a)*c^3+1/2/b/(a*b)^(
1/2)*arctan(x*b/(a*b)^(1/2))*d^3+3/2/a/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c*d^2
-9/2/a^2*b/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c^2*d+5/2/a^3*b^2/(a*b)^(1/2)*arc
tan(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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.241413, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left ({\left (5 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{5} +{\left (5 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (2 \, a^{2} b c^{3} - 3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{4} - 2 \,{\left (5 \, a b^{2} c^{3} - 9 \, a^{2} b c^{2} d\right )} x^{2}\right )} \sqrt{-a b}}{12 \,{\left (a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt{-a b}}, \frac{3 \,{\left ({\left (5 \, b^{4} c^{3} - 9 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} + a^{3} b d^{3}\right )} x^{5} +{\left (5 \, a b^{3} c^{3} - 9 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} + a^{4} d^{3}\right )} x^{3}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, a^{2} b c^{3} - 3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{4} - 2 \,{\left (5 \, a b^{2} c^{3} - 9 \, a^{2} b c^{2} d\right )} x^{2}\right )} \sqrt{a b}}{6 \,{\left (a^{3} b^{2} x^{5} + a^{4} b x^{3}\right )} \sqrt{a b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.50912, size = 321, normalized size = 2.18 \[ - \frac{\sqrt{- \frac{1}{a^{7} b^{3}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log{\left (- \frac{a^{4} b \sqrt{- \frac{1}{a^{7} b^{3}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right )}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{7} b^{3}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log{\left (\frac{a^{4} b \sqrt{- \frac{1}{a^{7} b^{3}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right )}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3}} + x \right )}}{4} - \frac{2 a^{2} b c^{3} + x^{4} \left (3 a^{3} d^{3} - 9 a^{2} b c d^{2} + 27 a b^{2} c^{2} d - 15 b^{3} c^{3}\right ) + x^{2} \left (18 a^{2} b c^{2} d - 10 a b^{2} c^{3}\right )}{6 a^{4} b x^{3} + 6 a^{3} b^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**3/x**4/(b*x**2+a)**2,x)

[Out]

-sqrt(-1/(a**7*b**3))*(a*d - b*c)**2*(a*d + 5*b*c)*log(-a**4*b*sqrt(-1/(a**7*b**
3))*(a*d - b*c)**2*(a*d + 5*b*c)/(a**3*d**3 + 3*a**2*b*c*d**2 - 9*a*b**2*c**2*d
+ 5*b**3*c**3) + x)/4 + sqrt(-1/(a**7*b**3))*(a*d - b*c)**2*(a*d + 5*b*c)*log(a*
*4*b*sqrt(-1/(a**7*b**3))*(a*d - b*c)**2*(a*d + 5*b*c)/(a**3*d**3 + 3*a**2*b*c*d
**2 - 9*a*b**2*c**2*d + 5*b**3*c**3) + x)/4 - (2*a**2*b*c**3 + x**4*(3*a**3*d**3
- 9*a**2*b*c*d**2 + 27*a*b**2*c**2*d - 15*b**3*c**3) + x**2*(18*a**2*b*c**2*d -
10*a*b**2*c**3))/(6*a**4*b*x**3 + 6*a**3*b**2*x**5)

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GIAC/XCAS [A]  time = 0.252069, size = 203, normalized size = 1.38 \[ \frac{{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3} b} + \frac{b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \,{\left (b x^{2} + a\right )} a^{3} b} + \frac{6 \, b c^{3} x^{2} - 9 \, a c^{2} d x^{2} - a c^{3}}{3 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x^2 + c)^3/((b*x^2 + a)^2*x^4),x, algorithm="giac")

[Out]

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