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

Optimal. Leaf size=107 \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{7/2}}-\frac{x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac{b^3 x^3}{3 d^2} \]

[Out]

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

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Rubi [A]  time = 0.215367, antiderivative size = 107, 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 \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{7/2}}-\frac{x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac{b^3 x^3}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0995282, size = 107, normalized size = 1. \[ -\frac{b^2 x (2 b c-3 a d)}{d^3}+\frac{(a d+5 b c) (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{2 c^{3/2} d^{7/2}}-\frac{x (b c-a d)^3}{2 c d^3 \left (c+d x^2\right )}+\frac{b^3 x^3}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 205, normalized size = 1.9 \[{\frac{{b}^{3}{x}^{3}}{3\,{d}^{2}}}+3\,{\frac{a{b}^{2}x}{{d}^{2}}}-2\,{\frac{{b}^{3}xc}{{d}^{3}}}+{\frac{x{a}^{3}}{2\,c \left ( d{x}^{2}+c \right ) }}-{\frac{3\,{a}^{2}bx}{2\,d \left ( d{x}^{2}+c \right ) }}+{\frac{3\,acx{b}^{2}}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{{c}^{2}x{b}^{3}}{2\,{d}^{3} \left ( d{x}^{2}+c \right ) }}+{\frac{{a}^{3}}{2\,c}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{3\,{a}^{2}b}{2\,d}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{9\,a{b}^{2}c}{2\,{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}+{\frac{5\,{b}^{3}{c}^{2}}{2\,{d}^{3}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/(d*x^2+c)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221784, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (5 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} +{\left (5 \, b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (2 \, b^{3} c d^{2} x^{5} - 2 \,{\left (5 \, b^{3} c^{2} d - 9 \, a b^{2} c d^{2}\right )} x^{3} - 3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x\right )} \sqrt{-c d}}{12 \,{\left (c d^{4} x^{2} + c^{2} d^{3}\right )} \sqrt{-c d}}, \frac{3 \,{\left (5 \, b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 3 \, a^{2} b c^{2} d^{2} + a^{3} c d^{3} +{\left (5 \, b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (2 \, b^{3} c d^{2} x^{5} - 2 \,{\left (5 \, b^{3} c^{2} d - 9 \, a b^{2} c d^{2}\right )} x^{3} - 3 \,{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x\right )} \sqrt{c d}}{6 \,{\left (c d^{4} x^{2} + c^{2} d^{3}\right )} \sqrt{c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.80084, size = 313, normalized size = 2.93 \[ \frac{b^{3} x^{3}}{3 d^{2}} + \frac{x \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{2 c^{2} d^{3} + 2 c d^{4} x^{2}} - \frac{\sqrt{- \frac{1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log{\left (- \frac{c^{2} d^{3} \sqrt{- \frac{1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right )}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right ) \log{\left (\frac{c^{2} d^{3} \sqrt{- \frac{1}{c^{3} d^{7}}} \left (a d - b c\right )^{2} \left (a d + 5 b c\right )}{a^{3} d^{3} + 3 a^{2} b c d^{2} - 9 a b^{2} c^{2} d + 5 b^{3} c^{3}} + x \right )}}{4} + \frac{x \left (3 a b^{2} d - 2 b^{3} c\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.235833, size = 205, normalized size = 1.92 \[ \frac{{\left (5 \, b^{3} c^{3} - 9 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{2 \, \sqrt{c d} c d^{3}} - \frac{b^{3} c^{3} x - 3 \, a b^{2} c^{2} d x + 3 \, a^{2} b c d^{2} x - a^{3} d^{3} x}{2 \,{\left (d x^{2} + c\right )} c d^{3}} + \frac{b^{3} d^{4} x^{3} - 6 \, b^{3} c d^{3} x + 9 \, a b^{2} d^{4} x}{3 \, d^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/(d*x^2 + c)^2,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(d*x/sqrt(c*d))/
(sqrt(c*d)*c*d^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)/((d*x^2 + c)*c*d^3) + 1/3*(b^3*d^4*x^3 - 6*b^3*c*d^3*x + 9*a*b^2*d^4*x)/d^6