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

Optimal. Leaf size=63 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{5/2}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^3}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0923967, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{5/2}}-\frac{b x (b c-2 a d)}{d^2}+\frac{b^2 x^3}{3 d} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.079031, size = 59, normalized size = 0.94 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{\sqrt{c} d^{5/2}}+\frac{b x \left (6 a d-3 b c+b d x^2\right )}{3 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.001, size = 95, normalized size = 1.5 \[{\frac{{b}^{2}{x}^{3}}{3\,d}}+2\,{\frac{abx}{d}}-{\frac{x{b}^{2}c}{{d}^{2}}}+{{a}^{2}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-2\,{\frac{abc}{d\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{{b}^{2}{c}^{2}}{{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/(d*x^2+c),x)

[Out]

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

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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)^2/(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.244299, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{2 \, c d x +{\left (d x^{2} - c\right )} \sqrt{-c d}}{d x^{2} + c}\right ) + 2 \,{\left (b^{2} d x^{3} - 3 \,{\left (b^{2} c - 2 \, a b d\right )} x\right )} \sqrt{-c d}}{6 \, \sqrt{-c d} d^{2}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) +{\left (b^{2} d x^{3} - 3 \,{\left (b^{2} c - 2 \, a b d\right )} x\right )} \sqrt{c d}}{3 \, \sqrt{c d} d^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.12824, size = 172, normalized size = 2.73 \[ \frac{b^{2} x^{3}}{3 d} - \frac{\sqrt{- \frac{1}{c d^{5}}} \left (a d - b c\right )^{2} \log{\left (- \frac{c d^{2} \sqrt{- \frac{1}{c d^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c d^{5}}} \left (a d - b c\right )^{2} \log{\left (\frac{c d^{2} \sqrt{- \frac{1}{c d^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{x \left (2 a b d - b^{2} c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.226779, size = 97, normalized size = 1.54 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{2}} + \frac{b^{2} d^{2} x^{3} - 3 \, b^{2} c d x + 6 \, a b d^{2} x}{3 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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