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

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

[Out]

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

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Rubi [A]  time = 0.0946087, 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 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}+\frac{d x (2 b c-a d)}{b^2}+\frac{d^2 x^3}{3 b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0833748, size = 59, normalized size = 0.94 \[ \frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} b^{5/2}}+\frac{d x \left (-3 a d+6 b c+b d x^2\right )}{3 b^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 95, normalized size = 1.5 \[{\frac{{d}^{2}{x}^{3}}{3\,b}}-{\frac{a{d}^{2}x}{{b}^{2}}}+2\,{\frac{dxc}{b}}+{\frac{{a}^{2}{d}^{2}}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-2\,{\frac{acd}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{{c}^{2}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^2/(b*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.210575, size = 1, normalized size = 0.02 \[ \left [\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (\frac{2 \, a b x +{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (b d^{2} x^{3} + 3 \,{\left (2 \, b c d - a d^{2}\right )} x\right )} \sqrt{-a b}}{6 \, \sqrt{-a b} b^{2}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (b d^{2} x^{3} + 3 \,{\left (2 \, b c d - a d^{2}\right )} x\right )} \sqrt{a b}}{3 \, \sqrt{a b} b^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.16255, size = 172, normalized size = 2.73 \[ - \frac{\sqrt{- \frac{1}{a b^{5}}} \left (a d - b c\right )^{2} \log{\left (- \frac{a b^{2} \sqrt{- \frac{1}{a b^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{a b^{5}}} \left (a d - b c\right )^{2} \log{\left (\frac{a b^{2} \sqrt{- \frac{1}{a b^{5}}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{d^{2} x^{3}}{3 b} - \frac{x \left (a d^{2} - 2 b c d\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23178, size = 97, normalized size = 1.54 \[ \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{2}} + \frac{b^{2} d^{2} x^{3} + 6 \, b^{2} c d x - 3 \, a b d^{2} x}{3 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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