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

Optimal. Leaf size=72 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0682524, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}-\frac{(d+e x) (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.27662, size = 60, normalized size = 0.83 \[ - \frac{\left (d + e x\right ) \left (a e - c d x\right )}{2 a c \left (a + c x^{2}\right )} + \frac{\left (a e^{2} + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)*(a*e - c*d*x)/(2*a*c*(a + c*x**2)) + (a*e**2 + c*d**2)*atan(sqrt(c)*x
/sqrt(a))/(2*a**(3/2)*c**(3/2))

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Mathematica [A]  time = 0.111341, size = 77, normalized size = 1.07 \[ \frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}+\frac{-2 a d e-a e^2 x+c d^2 x}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.009, size = 85, normalized size = 1.2 \[{\frac{1}{c{x}^{2}+a} \left ( -{\frac{ \left ( a{e}^{2}-c{d}^{2} \right ) x}{2\,ac}}-{\frac{de}{c}} \right ) }+{\frac{{e}^{2}}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{d}^{2}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^2+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.215002, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a c d^{2} + a^{2} e^{2} +{\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (2 \, a d e -{\left (c d^{2} - a e^{2}\right )} x\right )} \sqrt{-a c}}{4 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{-a c}}, \frac{{\left (a c d^{2} + a^{2} e^{2} +{\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (2 \, a d e -{\left (c d^{2} - a e^{2}\right )} x\right )} \sqrt{a c}}{2 \,{\left (a c^{2} x^{2} + a^{2} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*((a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)*log((2*a*c*x + (c*x^2 - a)*s
qrt(-a*c))/(c*x^2 + a)) - 2*(2*a*d*e - (c*d^2 - a*e^2)*x)*sqrt(-a*c))/((a*c^2*x^
2 + a^2*c)*sqrt(-a*c)), 1/2*((a*c*d^2 + a^2*e^2 + (c^2*d^2 + a*c*e^2)*x^2)*arcta
n(sqrt(a*c)*x/a) - (2*a*d*e - (c*d^2 - a*e^2)*x)*sqrt(a*c))/((a*c^2*x^2 + a^2*c)
*sqrt(a*c))]

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Sympy [A]  time = 2.64737, size = 129, normalized size = 1.79 \[ - \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log{\left (- a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log{\left (a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x \right )}}{4} - \frac{2 a d e + x \left (a e^{2} - c d^{2}\right )}{2 a^{2} c + 2 a c^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209431, size = 93, normalized size = 1.29 \[ \frac{{\left (c d^{2} + a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c} + \frac{c d^{2} x - a x e^{2} - 2 \, a d e}{2 \,{\left (c x^{2} + a\right )} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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