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

Optimal. Leaf size=59 \[ \frac{\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{d e \log \left (a+c x^2\right )}{c}+\frac{e^2 x}{c} \]

[Out]

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

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Rubi [A]  time = 0.12401, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ \frac{\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{d e \log \left (a+c x^2\right )}{c}+\frac{e^2 x}{c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(a + c*x^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ e^{2} \int \frac{1}{c}\, dx + \frac{d e \log{\left (a + c x^{2} \right )}}{c} - \frac{\left (a e^{2} - c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{a} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0736479, size = 56, normalized size = 0.95 \[ \frac{\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{e \left (d \log \left (a+c x^2\right )+e x\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 65, normalized size = 1.1 \[{\frac{{e}^{2}x}{c}}+{\frac{de\ln \left ( c{x}^{2}+a \right ) }{c}}-{\frac{a{e}^{2}}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{{d}^{2}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^2+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.21496, size = 1, normalized size = 0.02 \[ \left [-\frac{{\left (c d^{2} - a e^{2}\right )} \log \left (-\frac{2 \, a c x -{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (e^{2} x + d e \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{2 \, \sqrt{-a c} c}, \frac{{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (e^{2} x + d e \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{\sqrt{a c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.37397, size = 185, normalized size = 3.14 \[ \left (\frac{d e}{c} - \frac{\sqrt{- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{d e}{c} - \frac{\sqrt{- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) + 2 a d e}{a e^{2} - c d^{2}} \right )} + \left (\frac{d e}{c} + \frac{\sqrt{- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{d e}{c} + \frac{\sqrt{- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) + 2 a d e}{a e^{2} - c d^{2}} \right )} + \frac{e^{2} x}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d*e/c - sqrt(-a*c**3)*(a*e**2 - c*d**2)/(2*a*c**3))*log(x + (-2*a*c*(d*e/c - sq
rt(-a*c**3)*(a*e**2 - c*d**2)/(2*a*c**3)) + 2*a*d*e)/(a*e**2 - c*d**2)) + (d*e/c
+ sqrt(-a*c**3)*(a*e**2 - c*d**2)/(2*a*c**3))*log(x + (-2*a*c*(d*e/c + sqrt(-a*
c**3)*(a*e**2 - c*d**2)/(2*a*c**3)) + 2*a*d*e)/(a*e**2 - c*d**2)) + e**2*x/c

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GIAC/XCAS [A]  time = 0.212664, size = 70, normalized size = 1.19 \[ \frac{d e{\rm ln}\left (c x^{2} + a\right )}{c} + \frac{x e^{2}}{c} + \frac{{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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