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

Optimal. Leaf size=113 \[ \frac{\left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{2 a d e-x \left (a e^2+3 c d^2\right )}{8 a^2 c \left (a+c x^2\right )}-\frac{(d+e x) (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

[Out]

-((a*e - c*d*x)*(d + e*x))/(4*a*c*(a + c*x^2)^2) - (2*a*d*e - (3*c*d^2 + a*e^2)*
x)/(8*a^2*c*(a + c*x^2)) + ((3*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^
(5/2)*c^(3/2))

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Rubi [A]  time = 0.140779, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{\left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{2 a d e-x \left (a e^2+3 c d^2\right )}{8 a^2 c \left (a+c x^2\right )}-\frac{(d+e x) (a e-c d x)}{4 a c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-((a*e - c*d*x)*(d + e*x))/(4*a*c*(a + c*x^2)^2) - (2*a*d*e - (3*c*d^2 + a*e^2)*
x)/(8*a^2*c*(a + c*x^2)) + ((3*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^
(5/2)*c^(3/2))

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Rubi in Sympy [A]  time = 20.834, size = 97, normalized size = 0.86 \[ - \frac{\left (d + e x\right ) \left (a e - c d x\right )}{4 a c \left (a + c x^{2}\right )^{2}} + \frac{- 2 a d e + x \left (a e^{2} + 3 c d^{2}\right )}{8 a^{2} c \left (a + c x^{2}\right )} + \frac{\left (a e^{2} + 3 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 a^{\frac{5}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(d + e*x)*(a*e - c*d*x)/(4*a*c*(a + c*x**2)**2) + (-2*a*d*e + x*(a*e**2 + 3*c*d
**2))/(8*a**2*c*(a + c*x**2)) + (a*e**2 + 3*c*d**2)*atan(sqrt(c)*x/sqrt(a))/(8*a
**(5/2)*c**(3/2))

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Mathematica [A]  time = 0.133139, size = 101, normalized size = 0.89 \[ \frac{\left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}+\frac{-a^2 e (4 d+e x)+a c x \left (5 d^2+e^2 x^2\right )+3 c^2 d^2 x^3}{8 a^2 c \left (a+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(3*c^2*d^2*x^3 - a^2*e*(4*d + e*x) + a*c*x*(5*d^2 + e^2*x^2))/(8*a^2*c*(a + c*x^
2)^2) + ((3*c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(3/2))

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Maple [A]  time = 0.011, size = 108, normalized size = 1. \[{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( a{e}^{2}+3\,c{d}^{2} \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{ \left ( a{e}^{2}-5\,c{d}^{2} \right ) x}{8\,ac}}-{\frac{de}{2\,c}} \right ) }+{\frac{{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}}{8\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(c*x^2+a)^3,x)

[Out]

(1/8*(a*e^2+3*c*d^2)/a^2*x^3-1/8*(a*e^2-5*c*d^2)/a/c*x-1/2*d*e/c)/(c*x^2+a)^2+1/
8/a/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*e^2+3/8/a^2/(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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221997, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (3 \, a^{2} c d^{2} + a^{3} e^{2} +{\left (3 \, c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4} + 2 \,{\left (3 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (4 \, a^{2} d e -{\left (3 \, c^{2} d^{2} + a c e^{2}\right )} x^{3} -{\left (5 \, a c d^{2} - a^{2} e^{2}\right )} x\right )} \sqrt{-a c}}{16 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt{-a c}}, \frac{{\left (3 \, a^{2} c d^{2} + a^{3} e^{2} +{\left (3 \, c^{3} d^{2} + a c^{2} e^{2}\right )} x^{4} + 2 \,{\left (3 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (4 \, a^{2} d e -{\left (3 \, c^{2} d^{2} + a c e^{2}\right )} x^{3} -{\left (5 \, a c d^{2} - a^{2} e^{2}\right )} x\right )} \sqrt{a c}}{8 \,{\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*((3*a^2*c*d^2 + a^3*e^2 + (3*c^3*d^2 + a*c^2*e^2)*x^4 + 2*(3*a*c^2*d^2 + a
^2*c*e^2)*x^2)*log((2*a*c*x + (c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) - 2*(4*a^2*d*
e - (3*c^2*d^2 + a*c*e^2)*x^3 - (5*a*c*d^2 - a^2*e^2)*x)*sqrt(-a*c))/((a^2*c^3*x
^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(-a*c)), 1/8*((3*a^2*c*d^2 + a^3*e^2 + (3*c^3*d^
2 + a*c^2*e^2)*x^4 + 2*(3*a*c^2*d^2 + a^2*c*e^2)*x^2)*arctan(sqrt(a*c)*x/a) - (4
*a^2*d*e - (3*c^2*d^2 + a*c*e^2)*x^3 - (5*a*c*d^2 - a^2*e^2)*x)*sqrt(a*c))/((a^2
*c^3*x^4 + 2*a^3*c^2*x^2 + a^4*c)*sqrt(a*c))]

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Sympy [A]  time = 3.7177, size = 172, normalized size = 1.52 \[ - \frac{\sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + 3 c d^{2}\right ) \log{\left (- a^{3} c \sqrt{- \frac{1}{a^{5} c^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + 3 c d^{2}\right ) \log{\left (a^{3} c \sqrt{- \frac{1}{a^{5} c^{3}}} + x \right )}}{16} + \frac{- 4 a^{2} d e + x^{3} \left (a c e^{2} + 3 c^{2} d^{2}\right ) + x \left (- a^{2} e^{2} + 5 a c d^{2}\right )}{8 a^{4} c + 16 a^{3} c^{2} x^{2} + 8 a^{2} c^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-1/(a**5*c**3))*(a*e**2 + 3*c*d**2)*log(-a**3*c*sqrt(-1/(a**5*c**3)) + x)/
16 + sqrt(-1/(a**5*c**3))*(a*e**2 + 3*c*d**2)*log(a**3*c*sqrt(-1/(a**5*c**3)) +
x)/16 + (-4*a**2*d*e + x**3*(a*c*e**2 + 3*c**2*d**2) + x*(-a**2*e**2 + 5*a*c*d**
2))/(8*a**4*c + 16*a**3*c**2*x**2 + 8*a**2*c**3*x**4)

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GIAC/XCAS [A]  time = 0.212627, size = 128, normalized size = 1.13 \[ \frac{{\left (3 \, c d^{2} + a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c} + \frac{3 \, c^{2} d^{2} x^{3} + a c x^{3} e^{2} + 5 \, a c d^{2} x - a^{2} x e^{2} - 4 \, a^{2} d e}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(3*c*d^2 + a*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c) + 1/8*(3*c^2*d^2*x
^3 + a*c*x^3*e^2 + 5*a*c*d^2*x - a^2*x*e^2 - 4*a^2*d*e)/((c*x^2 + a)^2*a^2*c)