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

Optimal. Leaf size=109 \[ \frac{d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}+\frac{e^3 \log \left (a+c x^2\right )}{2 c^2}-\frac{d e^2 x}{2 a c}-\frac{(d+e x)^2 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.223479, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{3/2}}+\frac{e^3 \log \left (a+c x^2\right )}{2 c^2}-\frac{d e^2 x}{2 a c}-\frac{(d+e x)^2 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e**3*log(a + c*x**2)/(2*c**2) - e**2*Integral(d, x)/(2*a*c) - (d + e*x)**2*(a*e
- c*d*x)/(2*a*c*(a + c*x**2)) + d*(3*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 = 0.144818, size = 107, normalized size = 0.98 \[ \frac{\frac{\sqrt{a} \left (a^2 e^3-3 a c d e (d+e x)+a e^3 \left (a+c x^2\right ) \log \left (a+c x^2\right )+c^2 d^3 x\right )}{a+c x^2}+\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.013, size = 118, normalized size = 1.1 \[{\frac{1}{c{x}^{2}+a} \left ( -{\frac{d \left ( 3\,a{e}^{2}-c{d}^{2} \right ) x}{2\,ac}}+{\frac{e \left ( a{e}^{2}-3\,c{d}^{2} \right ) }{2\,{c}^{2}}} \right ) }+{\frac{{e}^{3}\ln \left ( ac \left ( c{x}^{2}+a \right ) \right ) }{2\,{c}^{2}}}+{\frac{3\,d{e}^{2}}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{d}^{3}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+a)^2,x)

[Out]

(-1/2*d*(3*a*e^2-c*d^2)/a/c*x+1/2*e*(a*e^2-3*c*d^2)/c^2)/(c*x^2+a)+1/2*e^3/c^2*l
n(a*c*(c*x^2+a))+3/2/c/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*d*e^2+1/2/a/(a*c)^(1/
2)*arctan(c*x/(a*c)^(1/2))*d^3

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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)^3/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.217105, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2} +{\left (c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (3 \, a c d^{2} e - a^{2} e^{3} -{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x -{\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{4 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{-a c}}, \frac{{\left (a c^{2} d^{3} + 3 \, a^{2} c d e^{2} +{\left (c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (3 \, a c d^{2} e - a^{2} e^{3} -{\left (c^{2} d^{3} - 3 \, a c d e^{2}\right )} x -{\left (a c e^{3} x^{2} + a^{2} e^{3}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.3561, size = 298, normalized size = 2.73 \[ \left (\frac{e^{3}}{2 c^{2}} - \frac{d \sqrt{- a^{3} c^{5}} \left (3 a e^{2} + c d^{2}\right )}{4 a^{3} c^{4}}\right ) \log{\left (x + \frac{4 a^{2} c^{2} \left (\frac{e^{3}}{2 c^{2}} - \frac{d \sqrt{- a^{3} c^{5}} \left (3 a e^{2} + c d^{2}\right )}{4 a^{3} c^{4}}\right ) - 2 a^{2} e^{3}}{3 a c d e^{2} + c^{2} d^{3}} \right )} + \left (\frac{e^{3}}{2 c^{2}} + \frac{d \sqrt{- a^{3} c^{5}} \left (3 a e^{2} + c d^{2}\right )}{4 a^{3} c^{4}}\right ) \log{\left (x + \frac{4 a^{2} c^{2} \left (\frac{e^{3}}{2 c^{2}} + \frac{d \sqrt{- a^{3} c^{5}} \left (3 a e^{2} + c d^{2}\right )}{4 a^{3} c^{4}}\right ) - 2 a^{2} e^{3}}{3 a c d e^{2} + c^{2} d^{3}} \right )} - \frac{- a^{2} e^{3} + 3 a c d^{2} e + x \left (3 a c d e^{2} - c^{2} d^{3}\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e**3/(2*c**2) - d*sqrt(-a**3*c**5)*(3*a*e**2 + c*d**2)/(4*a**3*c**4))*log(x + (
4*a**2*c**2*(e**3/(2*c**2) - d*sqrt(-a**3*c**5)*(3*a*e**2 + c*d**2)/(4*a**3*c**4
)) - 2*a**2*e**3)/(3*a*c*d*e**2 + c**2*d**3)) + (e**3/(2*c**2) + d*sqrt(-a**3*c*
*5)*(3*a*e**2 + c*d**2)/(4*a**3*c**4))*log(x + (4*a**2*c**2*(e**3/(2*c**2) + d*s
qrt(-a**3*c**5)*(3*a*e**2 + c*d**2)/(4*a**3*c**4)) - 2*a**2*e**3)/(3*a*c*d*e**2
+ c**2*d**3)) - (-a**2*e**3 + 3*a*c*d**2*e + x*(3*a*c*d*e**2 - c**2*d**3))/(2*a*
*2*c**2 + 2*a*c**3*x**2)

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GIAC/XCAS [A]  time = 0.214073, size = 140, normalized size = 1.28 \[ \frac{e^{3}{\rm ln}\left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{{\left (c d^{3} + 3 \, a d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c} + \frac{{\left (c d^{3} - 3 \, a d e^{2}\right )} x - \frac{3 \, a c d^{2} e - a^{2} e^{3}}{c}}{2 \,{\left (c x^{2} + a\right )} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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