3.495 \(\int \frac{(d+e x)^4}{\left (a+c x^2\right )^2} \, dx\)

Optimal. Leaf size=149 \[ \frac{\left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}-\frac{3 e^2 x \left (c d^2-a e^2\right )}{2 a c^2}+\frac{2 d e^3 \log \left (a+c x^2\right )}{c^2}-\frac{d e^3 x^2}{2 a c}-\frac{(d+e x)^3 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

[Out]

(-3*e^2*(c*d^2 - a*e^2)*x)/(2*a*c^2) - (d*e^3*x^2)/(2*a*c) - ((a*e - c*d*x)*(d +
e*x)^3)/(2*a*c*(a + c*x^2)) + ((c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(Sq
rt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(5/2)) + (2*d*e^3*Log[a + c*x^2])/c^2

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Rubi [A]  time = 0.309651, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294 \[ \frac{\left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}-\frac{3 e^2 x \left (c d^2-a e^2\right )}{2 a c^2}+\frac{2 d e^3 \log \left (a+c x^2\right )}{c^2}-\frac{d e^3 x^2}{2 a c}-\frac{(d+e x)^3 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(a + c*x^2)^2,x]

[Out]

(-3*e^2*(c*d^2 - a*e^2)*x)/(2*a*c^2) - (d*e^3*x^2)/(2*a*c) - ((a*e - c*d*x)*(d +
e*x)^3)/(2*a*c*(a + c*x^2)) + ((c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(Sq
rt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(5/2)) + (2*d*e^3*Log[a + c*x^2])/c^2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 d e^{3} \log{\left (a + c x^{2} \right )}}{c^{2}} - \frac{d e^{3} \int x\, dx}{a c} - \frac{\left (d + e x\right )^{3} \left (a e - c d x\right )}{2 a c \left (a + c x^{2}\right )} + \frac{3 e^{2} x \left (a e^{2} - c d^{2}\right )}{2 a c^{2}} - \frac{\left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**4/(c*x**2+a)**2,x)

[Out]

2*d*e**3*log(a + c*x**2)/c**2 - d*e**3*Integral(x, x)/(a*c) - (d + e*x)**3*(a*e
- c*d*x)/(2*a*c*(a + c*x**2)) + 3*e**2*x*(a*e**2 - c*d**2)/(2*a*c**2) - (3*a**2*
e**4 - 6*a*c*d**2*e**2 - c**2*d**4)*atan(sqrt(c)*x/sqrt(a))/(2*a**(3/2)*c**(5/2)
)

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Mathematica [A]  time = 0.156934, size = 137, normalized size = 0.92 \[ \frac{a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)+c^2 d^4 x}{2 a c^2 \left (a+c x^2\right )}+\frac{\left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}+\frac{2 d e^3 \log \left (a+c x^2\right )}{c^2}+\frac{e^4 x}{c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^4/(a + c*x^2)^2,x]

[Out]

(e^4*x)/c^2 + (c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x))/(2*a
*c^2*(a + c*x^2)) + ((c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sq
rt[a]])/(2*a^(3/2)*c^(5/2)) + (2*d*e^3*Log[a + c*x^2])/c^2

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Maple [A]  time = 0.011, size = 194, normalized size = 1.3 \[{\frac{{e}^{4}x}{{c}^{2}}}+{\frac{ax{e}^{4}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-3\,{\frac{x{d}^{2}{e}^{2}}{c \left ( c{x}^{2}+a \right ) }}+{\frac{x{d}^{4}}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}+2\,{\frac{ad{e}^{3}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-2\,{\frac{{d}^{3}e}{c \left ( c{x}^{2}+a \right ) }}+2\,{\frac{d{e}^{3}\ln \left ( a \left ( c{x}^{2}+a \right ) \right ) }{{c}^{2}}}-{\frac{3\,a{e}^{4}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+3\,{\frac{{d}^{2}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{{d}^{4}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+a)^2,x)

[Out]

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

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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)^4/(c*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224918, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (a c^{2} d^{4} + 6 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4} +{\left (c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} - 3 \, a^{2} c e^{4}\right )} x^{2}\right )} \log \left (-\frac{2 \, a c x -{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) - 2 \,{\left (2 \, a c e^{4} x^{3} - 4 \, a c d^{3} e + 4 \, a^{2} d e^{3} +{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} x + 4 \,{\left (a c d e^{3} x^{2} + a^{2} d e^{3}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{4 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{-a c}}, \frac{{\left (a c^{2} d^{4} + 6 \, a^{2} c d^{2} e^{2} - 3 \, a^{3} e^{4} +{\left (c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} - 3 \, a^{2} c e^{4}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (2 \, a c e^{4} x^{3} - 4 \, a c d^{3} e + 4 \, a^{2} d e^{3} +{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} x + 4 \,{\left (a c d e^{3} x^{2} + a^{2} d e^{3}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + a)^2,x, algorithm="fricas")

[Out]

[-1/4*((a*c^2*d^4 + 6*a^2*c*d^2*e^2 - 3*a^3*e^4 + (c^3*d^4 + 6*a*c^2*d^2*e^2 - 3
*a^2*c*e^4)*x^2)*log(-(2*a*c*x - (c*x^2 - a)*sqrt(-a*c))/(c*x^2 + a)) - 2*(2*a*c
*e^4*x^3 - 4*a*c*d^3*e + 4*a^2*d*e^3 + (c^2*d^4 - 6*a*c*d^2*e^2 + 3*a^2*e^4)*x +
4*(a*c*d*e^3*x^2 + a^2*d*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^4 + 6*a^2*c*d^2*e^2 - 3*a^3*e^4 + (c^3*d^4 + 6*a*c^
2*d^2*e^2 - 3*a^2*c*e^4)*x^2)*arctan(sqrt(a*c)*x/a) + (2*a*c*e^4*x^3 - 4*a*c*d^3
*e + 4*a^2*d*e^3 + (c^2*d^4 - 6*a*c*d^2*e^2 + 3*a^2*e^4)*x + 4*(a*c*d*e^3*x^2 +
a^2*d*e^3)*log(c*x^2 + a))*sqrt(a*c))/((a*c^3*x^2 + a^2*c^2)*sqrt(a*c))]

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Sympy [A]  time = 6.26926, size = 403, normalized size = 2.7 \[ \left (\frac{2 d e^{3}}{c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) \log{\left (x + \frac{- 4 a^{2} c^{2} \left (\frac{2 d e^{3}}{c^{2}} - \frac{\sqrt{- a^{3} c^{5}} \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) + 8 a^{2} d e^{3}}{3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}} \right )} + \left (\frac{2 d e^{3}}{c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) \log{\left (x + \frac{- 4 a^{2} c^{2} \left (\frac{2 d e^{3}}{c^{2}} + \frac{\sqrt{- a^{3} c^{5}} \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) + 8 a^{2} d e^{3}}{3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}} \right )} + \frac{4 a^{2} d e^{3} - 4 a c d^{3} e + x \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} + \frac{e^{4} x}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**4/(c*x**2+a)**2,x)

[Out]

(2*d*e**3/c**2 - sqrt(-a**3*c**5)*(3*a**2*e**4 - 6*a*c*d**2*e**2 - c**2*d**4)/(4
*a**3*c**5))*log(x + (-4*a**2*c**2*(2*d*e**3/c**2 - sqrt(-a**3*c**5)*(3*a**2*e**
4 - 6*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**5)) + 8*a**2*d*e**3)/(3*a**2*e**4 -
6*a*c*d**2*e**2 - c**2*d**4)) + (2*d*e**3/c**2 + sqrt(-a**3*c**5)*(3*a**2*e**4 -
6*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**5))*log(x + (-4*a**2*c**2*(2*d*e**3/c**
2 + sqrt(-a**3*c**5)*(3*a**2*e**4 - 6*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**5))
+ 8*a**2*d*e**3)/(3*a**2*e**4 - 6*a*c*d**2*e**2 - c**2*d**4)) + (4*a**2*d*e**3 -
4*a*c*d**3*e + x*(a**2*e**4 - 6*a*c*d**2*e**2 + c**2*d**4))/(2*a**2*c**2 + 2*a*
c**3*x**2) + e**4*x/c**2

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GIAC/XCAS [A]  time = 0.212965, size = 177, normalized size = 1.19 \[ \frac{2 \, d e^{3}{\rm ln}\left (c x^{2} + a\right )}{c^{2}} + \frac{x e^{4}}{c^{2}} + \frac{{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c^{2}} - \frac{4 \, a c d^{3} e - 4 \, a^{2} d e^{3} -{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^4/(c*x^2 + a)^2,x, algorithm="giac")

[Out]

2*d*e^3*ln(c*x^2 + a)/c^2 + x*e^4/c^2 + 1/2*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4
)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a*c^2) - 1/2*(4*a*c*d^3*e - 4*a^2*d*e^3 - (c^
2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*x)/((c*x^2 + a)*a*c^2)