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

Optimal. Leaf size=123 \[ \frac{\left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2}+\frac{e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac{2 d e^3 x^2}{c}+\frac{e^4 x^3}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.24028, antiderivative size = 123, 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 \[ \frac{\left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{2 d e \left (c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^2}+\frac{e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac{2 d e^3 x^2}{c}+\frac{e^4 x^3}{3 c} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^4/(a + c*x^2),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - e^{2} \left (a e^{2} - 6 c d^{2}\right ) \int \frac{1}{c^{2}}\, dx + \frac{4 d e^{3} \int x\, dx}{c} + \frac{e^{4} x^{3}}{3 c} - \frac{2 d e \left (a e^{2} - c d^{2}\right ) \log{\left (a + c x^{2} \right )}}{c^{2}} + \frac{\left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{\sqrt{a} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.169623, size = 111, normalized size = 0.9 \[ \frac{\left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{e \left (6 \left (c d^3-a d e^2\right ) \log \left (a+c x^2\right )-3 a e^3 x+c e x \left (18 d^2+6 d e x+e^2 x^2\right )\right )}{3 c^2} \]

Antiderivative was successfully verified.

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

[Out]

((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*c^(5/
2)) + (e*(-3*a*e^3*x + c*e*x*(18*d^2 + 6*d*e*x + e^2*x^2) + 6*(c*d^3 - a*d*e^2)*
Log[a + c*x^2]))/(3*c^2)

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Maple [A]  time = 0.008, size = 150, normalized size = 1.2 \[{\frac{{e}^{4}{x}^{3}}{3\,c}}+2\,{\frac{d{e}^{3}{x}^{2}}{c}}-{\frac{{e}^{4}ax}{{c}^{2}}}+6\,{\frac{{d}^{2}{e}^{2}x}{c}}-2\,{\frac{\ln \left ( c{x}^{2}+a \right ) ad{e}^{3}}{{c}^{2}}}+2\,{\frac{\ln \left ( c{x}^{2}+a \right ){d}^{3}e}{c}}+{\frac{{a}^{2}{e}^{4}}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-6\,{\frac{{d}^{2}{e}^{2}a}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{{d}^{4}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^4/(c*x^2+a),x)

[Out]

1/3*e^4*x^3/c+2*d*e^3*x^2/c-e^4/c^2*a*x+6*e^2/c*d^2*x-2/c^2*ln(c*x^2+a)*a*d*e^3+
2/c*ln(c*x^2+a)*d^3*e+1/c^2/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))*a^2*e^4-6/c/(a*c
)^(1/2)*arctan(c*x/(a*c)^(1/2))*a*d^2*e^2+1/(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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.226735, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (\frac{2 \, a c x +{\left (c x^{2} - a\right )} \sqrt{-a c}}{c x^{2} + a}\right ) + 2 \,{\left (c e^{4} x^{3} + 6 \, c d e^{3} x^{2} + 3 \,{\left (6 \, c d^{2} e^{2} - a e^{4}\right )} x + 6 \,{\left (c d^{3} e - a d e^{3}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{-a c}}{6 \, \sqrt{-a c} c^{2}}, \frac{3 \,{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (c e^{4} x^{3} + 6 \, c d e^{3} x^{2} + 3 \,{\left (6 \, c d^{2} e^{2} - a e^{4}\right )} x + 6 \,{\left (c d^{3} e - a d e^{3}\right )} \log \left (c x^{2} + a\right )\right )} \sqrt{a c}}{3 \, \sqrt{a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 4.14388, size = 401, normalized size = 3.26 \[ \left (- \frac{2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} - \frac{\sqrt{- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) \log{\left (x + \frac{4 a^{2} d e^{3} + 2 a c^{2} \left (- \frac{2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} - \frac{\sqrt{- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) - 4 a c d^{3} e}{a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}} \right )} + \left (- \frac{2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} + \frac{\sqrt{- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) \log{\left (x + \frac{4 a^{2} d e^{3} + 2 a c^{2} \left (- \frac{2 d e \left (a e^{2} - c d^{2}\right )}{c^{2}} + \frac{\sqrt{- a c^{5}} \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a c^{5}}\right ) - 4 a c d^{3} e}{a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}} \right )} + \frac{2 d e^{3} x^{2}}{c} + \frac{e^{4} x^{3}}{3 c} - \frac{x \left (a e^{4} - 6 c d^{2} e^{2}\right )}{c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.213275, size = 153, normalized size = 1.24 \[ \frac{2 \,{\left (c d^{3} e - a d e^{3}\right )}{\rm ln}\left (c x^{2} + a\right )}{c^{2}} + \frac{{\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{c^{2} x^{3} e^{4} + 6 \, c^{2} d x^{2} e^{3} + 18 \, c^{2} d^{2} x e^{2} - 3 \, a c x e^{4}}{3 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*d^3*e - a*d*e^3)*ln(c*x^2 + a)/c^2 + (c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*ar
ctan(c*x/sqrt(a*c))/(sqrt(a*c)*c^2) + 1/3*(c^2*x^3*e^4 + 6*c^2*d*x^2*e^3 + 18*c^
2*d^2*x*e^2 - 3*a*c*x*e^4)/c^3