3.2172 \(\int \frac{(d+e x)^3}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=151 \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^2}{2 c} \]

[Out]

(e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x^2)/(2*c) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2
- c*e*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a
*c]) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^3
)

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Rubi [A]  time = 0.395879, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}+\frac{e^2 x (3 c d-b e)}{c^2}+\frac{e^3 x^2}{2 c} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*(3*c*d - b*e)*x)/c^2 + (e^3*x^2)/(2*c) - ((2*c*d - b*e)*(c^2*d^2 + b^2*e^2
- c*e*(b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^3*Sqrt[b^2 - 4*a
*c]) + (e*(3*c^2*d^2 + b^2*e^2 - c*e*(3*b*d + a*e))*Log[a + b*x + c*x^2])/(2*c^3
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**2*(b*e - 3*c*d)*Integral(c**(-2), x) + e**3*Integral(x, x)/c + e*(-a*c*e**2
+ b**2*e**2 - 3*b*c*d*e + 3*c**2*d**2)*log(a + b*x + c*x**2)/(2*c**3) + (b*e - 2
*c*d)*(-3*a*c*e**2 + b**2*e**2 - b*c*d*e + c**2*d**2)*atanh((b + 2*c*x)/sqrt(-4*
a*c + b**2))/(c**3*sqrt(-4*a*c + b**2))

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Mathematica [A]  time = 0.233673, size = 148, normalized size = 0.98 \[ \frac{e \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac{2 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+2 c e^2 x (3 c d-b e)+c^2 e^3 x^2}{2 c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 366, normalized size = 2.4 \[{\frac{{e}^{3}{x}^{2}}{2\,c}}-{\frac{b{e}^{3}x}{{c}^{2}}}+3\,{\frac{d{e}^{2}x}{c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) a{e}^{3}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{e}^{3}}{2\,{c}^{3}}}-{\frac{3\,d\ln \left ( c{x}^{2}+bx+a \right ){e}^{2}b}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ){d}^{2}e}{2\,c}}+3\,{\frac{ab{e}^{3}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-6\,{\frac{ad{e}^{2}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}{e}^{3}}{{c}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+3\,{\frac{{b}^{2}d{e}^{2}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-3\,{\frac{b{d}^{2}e}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3/(c*x^2+b*x+a),x)

[Out]

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

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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 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229412, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} -{\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (c^{2} e^{3} x^{2} + 2 \,{\left (3 \, c^{2} d e^{2} - b c e^{3}\right )} x +{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} +{\left (b^{2} - a c\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} c^{3}}, \frac{2 \,{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d e^{2} -{\left (b^{3} - 3 \, a b c\right )} e^{3}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (c^{2} e^{3} x^{2} + 2 \,{\left (3 \, c^{2} d e^{2} - b c e^{3}\right )} x +{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} +{\left (b^{2} - a c\right )} e^{3}\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 11.3155, size = 892, normalized size = 5.91 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.206756, size = 217, normalized size = 1.44 \[ \frac{c x^{2} e^{3} + 6 \, c d x e^{2} - 2 \, b x e^{3}}{2 \, c^{2}} + \frac{{\left (3 \, c^{2} d^{2} e - 3 \, b c d e^{2} + b^{2} e^{3} - a c e^{3}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - 6 \, a c^{2} d e^{2} - b^{3} e^{3} + 3 \, a b c e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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