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

Optimal. Leaf size=169 \[ \frac{\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac{\left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{x^2 (c d-b e)}{2 c^2}+\frac{e x^3}{3 c} \]

[Out]

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

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Rubi [A]  time = 0.4691, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{\left (-2 a^2 c^2 e+4 a b^2 c e-3 a b c^2 d+b^4 (-e)+b^3 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^4 \sqrt{b^2-4 a c}}-\frac{x \left (a c e+b^2 (-e)+b c d\right )}{c^3}+\frac{\left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac{x^2 (c d-b e)}{2 c^2}+\frac{e x^3}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.179093, size = 165, normalized size = 0.98 \[ \frac{\frac{6 \left (2 a^2 c^2 e-4 a b^2 c e+3 a b c^2 d+b^4 e-b^3 c d\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-6 c x \left (a c e+b^2 (-e)+b c d\right )-3 \left (-2 a b c e+a c^2 d+b^3 e-b^2 c d\right ) \log (a+x (b+c x))+3 c^2 x^2 (c d-b e)+2 c^3 e x^3}{6 c^4} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.007, size = 335, normalized size = 2. \[{\frac{e{x}^{3}}{3\,c}}-{\frac{b{x}^{2}e}{2\,{c}^{2}}}+{\frac{d{x}^{2}}{2\,c}}-{\frac{aex}{{c}^{2}}}+{\frac{{b}^{2}ex}{{c}^{3}}}-{\frac{bdx}{{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) abe}{{c}^{3}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) ad}{2\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}e}{2\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}d}{2\,{c}^{3}}}+2\,{\frac{{a}^{2}e}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{ae{b}^{2}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+3\,{\frac{bda}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}e}{{c}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{d{b}^{3}}{{c}^{3}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.298734, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d -{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (2 \, c^{3} e x^{3} + 3 \,{\left (c^{3} d - b c^{2} e\right )} x^{2} - 6 \,{\left (b c^{2} d -{\left (b^{2} c - a c^{2}\right )} e\right )} x + 3 \,{\left ({\left (b^{2} c - a c^{2}\right )} d -{\left (b^{3} - 2 \, a b c\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{6 \, \sqrt{b^{2} - 4 \, a c} c^{4}}, -\frac{6 \,{\left ({\left (b^{3} c - 3 \, a b c^{2}\right )} d -{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, c^{3} e x^{3} + 3 \,{\left (c^{3} d - b c^{2} e\right )} x^{2} - 6 \,{\left (b c^{2} d -{\left (b^{2} c - a c^{2}\right )} e\right )} x + 3 \,{\left ({\left (b^{2} c - a c^{2}\right )} d -{\left (b^{3} - 2 \, a b c\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{6 \, \sqrt{-b^{2} + 4 \, a c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.36567, size = 835, normalized size = 4.94 \[ \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log{\left (x + \frac{- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) \log{\left (x + \frac{- 3 a^{2} b c e + 2 a^{2} c^{2} d + a b^{3} e - a b^{2} c d + 4 a c^{4} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right ) - b^{2} c^{3} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d\right )}{2 c^{4} \left (4 a c - b^{2}\right )} + \frac{2 a b c e - a c^{2} d - b^{3} e + b^{2} c d}{2 c^{4}}\right )}{2 a^{2} c^{2} e - 4 a b^{2} c e + 3 a b c^{2} d + b^{4} e - b^{3} c d} \right )} + \frac{e x^{3}}{3 c} - \frac{x^{2} \left (b e - c d\right )}{2 c^{2}} - \frac{x \left (a c e - b^{2} e + b c d\right )}{c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.273742, size = 240, normalized size = 1.42 \[ \frac{2 \, c^{2} x^{3} e + 3 \, c^{2} d x^{2} - 3 \, b c x^{2} e - 6 \, b c d x + 6 \, b^{2} x e - 6 \, a c x e}{6 \, c^{3}} + \frac{{\left (b^{2} c d - a c^{2} d - b^{3} e + 2 \, a b c e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{4}} - \frac{{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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