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

Optimal. Leaf size=229 \[ -\frac{\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{\left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}-\frac{x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.808237, antiderivative size = 229, 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 \[ -\frac{\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{\left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}-\frac{x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac{x \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}+\frac{x^3 (c d-b e)}{3 c^2}+\frac{e x^4}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.25541, size = 222, normalized size = 0.97 \[ \frac{6 \left (a^2 c^2 e-3 a b^2 c e+2 a b c^2 d+b^4 e-b^3 c d\right ) \log (a+x (b+c x))+\frac{12 \left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-6 c^2 x^2 \left (a c e+b^2 (-e)+b c d\right )-12 c x \left (-2 a b c e+a c^2 d+b^3 e-b^2 c d\right )+4 c^3 x^3 (c d-b e)+3 c^4 e x^4}{12 c^5} \]

Antiderivative was successfully verified.

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

[Out]

(-12*c*(-(b^2*c*d) + a*c^2*d + b^3*e - 2*a*b*c*e)*x - 6*c^2*(b*c*d - b^2*e + a*c
*e)*x^2 + 4*c^3*(c*d - b*e)*x^3 + 3*c^4*e*x^4 + (12*(b^4*c*d - 4*a*b^2*c^2*d + 2
*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*ArcTan[(b + 2*c*x)/Sqrt[-b^2 +
4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(-(b^3*c*d) + 2*a*b*c^2*d + b^4*e - 3*a*b^2*c*e
+ a^2*c^2*e)*Log[a + x*(b + c*x)])/(12*c^5)

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Maple [B]  time = 0.009, size = 445, normalized size = 1.9 \[{\frac{e{x}^{4}}{4\,c}}-{\frac{b{x}^{3}e}{3\,{c}^{2}}}+{\frac{d{x}^{3}}{3\,c}}-{\frac{ae{x}^{2}}{2\,{c}^{2}}}+{\frac{{b}^{2}e{x}^{2}}{2\,{c}^{3}}}-{\frac{{x}^{2}bd}{2\,{c}^{2}}}+2\,{\frac{beax}{{c}^{3}}}-{\frac{adx}{{c}^{2}}}-{\frac{{b}^{3}ex}{{c}^{4}}}+{\frac{{b}^{2}dx}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){a}^{2}e}{2\,{c}^{3}}}-{\frac{3\,\ln \left ( c{x}^{2}+bx+a \right ) a{b}^{2}e}{2\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) abd}{{c}^{3}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{4}e}{2\,{c}^{5}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{3}d}{2\,{c}^{4}}}-5\,{\frac{{a}^{2}be}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{a}^{2}d}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+5\,{\frac{a{b}^{3}e}{{c}^{4}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{a{b}^{2}d}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{5}e}{{c}^{5}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{d{b}^{4}}{{c}^{4}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*e*x^4/c-1/3/c^2*b*x^3*e+1/3*d*x^3/c-1/2*a*e*x^2/c^2+1/2/c^3*b^2*e*x^2-1/2/c^
2*b*d*x^2+2/c^3*a*b*e*x-a*d*x/c^2-1/c^4*b^3*e*x+1/c^3*b^2*d*x+1/2/c^3*ln(c*x^2+b
*x+a)*a^2*e-3/2/c^4*ln(c*x^2+b*x+a)*a*b^2*e+1/c^3*ln(c*x^2+b*x+a)*a*b*d+1/2/c^5*
ln(c*x^2+b*x+a)*b^4*e-1/2/c^4*ln(c*x^2+b*x+a)*b^3*d-5/c^3/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*e+2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)
/(4*a*c-b^2)^(1/2))*a^2*d+5/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(
1/2))*a*b^3*e-4/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*
d-1/c^5/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*e+1/c^4/(4*a*c
-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d*b^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)*x^4/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.296476, size = 1, normalized size = 0. \[ \left [-\frac{6 \,{\left ({\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d -{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (3 \, c^{4} e x^{4} + 4 \,{\left (c^{4} d - b c^{3} e\right )} x^{3} - 6 \,{\left (b c^{3} d -{\left (b^{2} c^{2} - a c^{3}\right )} e\right )} x^{2} + 12 \,{\left ({\left (b^{2} c^{2} - a c^{3}\right )} d -{\left (b^{3} c - 2 \, a b c^{2}\right )} e\right )} x - 6 \,{\left ({\left (b^{3} c - 2 \, a b c^{2}\right )} d -{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{12 \, \sqrt{b^{2} - 4 \, a c} c^{5}}, \frac{12 \,{\left ({\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d -{\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (3 \, c^{4} e x^{4} + 4 \,{\left (c^{4} d - b c^{3} e\right )} x^{3} - 6 \,{\left (b c^{3} d -{\left (b^{2} c^{2} - a c^{3}\right )} e\right )} x^{2} + 12 \,{\left ({\left (b^{2} c^{2} - a c^{3}\right )} d -{\left (b^{3} c - 2 \, a b c^{2}\right )} e\right )} x - 6 \,{\left ({\left (b^{3} c - 2 \, a b c^{2}\right )} d -{\left (b^{4} - 3 \, a b^{2} c + a^{2} c^{2}\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{12 \, \sqrt{-b^{2} + 4 \, a c} c^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/12*(6*((b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*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)) - (3*c^4*e*x^4 + 4*(c^4*d - b*c^3*e)*
x^3 - 6*(b*c^3*d - (b^2*c^2 - a*c^3)*e)*x^2 + 12*((b^2*c^2 - a*c^3)*d - (b^3*c -
2*a*b*c^2)*e)*x - 6*((b^3*c - 2*a*b*c^2)*d - (b^4 - 3*a*b^2*c + a^2*c^2)*e)*log
(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/(sqrt(b^2 - 4*a*c)*c^5), 1/12*(12*((b^4*c
- 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e)*arctan(-sqrt(-
b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (3*c^4*e*x^4 + 4*(c^4*d - b*c^3*e)*x^3
- 6*(b*c^3*d - (b^2*c^2 - a*c^3)*e)*x^2 + 12*((b^2*c^2 - a*c^3)*d - (b^3*c - 2*
a*b*c^2)*e)*x - 6*((b^3*c - 2*a*b*c^2)*d - (b^4 - 3*a*b^2*c + a^2*c^2)*e)*log(c*
x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)]

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Sympy [A]  time = 10.639, size = 1088, normalized size = 4.75 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.272872, size = 333, normalized size = 1.45 \[ \frac{3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 4 \, b c^{2} x^{3} e - 6 \, b c^{2} d x^{2} + 6 \, b^{2} c x^{2} e - 6 \, a c^{2} x^{2} e + 12 \, b^{2} c d x - 12 \, a c^{2} d x - 12 \, b^{3} x e + 24 \, a b c x e}{12 \, c^{4}} - \frac{{\left (b^{3} c d - 2 \, a b c^{2} d - b^{4} e + 3 \, a b^{2} c e - a^{2} c^{2} e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac{{\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b c^{2} e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(3*c^3*x^4*e + 4*c^3*d*x^3 - 4*b*c^2*x^3*e - 6*b*c^2*d*x^2 + 6*b^2*c*x^2*e
- 6*a*c^2*x^2*e + 12*b^2*c*d*x - 12*a*c^2*d*x - 12*b^3*x*e + 24*a*b*c*x*e)/c^4 -
1/2*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*e)*ln(c*x^2 + b*x +
a)/c^5 + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*
c^2*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)