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

Optimal. Leaf size=132 \[ \frac{x \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (2 a c (2 c d-3 b e)+b^3 e\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{e \log \left (a+b x+c x^2\right )}{2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.248801, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{x \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (2 a c (2 c d-3 b e)+b^3 e\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{e \log \left (a+b x+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.9066, size = 126, normalized size = 0.95 \[ - \frac{x \left (a \left (b e - 2 c d\right ) + x \left (- 2 a c e + b^{2} e - b c d\right )\right )}{c \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} + \frac{e \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} + \frac{\left (- 6 a b c e + 4 a c^{2} d + b^{3} e\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.386851, size = 146, normalized size = 1.11 \[ \frac{-\frac{2 \left (2 a^2 c e+a \left (b^2 (-e)+b c (d+3 e x)-2 c^2 d x\right )+b^2 x (c d-b e)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{2 \left (2 a c (2 c d-3 b e)+b^3 e\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+e \log (a+x (b+c x))}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 526, normalized size = 4. \[{\frac{1}{c{x}^{2}+bx+a} \left ({\frac{ \left ( 3\,abce-2\,a{c}^{2}d-{b}^{3}e+{b}^{2}cd \right ) x}{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{a \left ( 2\,ace-{b}^{2}e+bcd \right ) }{{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+2\,{\frac{\ln \left ( c \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ) ae}{c \left ( 4\,ac-{b}^{2} \right ) }}-{\frac{\ln \left ( c \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) \right ){b}^{2}e}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }}-6\,{\frac{bea}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}\arctan \left ({\frac{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) x+bc \left ( 4\,ac-{b}^{2} \right ) }{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}} \right ) }+4\,{\frac{acd}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}\arctan \left ({\frac{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) x+bc \left ( 4\,ac-{b}^{2} \right ) }{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}} \right ) }+{\frac{{b}^{3}e}{c}\arctan \left ({(2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) x+bc \left ( 4\,ac-{b}^{2} \right ) ){\frac{1}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}}} \right ){\frac{1}{\sqrt{64\,{a}^{3}{c}^{5}-48\,{a}^{2}{b}^{2}{c}^{4}+12\,a{b}^{4}{c}^{3}-{b}^{6}{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(1/c^2*(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(4*a*c-b^2)*x+a*(2*a*c*e-b^2*e+b*c*d)
/c^2/(4*a*c-b^2))/(c*x^2+b*x+a)+2/c/(4*a*c-b^2)*ln(c*(4*a*c-b^2)*(c*x^2+b*x+a))*
a*e-1/2/c^2/(4*a*c-b^2)*ln(c*(4*a*c-b^2)*(c*x^2+b*x+a))*b^2*e-6/(64*a^3*c^5-48*a
^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x+b*c*(4*a*c-b^
2))/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*b*e*a+4/(64*a^3*c^5-
48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x+b*c*(4*a*
c-b^2))/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*a*c*d+1/(64*a^3*
c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2)*arctan((2*c^2*(4*a*c-b^2)*x+b*c*(
4*a*c-b^2))/(64*a^3*c^5-48*a^2*b^2*c^4+12*a*b^4*c^3-b^6*c^2)^(1/2))*b^3/c*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)*x^2/(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.28655, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (4 \, a^{2} c^{2} d +{\left (4 \, a c^{3} d +{\left (b^{3} c - 6 \, a b c^{2}\right )} e\right )} x^{2} +{\left (a b^{3} - 6 \, a^{2} b c\right )} e +{\left (4 \, a b c^{2} d +{\left (b^{4} - 6 \, a b^{2} c\right )} e\right )} x\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (2 \, a b c d - 2 \,{\left (a b^{2} - 2 \, a^{2} c\right )} e + 2 \,{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d -{\left (b^{3} - 3 \, a b c\right )} e\right )} x -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e x^{2} +{\left (b^{3} - 4 \, a b c\right )} e x +{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left (4 \, a^{2} c^{2} d +{\left (4 \, a c^{3} d +{\left (b^{3} c - 6 \, a b c^{2}\right )} e\right )} x^{2} +{\left (a b^{3} - 6 \, a^{2} b c\right )} e +{\left (4 \, a b c^{2} d +{\left (b^{4} - 6 \, a b^{2} c\right )} e\right )} x\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, a b c d - 2 \,{\left (a b^{2} - 2 \, a^{2} c\right )} e + 2 \,{\left ({\left (b^{2} c - 2 \, a c^{2}\right )} d -{\left (b^{3} - 3 \, a b c\right )} e\right )} x -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e x^{2} +{\left (b^{3} - 4 \, a b c\right )} e x +{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (a b^{2} c^{2} - 4 \, a^{2} c^{3} +{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} x^{2} +{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 9.00903, size = 901, normalized size = 6.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(e/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(6*a*b*c*e - 4*a*c**2*d - b**3*e)/(2*c**2
*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-16*a**2*c**
3*(e/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(6*a*b*c*e - 4*a*c**2*d - b**3*e)/(2*c*
*2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 8*a**2*c*e + 8*a*
b**2*c**2*(e/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(6*a*b*c*e - 4*a*c**2*d - b**3*
e)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) - a*b**2*e
- 2*a*b*c*d - b**4*c*(e/(2*c**2) - sqrt(-(4*a*c - b**2)**3)*(6*a*b*c*e - 4*a*c**
2*d - b**3*e)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))))
/(6*a*b*c*e - 4*a*c**2*d - b**3*e)) + (e/(2*c**2) + sqrt(-(4*a*c - b**2)**3)*(6*
a*b*c*e - 4*a*c**2*d - b**3*e)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*
b**4*c - b**6)))*log(x + (-16*a**2*c**3*(e/(2*c**2) + sqrt(-(4*a*c - b**2)**3)*(
6*a*b*c*e - 4*a*c**2*d - b**3*e)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*
a*b**4*c - b**6))) + 8*a**2*c*e + 8*a*b**2*c**2*(e/(2*c**2) + sqrt(-(4*a*c - b**
2)**3)*(6*a*b*c*e - 4*a*c**2*d - b**3*e)/(2*c**2*(64*a**3*c**3 - 48*a**2*b**2*c*
*2 + 12*a*b**4*c - b**6))) - a*b**2*e - 2*a*b*c*d - b**4*c*(e/(2*c**2) + sqrt(-(
4*a*c - b**2)**3)*(6*a*b*c*e - 4*a*c**2*d - b**3*e)/(2*c**2*(64*a**3*c**3 - 48*a
**2*b**2*c**2 + 12*a*b**4*c - b**6))))/(6*a*b*c*e - 4*a*c**2*d - b**3*e)) + (2*a
**2*c*e - a*b**2*e + a*b*c*d + x*(3*a*b*c*e - 2*a*c**2*d - b**3*e + b**2*c*d))/(
4*a**2*c**3 - a*b**2*c**2 + x**2*(4*a*c**4 - b**2*c**3) + x*(4*a*b*c**3 - b**3*c
**2))

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GIAC/XCAS [A]  time = 0.274411, size = 228, normalized size = 1.73 \[ -\frac{{\left (4 \, a c^{2} d + b^{3} e - 6 \, a b c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{e{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac{a b c d - a b^{2} e + 2 \, a^{2} c e +{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} x}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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