3.891 \(\int \frac{x^3 (d+e x)}{\left (a+b x+c x^2\right )^2} \, dx\)
Optimal. Leaf size=192 \[ \frac{\left (-12 a^2 c^2 e+12 a b^2 c e-6 a b c^2 d-2 b^4 e+b^3 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac{x \left (6 a c e-2 b^2 e+b c d\right )}{c^2 \left (b^2-4 a c\right )}+\frac{x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3} \]
[Out]
-(((b*c*d - 2*b^2*e + 6*a*c*e)*x)/(c^2*(b^2 - 4*a*c))) + (x^2*(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*c*d -
6*a*b*c^2*d - 2*b^4*e + 12*a*b^2*c*e - 12*a^2*c^2*e)*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) + ((c*d - 2*b*e)*Log[a + b*x + c*x^2])/(
2*c^3)
_______________________________________________________________________________________
Rubi [A] time = 0.615897, antiderivative size = 192, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286
\[ \frac{\left (-12 a^2 c^2 e+12 a b^2 c e-6 a b c^2 d-2 b^4 e+b^3 c d\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}-\frac{x \left (6 a c e-2 b^2 e+b c d\right )}{c^2 \left (b^2-4 a c\right )}+\frac{x^2 \left (x \left (2 a c e+b^2 (-e)+b c d\right )+a (2 c d-b e)\right )}{c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{(c d-2 b e) \log \left (a+b x+c x^2\right )}{2 c^3} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(d + e*x))/(a + b*x + c*x^2)^2,x]
[Out]
-(((b*c*d - 2*b^2*e + 6*a*c*e)*x)/(c^2*(b^2 - 4*a*c))) + (x^2*(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*c*d -
6*a*b*c^2*d - 2*b^4*e + 12*a*b^2*c*e - 12*a^2*c^2*e)*ArcTanh[(b + 2*c*x)/Sqrt[b
^2 - 4*a*c]])/(c^3*(b^2 - 4*a*c)^(3/2)) + ((c*d - 2*b*e)*Log[a + b*x + c*x^2])/(
2*c^3)
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{6 a e x}{c \left (- 4 a c + b^{2}\right )} + \frac{2 b^{2} e x}{c^{2} \left (- 4 a c + b^{2}\right )} - \frac{d \int b\, dx}{c \left (- 4 a c + b^{2}\right )} - \frac{x^{2} \left (a \left (b e - 2 c d\right ) + x \left (- 2 a c e + b^{2} e - b c d\right )\right )}{c \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{\left (2 b e - c d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{3}} - \frac{\left (12 a^{2} c^{2} e - 12 a b^{2} c e + 6 a b c^{2} d + 2 b^{4} e - b^{3} c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{3} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(e*x+d)/(c*x**2+b*x+a)**2,x)
[Out]
-6*a*e*x/(c*(-4*a*c + b**2)) + 2*b**2*e*x/(c**2*(-4*a*c + b**2)) - d*Integral(b,
x)/(c*(-4*a*c + b**2)) - x**2*(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)) - (2*b*e - c*d)*log(a + b*x + c*x**2)/(2
*c**3) - (12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)*a
tanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(c**3*(-4*a*c + b**2)**(3/2))
_______________________________________________________________________________________
Mathematica [A] time = 0.572036, size = 190, normalized size = 0.99 \[ \frac{\frac{2 \left (a^2 c (3 b e-2 c (d+e x))+a b \left (b^2 (-e)+b c (d+4 e x)-3 c^2 d x\right )+b^3 x (c d-b e)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{2 \left (12 a^2 c^2 e-12 a b^2 c e+6 a b c^2 d+2 b^4 e-b^3 c d\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+(c d-2 b e) \log (a+x (b+c x))+2 c e x}{2 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(d + e*x))/(a + b*x + c*x^2)^2,x]
[Out]
(2*c*e*x + (2*(b^3*(c*d - b*e)*x + a^2*c*(3*b*e - 2*c*(d + e*x)) + a*b*(-(b^2*e)
- 3*c^2*d*x + b*c*(d + 4*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) - (2*(-(b^3*
c*d) + 6*a*b*c^2*d + 2*b^4*e - 12*a*b^2*c*e + 12*a^2*c^2*e)*ArcTan[(b + 2*c*x)/S
qrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (c*d - 2*b*e)*Log[a + x*(b + c*x)])/(
2*c^3)
_______________________________________________________________________________________
Maple [B] time = 0.016, size = 1008, normalized size = 5.3 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(e*x+d)/(c*x^2+b*x+a)^2,x)
[Out]
e*x/c^2+2/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a^2*e-4/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*
a*b^2*e+3/c/(c*x^2+b*x+a)/(4*a*c-b^2)*x*a*b*d+1/c^3/(c*x^2+b*x+a)/(4*a*c-b^2)*x*
b^4*e-1/c^2/(c*x^2+b*x+a)/(4*a*c-b^2)*x*b^3*d-3/c^2/(c*x^2+b*x+a)*a^2/(4*a*c-b^2
)*b*e+2/c/(c*x^2+b*x+a)*a^2/(4*a*c-b^2)*d+1/c^3/(c*x^2+b*x+a)*a/(4*a*c-b^2)*b^3*
e-1/c^2/(c*x^2+b*x+a)*a/(4*a*c-b^2)*b^2*d-4/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^
2+b*x+a))*a*b*e+2/c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*a*d+1/c^3/(4*a*c-b
^2)*ln((4*a*c-b^2)*(c*x^2+b*x+a))*b^3*e-1/2/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^
2+b*x+a))*b^2*d-12/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*
c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))
*a^2*e+12/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*
c-b^2)*x+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*e*b^
2-6/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x
+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b*d-2/c^3/(6
4*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-
b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^4*e+1/c^2/(64*a^3*c^
3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*c*(4*a*c-b^2)*x+(4*a*c-b^2)*b)/
(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*d*b^3
_______________________________________________________________________________________
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)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.293963, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^3/(c*x^2 + b*x + a)^2,x, algorithm="fricas")
[Out]
[1/2*((((b^3*c^2 - 6*a*b*c^3)*d - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c^3)*e)*x^2 + (
a*b^3*c - 6*a^2*b*c^2)*d - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e + ((b^4*c - 6*a
*b^2*c^2)*d - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*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*(b^2*c^2 - 4*a*c^3)*e*x^3 + 2*(b^3*c - 4*a*b*c^2)*e*x^2 + 2*(a
*b^2*c - 2*a^2*c^2)*d - 2*(a*b^3 - 3*a^2*b*c)*e + 2*((b^3*c - 3*a*b*c^2)*d - (b^
4 - 5*a*b^2*c + 6*a^2*c^2)*e)*x + (((b^2*c^2 - 4*a*c^3)*d - 2*(b^3*c - 4*a*b*c^2
)*e)*x^2 + (a*b^2*c - 4*a^2*c^2)*d - 2*(a*b^3 - 4*a^2*b*c)*e + ((b^3*c - 4*a*b*c
^2)*d - 2*(b^4 - 4*a*b^2*c)*e)*x)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((a*b
^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^3*c^3 - 4*a*b*c^4)*x)*sqrt(b^2
- 4*a*c)), -1/2*(2*(((b^3*c^2 - 6*a*b*c^3)*d - 2*(b^4*c - 6*a*b^2*c^2 + 6*a^2*c
^3)*e)*x^2 + (a*b^3*c - 6*a^2*b*c^2)*d - 2*(a*b^4 - 6*a^2*b^2*c + 6*a^3*c^2)*e +
((b^4*c - 6*a*b^2*c^2)*d - 2*(b^5 - 6*a*b^3*c + 6*a^2*b*c^2)*e)*x)*arctan(-sqrt
(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (2*(b^2*c^2 - 4*a*c^3)*e*x^3 + 2*(b^
3*c - 4*a*b*c^2)*e*x^2 + 2*(a*b^2*c - 2*a^2*c^2)*d - 2*(a*b^3 - 3*a^2*b*c)*e + 2
*((b^3*c - 3*a*b*c^2)*d - (b^4 - 5*a*b^2*c + 6*a^2*c^2)*e)*x + (((b^2*c^2 - 4*a*
c^3)*d - 2*(b^3*c - 4*a*b*c^2)*e)*x^2 + (a*b^2*c - 4*a^2*c^2)*d - 2*(a*b^3 - 4*a
^2*b*c)*e + ((b^3*c - 4*a*b*c^2)*d - 2*(b^4 - 4*a*b^2*c)*e)*x)*log(c*x^2 + b*x +
a))*sqrt(-b^2 + 4*a*c))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^2 + (b^
3*c^3 - 4*a*b*c^4)*x)*sqrt(-b^2 + 4*a*c))]
_______________________________________________________________________________________
Sympy [A] time = 15.2879, size = 1248, normalized size = 6.5 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(e*x+d)/(c*x**2+b*x+a)**2,x)
[Out]
(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b*
*4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)
) - (2*b*e - c*d)/(2*c**3))*log(x + (-10*a**2*b*c*e - 16*a**2*c**4*(-sqrt(-(4*a*
c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c
*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e -
c*d)/(2*c**3)) + 8*a**2*c**2*d + 2*a*b**3*e + 8*a*b**2*c**3*(-sqrt(-(4*a*c - b**
2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*
c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2
*c**3)) - a*b**2*c*d - b**4*c**2*(-sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12
*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**
2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)))/(12*a**2*c**2*e -
12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)) + (sqrt(-(4*a*c - b**2)**3)
*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(
64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)
)*log(x + (-10*a**2*b*c*e - 16*a**2*c**4*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2
*e - 12*a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 -
48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) + 8*a**2*c**
2*d + 2*a*b**3*e + 8*a*b**2*c**3*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*
a*b**2*c*e + 6*a*b*c**2*d + 2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2
*b**2*c**2 + 12*a*b**4*c - b**6)) - (2*b*e - c*d)/(2*c**3)) - a*b**2*c*d - b**4*
c**2*(sqrt(-(4*a*c - b**2)**3)*(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d +
2*b**4*e - b**3*c*d)/(2*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b
**6)) - (2*b*e - c*d)/(2*c**3)))/(12*a**2*c**2*e - 12*a*b**2*c*e + 6*a*b*c**2*d
+ 2*b**4*e - b**3*c*d)) + (-3*a**2*b*c*e + 2*a**2*c**2*d + a*b**3*e - a*b**2*c*d
+ x*(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))/(4*a**2*
c**4 - a*b**2*c**3 + x**2*(4*a*c**5 - b**2*c**4) + x*(4*a*b*c**4 - b**3*c**3)) +
e*x/c**2
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.271245, size = 317, normalized size = 1.65 \[ -\frac{{\left (b^{3} c d - 6 \, a b c^{2} d - 2 \, b^{4} e + 12 \, a b^{2} c e - 12 \, a^{2} c^{2} e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{x e}{c^{2}} + \frac{{\left (c d - 2 \, b e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{3}} + \frac{\frac{{\left (b^{3} c d - 3 \, a b c^{2} d - b^{4} e + 4 \, a b^{2} c e - 2 \, a^{2} c^{2} e\right )} x}{c} + \frac{a b^{2} c d - 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e}{c}}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*x^3/(c*x^2 + b*x + a)^2,x, algorithm="giac")
[Out]
-(b^3*c*d - 6*a*b*c^2*d - 2*b^4*e + 12*a*b^2*c*e - 12*a^2*c^2*e)*arctan((2*c*x +
b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + x*e/c^2 + 1/2
*(c*d - 2*b*e)*ln(c*x^2 + b*x + a)/c^3 + ((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)*x/c + (a*b^2*c*d - 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)/c)
/((c*x^2 + b*x + a)*(b^2 - 4*a*c)*c^2)