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3.2206 \(\int \frac{d+e x}{\left (a+b x+c x^2\right )^4} \, dx\)

Optimal. Leaf size=173 \[ \frac{20 c^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{5 c (b+2 c x) (2 c d-b e)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{5 (b+2 c x) (2 c d-b e)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{-2 a e+x (2 c d-b e)+b d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]

[Out]

-(b*d - 2*a*e + (2*c*d - b*e)*x)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (5*(2*c
*d - b*e)*(b + 2*c*x))/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) - (5*c*(2*c*d - b
*e)*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (20*c^2*(2*c*d - b*e)*Arc
Tanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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Rubi [A]  time = 0.206034, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{20 c^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{7/2}}-\frac{5 c (b+2 c x) (2 c d-b e)}{\left (b^2-4 a c\right )^3 \left (a+b x+c x^2\right )}+\frac{5 (b+2 c x) (2 c d-b e)}{6 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^2}-\frac{-2 a e+x (2 c d-b e)+b d}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

-(b*d - 2*a*e + (2*c*d - b*e)*x)/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^3) + (5*(2*c
*d - b*e)*(b + 2*c*x))/(6*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^2) - (5*c*(2*c*d - b
*e)*(b + 2*c*x))/((b^2 - 4*a*c)^3*(a + b*x + c*x^2)) + (20*c^2*(2*c*d - b*e)*Arc
Tanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(7/2)

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Rubi in Sympy [A]  time = 28.1269, size = 165, normalized size = 0.95 \[ - \frac{20 c^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{7}{2}}} + \frac{5 c \left (b + 2 c x\right ) \left (b e - 2 c d\right )}{\left (- 4 a c + b^{2}\right )^{3} \left (a + b x + c x^{2}\right )} - \frac{5 \left (b + 2 c x\right ) \left (b e - 2 c d\right )}{6 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )^{2}} + \frac{2 a e - b d + x \left (b e - 2 c d\right )}{3 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*c**2*(b*e - 2*c*d)*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(
7/2) + 5*c*(b + 2*c*x)*(b*e - 2*c*d)/((-4*a*c + b**2)**3*(a + b*x + c*x**2)) - 5
*(b + 2*c*x)*(b*e - 2*c*d)/(6*(-4*a*c + b**2)**2*(a + b*x + c*x**2)**2) + (2*a*e
 - b*d + x*(b*e - 2*c*d))/(3*(-4*a*c + b**2)*(a + b*x + c*x**2)**3)

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Mathematica [A]  time = 0.412586, size = 168, normalized size = 0.97 \[ \frac{\frac{120 c^2 (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{5 \left (b^2-4 a c\right ) (b+2 c x) (b e-2 c d)}{(a+x (b+c x))^2}+\frac{2 \left (b^2-4 a c\right )^2 (2 a e-b d+b e x-2 c d x)}{(a+x (b+c x))^3}+\frac{30 c (b+2 c x) (b e-2 c d)}{a+x (b+c x)}}{6 \left (b^2-4 a c\right )^3} \]

Antiderivative was successfully verified.

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

[Out]

((2*(b^2 - 4*a*c)^2*(-(b*d) + 2*a*e - 2*c*d*x + b*e*x))/(a + x*(b + c*x))^3 - (5
*(b^2 - 4*a*c)*(-2*c*d + b*e)*(b + 2*c*x))/(a + x*(b + c*x))^2 + (30*c*(-2*c*d +
 b*e)*(b + 2*c*x))/(a + x*(b + c*x)) + (120*c^2*(-2*c*d + b*e)*ArcTan[(b + 2*c*x
)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(6*(b^2 - 4*a*c)^3)

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Maple [B]  time = 0.008, size = 369, normalized size = 2.1 \[{\frac{bd-2\,ae+ \left ( -be+2\,cd \right ) x}{ \left ( 12\,ac-3\,{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) ^{3}}}-{\frac{5\,xbce}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{10\,x{c}^{2}d}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{5\,{b}^{2}e}{6\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+{\frac{5\,bcd}{3\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-10\,{\frac{{c}^{2}xbe}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) }}+20\,{\frac{{c}^{3}dx}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) }}-5\,{\frac{{b}^{2}ce}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) }}+10\,{\frac{{c}^{2}bd}{ \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) }}-20\,{\frac{b{c}^{2}e}{ \left ( 4\,ac-{b}^{2} \right ) ^{7/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+40\,{\frac{{c}^{3}d}{ \left ( 4\,ac-{b}^{2} \right ) ^{7/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)/(c*x^2+b*x+a)^4,x)

[Out]

1/3*(b*d-2*a*e+(-b*e+2*c*d)*x)/(4*a*c-b^2)/(c*x^2+b*x+a)^3-5/3/(4*a*c-b^2)^2/(c*
x^2+b*x+a)^2*c*x*b*e+10/3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^2*c^2*x*d-5/6/(4*a*c-b^2)^
2/(c*x^2+b*x+a)^2*b^2*e+5/3/(4*a*c-b^2)^2/(c*x^2+b*x+a)^2*b*c*d-10/(4*a*c-b^2)^3
*c^2/(c*x^2+b*x+a)*x*b*e+20/(4*a*c-b^2)^3*c^3/(c*x^2+b*x+a)*x*d-5/(4*a*c-b^2)^3*
c/(c*x^2+b*x+a)*b^2*e+10/(4*a*c-b^2)^3*c^2/(c*x^2+b*x+a)*b*d-20/(4*a*c-b^2)^(7/2
)*c^2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*e+40/(4*a*c-b^2)^(7/2)*c^3*arctan((2
*c*x+b)/(4*a*c-b^2)^(1/2))*d

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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)/(c*x^2 + b*x + a)^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22626, 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)/(c*x^2 + b*x + a)^4,x, algorithm="fricas")

[Out]

[1/6*(60*(2*a^3*c^3*d - a^3*b*c^2*e + (2*c^6*d - b*c^5*e)*x^6 + 3*(2*b*c^5*d - b
^2*c^4*e)*x^5 + 3*(2*(b^2*c^4 + a*c^5)*d - (b^3*c^3 + a*b*c^4)*e)*x^4 + (2*(b^3*
c^3 + 6*a*b*c^4)*d - (b^4*c^2 + 6*a*b^2*c^3)*e)*x^3 + 3*(2*(a*b^2*c^3 + a^2*c^4)
*d - (a*b^3*c^2 + a^2*b*c^3)*e)*x^2 + 3*(2*a^2*b*c^3*d - a^2*b^2*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)) - (60*(2*c^5*d - b*c^4*e)*x^5 + 150*(2*b*c^4*d
 - b^2*c^3*e)*x^4 + 10*(2*(11*b^2*c^3 + 16*a*c^4)*d - (11*b^3*c^2 + 16*a*b*c^3)*
e)*x^3 + 15*(2*(b^3*c^2 + 16*a*b*c^3)*d - (b^4*c + 16*a*b^2*c^2)*e)*x^2 + 2*(b^5
 - 13*a*b^3*c + 66*a^2*b*c^2)*d + (a*b^4 - 18*a^2*b^2*c - 64*a^3*c^2)*e - 3*(2*(
b^4*c - 18*a*b^2*c^2 - 44*a^2*c^3)*d - (b^5 - 18*a*b^3*c - 44*a^2*b*c^2)*e)*x)*s
qrt(b^2 - 4*a*c))/((a^3*b^6 - 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*
c^3 - 12*a*b^4*c^4 + 48*a^2*b^2*c^5 - 64*a^3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^
3 + 48*a^2*b^3*c^4 - 64*a^3*b*c^5)*x^5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^
3 - 16*a^3*b^2*c^4 - 64*a^4*c^5)*x^4 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a
^3*b^3*c^3 - 384*a^4*b*c^4)*x^3 + 3*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*
a^4*b^2*c^3 - 64*a^5*c^4)*x^2 + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*
a^5*b*c^3)*x)*sqrt(b^2 - 4*a*c)), -1/6*(120*(2*a^3*c^3*d - a^3*b*c^2*e + (2*c^6*
d - b*c^5*e)*x^6 + 3*(2*b*c^5*d - b^2*c^4*e)*x^5 + 3*(2*(b^2*c^4 + a*c^5)*d - (b
^3*c^3 + a*b*c^4)*e)*x^4 + (2*(b^3*c^3 + 6*a*b*c^4)*d - (b^4*c^2 + 6*a*b^2*c^3)*
e)*x^3 + 3*(2*(a*b^2*c^3 + a^2*c^4)*d - (a*b^3*c^2 + a^2*b*c^3)*e)*x^2 + 3*(2*a^
2*b*c^3*d - a^2*b^2*c^2*e)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*
c)) + (60*(2*c^5*d - b*c^4*e)*x^5 + 150*(2*b*c^4*d - b^2*c^3*e)*x^4 + 10*(2*(11*
b^2*c^3 + 16*a*c^4)*d - (11*b^3*c^2 + 16*a*b*c^3)*e)*x^3 + 15*(2*(b^3*c^2 + 16*a
*b*c^3)*d - (b^4*c + 16*a*b^2*c^2)*e)*x^2 + 2*(b^5 - 13*a*b^3*c + 66*a^2*b*c^2)*
d + (a*b^4 - 18*a^2*b^2*c - 64*a^3*c^2)*e - 3*(2*(b^4*c - 18*a*b^2*c^2 - 44*a^2*
c^3)*d - (b^5 - 18*a*b^3*c - 44*a^2*b*c^2)*e)*x)*sqrt(-b^2 + 4*a*c))/((a^3*b^6 -
 12*a^4*b^4*c + 48*a^5*b^2*c^2 - 64*a^6*c^3 + (b^6*c^3 - 12*a*b^4*c^4 + 48*a^2*b
^2*c^5 - 64*a^3*c^6)*x^6 + 3*(b^7*c^2 - 12*a*b^5*c^3 + 48*a^2*b^3*c^4 - 64*a^3*b
*c^5)*x^5 + 3*(b^8*c - 11*a*b^6*c^2 + 36*a^2*b^4*c^3 - 16*a^3*b^2*c^4 - 64*a^4*c
^5)*x^4 + (b^9 - 6*a*b^7*c - 24*a^2*b^5*c^2 + 224*a^3*b^3*c^3 - 384*a^4*b*c^4)*x
^3 + 3*(a*b^8 - 11*a^2*b^6*c + 36*a^3*b^4*c^2 - 16*a^4*b^2*c^3 - 64*a^5*c^4)*x^2
 + 3*(a^2*b^7 - 12*a^3*b^5*c + 48*a^4*b^3*c^2 - 64*a^5*b*c^3)*x)*sqrt(-b^2 + 4*a
*c))]

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Sympy [A]  time = 15.4783, size = 1062, normalized size = 6.14 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

10*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*log(x + (-2560*a**4*c**6*sqrt(-
1/(4*a*c - b**2)**7)*(b*e - 2*c*d) + 2560*a**3*b**2*c**5*sqrt(-1/(4*a*c - b**2)*
*7)*(b*e - 2*c*d) - 960*a**2*b**4*c**4*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)
+ 160*a*b**6*c**3*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) - 10*b**8*c**2*sqrt(-
1/(4*a*c - b**2)**7)*(b*e - 2*c*d) + 10*b**2*c**2*e - 20*b*c**3*d)/(20*b*c**3*e
- 40*c**4*d)) - 10*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d)*log(x + (2560*a
**4*c**6*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) - 2560*a**3*b**2*c**5*sqrt(-1/
(4*a*c - b**2)**7)*(b*e - 2*c*d) + 960*a**2*b**4*c**4*sqrt(-1/(4*a*c - b**2)**7)
*(b*e - 2*c*d) - 160*a*b**6*c**3*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) + 10*b
**8*c**2*sqrt(-1/(4*a*c - b**2)**7)*(b*e - 2*c*d) + 10*b**2*c**2*e - 20*b*c**3*d
)/(20*b*c**3*e - 40*c**4*d)) - (64*a**3*c**2*e + 18*a**2*b**2*c*e - 132*a**2*b*c
**2*d - a*b**4*e + 26*a*b**3*c*d - 2*b**5*d + x**5*(60*b*c**4*e - 120*c**5*d) +
x**4*(150*b**2*c**3*e - 300*b*c**4*d) + x**3*(160*a*b*c**3*e - 320*a*c**4*d + 11
0*b**3*c**2*e - 220*b**2*c**3*d) + x**2*(240*a*b**2*c**2*e - 480*a*b*c**3*d + 15
*b**4*c*e - 30*b**3*c**2*d) + x*(132*a**2*b*c**2*e - 264*a**2*c**3*d + 54*a*b**3
*c*e - 108*a*b**2*c**2*d - 3*b**5*e + 6*b**4*c*d))/(384*a**6*c**3 - 288*a**5*b**
2*c**2 + 72*a**4*b**4*c - 6*a**3*b**6 + x**6*(384*a**3*c**6 - 288*a**2*b**2*c**5
 + 72*a*b**4*c**4 - 6*b**6*c**3) + x**5*(1152*a**3*b*c**5 - 864*a**2*b**3*c**4 +
 216*a*b**5*c**3 - 18*b**7*c**2) + x**4*(1152*a**4*c**5 + 288*a**3*b**2*c**4 - 6
48*a**2*b**4*c**3 + 198*a*b**6*c**2 - 18*b**8*c) + x**3*(2304*a**4*b*c**4 - 1344
*a**3*b**3*c**3 + 144*a**2*b**5*c**2 + 36*a*b**7*c - 6*b**9) + x**2*(1152*a**5*c
**4 + 288*a**4*b**2*c**3 - 648*a**3*b**4*c**2 + 198*a**2*b**6*c - 18*a*b**8) + x
*(1152*a**5*b*c**3 - 864*a**4*b**3*c**2 + 216*a**3*b**5*c - 18*a**2*b**7))

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GIAC/XCAS [A]  time = 0.205369, size = 510, normalized size = 2.95 \[ -\frac{20 \,{\left (2 \, c^{3} d - b c^{2} e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{120 \, c^{5} d x^{5} - 60 \, b c^{4} x^{5} e + 300 \, b c^{4} d x^{4} - 150 \, b^{2} c^{3} x^{4} e + 220 \, b^{2} c^{3} d x^{3} + 320 \, a c^{4} d x^{3} - 110 \, b^{3} c^{2} x^{3} e - 160 \, a b c^{3} x^{3} e + 30 \, b^{3} c^{2} d x^{2} + 480 \, a b c^{3} d x^{2} - 15 \, b^{4} c x^{2} e - 240 \, a b^{2} c^{2} x^{2} e - 6 \, b^{4} c d x + 108 \, a b^{2} c^{2} d x + 264 \, a^{2} c^{3} d x + 3 \, b^{5} x e - 54 \, a b^{3} c x e - 132 \, a^{2} b c^{2} x e + 2 \, b^{5} d - 26 \, a b^{3} c d + 132 \, a^{2} b c^{2} d + a b^{4} e - 18 \, a^{2} b^{2} c e - 64 \, a^{3} c^{2} e}{6 \,{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )}{\left (c x^{2} + b x + a\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-20*(2*c^3*d - b*c^2*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^6 - 12*a*b^4*
c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*sqrt(-b^2 + 4*a*c)) - 1/6*(120*c^5*d*x^5 - 60*b
*c^4*x^5*e + 300*b*c^4*d*x^4 - 150*b^2*c^3*x^4*e + 220*b^2*c^3*d*x^3 + 320*a*c^4
*d*x^3 - 110*b^3*c^2*x^3*e - 160*a*b*c^3*x^3*e + 30*b^3*c^2*d*x^2 + 480*a*b*c^3*
d*x^2 - 15*b^4*c*x^2*e - 240*a*b^2*c^2*x^2*e - 6*b^4*c*d*x + 108*a*b^2*c^2*d*x +
 264*a^2*c^3*d*x + 3*b^5*x*e - 54*a*b^3*c*x*e - 132*a^2*b*c^2*x*e + 2*b^5*d - 26
*a*b^3*c*d + 132*a^2*b*c^2*d + a*b^4*e - 18*a^2*b^2*c*e - 64*a^3*c^2*e)/((b^6 -
12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)*(c*x^2 + b*x + a)^3)

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