3.1540 \(\int \frac{(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=197 \[ \frac{2 e (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{2 e^3 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{2 e (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^4 \log \left (a+b x+c x^2\right )}{c^2}-\frac{(d+e x)^4}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.680172, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ \frac{2 e (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac{2 e^3 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{2 e (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{e^4 \log \left (a+b x+c x^2\right )}{c^2}-\frac{(d+e x)^4}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{2 e^{3} \left (b e - 2 c d\right ) \int \frac{1}{c}\, dx}{- 4 a c + b^{2}} + \frac{2 e \left (d + e x\right )^{2} \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} - \frac{\left (d + e x\right )^{4}}{2 \left (a + b x + c x^{2}\right )^{2}} + \frac{e^{4} \log{\left (a + b x + c x^{2} \right )}}{c^{2}} + \frac{2 e \left (b e - 2 c d\right ) \left (- 6 a c e^{2} + b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*x+b)*(e*x+d)**4/(c*x**2+b*x+a)**3,x)

[Out]

-2*e**3*(b*e - 2*c*d)*Integral(1/c, x)/(-4*a*c + b**2) + 2*e*(d + e*x)**2*(2*a*e
- b*d + x*(b*e - 2*c*d))/((-4*a*c + b**2)*(a + b*x + c*x**2)) - (d + e*x)**4/(2
*(a + b*x + c*x**2)**2) + e**4*log(a + b*x + c*x**2)/c**2 + 2*e*(b*e - 2*c*d)*(-
6*a*c*e**2 + b**2*e**2 + 2*b*c*d*e - 2*c**2*d**2)*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 = 1.15318, size = 334, normalized size = 1.7 \[ \frac{\frac{-c e^3 \left (a^2 e+2 a b (2 d+e x)+4 b^2 d x\right )+b^2 e^4 (a+b x)+2 c^2 d e^2 (3 a d+2 a e x+3 b d x)-c^3 d^3 (d+4 e x)}{(a+x (b+c x))^2}-\frac{e \left (8 c^2 \left (2 a^2 e^3-a c d e (6 d+5 e x)+c^2 d^3 x\right )+2 b^2 c e \left (c d (3 d+8 e x)-5 a e^2\right )+4 b c^2 \left (a e^2 (7 d+5 e x)+c d^2 (d-3 e x)\right )+b^4 e^3-2 b^3 c e^2 (2 d+3 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac{4 c e (b e-2 c d) \left (2 c e (b d-3 a e)+b^2 e^2-2 c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+2 c e^4 \log (a+x (b+c x))}{2 c^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^3,x]

[Out]

((b^2*e^4*(a + b*x) - c^3*d^3*(d + 4*e*x) + 2*c^2*d*e^2*(3*a*d + 3*b*d*x + 2*a*e
*x) - c*e^3*(a^2*e + 4*b^2*d*x + 2*a*b*(2*d + e*x)))/(a + x*(b + c*x))^2 - (e*(b
^4*e^3 - 2*b^3*c*e^2*(2*d + 3*e*x) + 8*c^2*(2*a^2*e^3 + c^2*d^3*x - a*c*d*e*(6*d
+ 5*e*x)) + 4*b*c^2*(c*d^2*(d - 3*e*x) + a*e^2*(7*d + 5*e*x)) + 2*b^2*c*e*(-5*a
*e^2 + c*d*(3*d + 8*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (4*c*e*(-2*c*d +
b*e)*(-2*c^2*d^2 + b^2*e^2 + 2*c*e*(b*d - 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2
+ 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*c*e^4*Log[a + x*(b + c*x)])/(2*c^3)

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Maple [B]  time = 0.022, size = 1064, normalized size = 5.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*x+b)*(e*x+d)^4/(c*x^2+b*x+a)^3,x)

[Out]

(1/c*e*(10*a*b*c*e^3-20*a*c^2*d*e^2-3*b^3*e^3+8*b^2*c*d*e^2-6*b*c^2*d^2*e+4*c^3*
d^3)/(4*a*c-b^2)*x^3+1/2*e*(16*a^2*c^2*e^3+10*a*b^2*c*e^3-12*a*b*c^2*d*e^2-48*a*
c^3*d^2*e-5*b^4*e^3+12*b^3*c*d*e^2-6*b^2*c^2*d^2*e+12*b*c^3*d^3)/c^2/(4*a*c-b^2)
*x^2+e*(14*a^2*b*c*e^3-12*a^2*c^2*d*e^2-5*a*b^3*e^3+12*a*b^2*c*d*e^2-18*a*b*c^2*
d^2*e-4*a*c^3*d^3+4*b^2*c^2*d^3)/(4*a*c-b^2)/c^2*x+1/2*(12*a^3*c*e^4-5*a^2*b^2*e
^4+12*a^2*b*c*d*e^3-24*a^2*c^2*d^2*e^2+4*a*b*c^2*d^3*e-4*a*c^3*d^4+b^2*c^2*d^4)/
c^2/(4*a*c-b^2))/(c*x^2+b*x+a)^2+4*e^4/c/(4*a*c-b^2)*ln(c*(4*a*c-b^2)*(c*x^2+b*x
+a))*a-e^4/c^2/(4*a*c-b^2)*ln(c*(4*a*c-b^2)*(c*x^2+b*x+a))*b^2-12/(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*b*e^4+24/(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*e^3-1
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)*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*
c*d^2*e^2+8/(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))*c^2*d^3*e+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)*arcta
n((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*e^4/c

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.323513, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^4/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[1/2*(2*(4*a^2*c^3*d^3*e - 6*a^2*b*c^2*d^2*e^2 + 12*a^3*c^2*d*e^3 + (a^2*b^3 - 6
*a^3*b*c)*e^4 + (4*c^5*d^3*e - 6*b*c^4*d^2*e^2 + 12*a*c^4*d*e^3 + (b^3*c^2 - 6*a
*b*c^3)*e^4)*x^4 + 2*(4*b*c^4*d^3*e - 6*b^2*c^3*d^2*e^2 + 12*a*b*c^3*d*e^3 + (b^
4*c - 6*a*b^2*c^2)*e^4)*x^3 + (4*(b^2*c^3 + 2*a*c^4)*d^3*e - 6*(b^3*c^2 + 2*a*b*
c^3)*d^2*e^2 + 12*(a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 4*a*b^3*c - 12*a^2*b*c^
2)*e^4)*x^2 + 2*(4*a*b*c^3*d^3*e - 6*a*b^2*c^2*d^2*e^2 + 12*a^2*b*c^2*d*e^3 + (a
*b^4 - 6*a^2*b^2*c)*e^4)*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)) - (4*a*b*c^2*
d^3*e - 24*a^2*c^2*d^2*e^2 + 12*a^2*b*c*d*e^3 + (b^2*c^2 - 4*a*c^3)*d^4 - (5*a^2
*b^2 - 12*a^3*c)*e^4 + 2*(4*c^4*d^3*e - 6*b*c^3*d^2*e^2 + 4*(2*b^2*c^2 - 5*a*c^3
)*d*e^3 - (3*b^3*c - 10*a*b*c^2)*e^4)*x^3 + (12*b*c^3*d^3*e - 6*(b^2*c^2 + 8*a*c
^3)*d^2*e^2 + 12*(b^3*c - a*b*c^2)*d*e^3 - (5*b^4 - 10*a*b^2*c - 16*a^2*c^2)*e^4
)*x^2 - 2*(18*a*b*c^2*d^2*e^2 - 4*(b^2*c^2 - a*c^3)*d^3*e - 12*(a*b^2*c - a^2*c^
2)*d*e^3 + (5*a*b^3 - 14*a^2*b*c)*e^4)*x - 2*((b^2*c^2 - 4*a*c^3)*e^4*x^4 + 2*(b
^3*c - 4*a*b*c^2)*e^4*x^3 + (b^4 - 2*a*b^2*c - 8*a^2*c^2)*e^4*x^2 + 2*(a*b^3 - 4
*a^2*b*c)*e^4*x + (a^2*b^2 - 4*a^3*c)*e^4)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*
c))/((a^2*b^2*c^2 - 4*a^3*c^3 + (b^2*c^4 - 4*a*c^5)*x^4 + 2*(b^3*c^3 - 4*a*b*c^4
)*x^3 + (b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)*x^2 + 2*(a*b^3*c^2 - 4*a^2*b*c^3)*x)
*sqrt(b^2 - 4*a*c)), -1/2*(4*(4*a^2*c^3*d^3*e - 6*a^2*b*c^2*d^2*e^2 + 12*a^3*c^2
*d*e^3 + (a^2*b^3 - 6*a^3*b*c)*e^4 + (4*c^5*d^3*e - 6*b*c^4*d^2*e^2 + 12*a*c^4*d
*e^3 + (b^3*c^2 - 6*a*b*c^3)*e^4)*x^4 + 2*(4*b*c^4*d^3*e - 6*b^2*c^3*d^2*e^2 + 1
2*a*b*c^3*d*e^3 + (b^4*c - 6*a*b^2*c^2)*e^4)*x^3 + (4*(b^2*c^3 + 2*a*c^4)*d^3*e
- 6*(b^3*c^2 + 2*a*b*c^3)*d^2*e^2 + 12*(a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 4*
a*b^3*c - 12*a^2*b*c^2)*e^4)*x^2 + 2*(4*a*b*c^3*d^3*e - 6*a*b^2*c^2*d^2*e^2 + 12
*a^2*b*c^2*d*e^3 + (a*b^4 - 6*a^2*b^2*c)*e^4)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c
*x + b)/(b^2 - 4*a*c)) + (4*a*b*c^2*d^3*e - 24*a^2*c^2*d^2*e^2 + 12*a^2*b*c*d*e^
3 + (b^2*c^2 - 4*a*c^3)*d^4 - (5*a^2*b^2 - 12*a^3*c)*e^4 + 2*(4*c^4*d^3*e - 6*b*
c^3*d^2*e^2 + 4*(2*b^2*c^2 - 5*a*c^3)*d*e^3 - (3*b^3*c - 10*a*b*c^2)*e^4)*x^3 +
(12*b*c^3*d^3*e - 6*(b^2*c^2 + 8*a*c^3)*d^2*e^2 + 12*(b^3*c - a*b*c^2)*d*e^3 - (
5*b^4 - 10*a*b^2*c - 16*a^2*c^2)*e^4)*x^2 - 2*(18*a*b*c^2*d^2*e^2 - 4*(b^2*c^2 -
a*c^3)*d^3*e - 12*(a*b^2*c - a^2*c^2)*d*e^3 + (5*a*b^3 - 14*a^2*b*c)*e^4)*x - 2
*((b^2*c^2 - 4*a*c^3)*e^4*x^4 + 2*(b^3*c - 4*a*b*c^2)*e^4*x^3 + (b^4 - 2*a*b^2*c
- 8*a^2*c^2)*e^4*x^2 + 2*(a*b^3 - 4*a^2*b*c)*e^4*x + (a^2*b^2 - 4*a^3*c)*e^4)*l
og(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((a^2*b^2*c^2 - 4*a^3*c^3 + (b^2*c^4 -
4*a*c^5)*x^4 + 2*(b^3*c^3 - 4*a*b*c^4)*x^3 + (b^4*c^2 - 2*a*b^2*c^3 - 8*a^2*c^4)
*x^2 + 2*(a*b^3*c^2 - 4*a^2*b*c^3)*x)*sqrt(-b^2 + 4*a*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x+b)*(e*x+d)**4/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.276573, size = 593, normalized size = 3.01 \[ -\frac{2 \,{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 12 \, a c^{2} d e^{3} + b^{3} e^{4} - 6 \, a b c e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{e^{4}{\rm ln}\left (c x^{2} + b x + a\right )}{c^{2}} - \frac{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} + 4 \, a b c^{2} d^{3} e - 24 \, a^{2} c^{2} d^{2} e^{2} + 12 \, a^{2} b c d e^{3} - 5 \, a^{2} b^{2} e^{4} + 12 \, a^{3} c e^{4} + 2 \,{\left (4 \, c^{4} d^{3} e - 6 \, b c^{3} d^{2} e^{2} + 8 \, b^{2} c^{2} d e^{3} - 20 \, a c^{3} d e^{3} - 3 \, b^{3} c e^{4} + 10 \, a b c^{2} e^{4}\right )} x^{3} +{\left (12 \, b c^{3} d^{3} e - 6 \, b^{2} c^{2} d^{2} e^{2} - 48 \, a c^{3} d^{2} e^{2} + 12 \, b^{3} c d e^{3} - 12 \, a b c^{2} d e^{3} - 5 \, b^{4} e^{4} + 10 \, a b^{2} c e^{4} + 16 \, a^{2} c^{2} e^{4}\right )} x^{2} + 2 \,{\left (4 \, b^{2} c^{2} d^{3} e - 4 \, a c^{3} d^{3} e - 18 \, a b c^{2} d^{2} e^{2} + 12 \, a b^{2} c d e^{3} - 12 \, a^{2} c^{2} d e^{3} - 5 \, a b^{3} e^{4} + 14 \, a^{2} b c e^{4}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)*(e*x + d)^4/(c*x^2 + b*x + a)^3,x, algorithm="giac")

[Out]

-2*(4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 12*a*c^2*d*e^3 + b^3*e^4 - 6*a*b*c*e^4)*arct
an((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^2 - 4*a*c^3)*sqrt(-b^2 + 4*a*c)) + e^
4*ln(c*x^2 + b*x + a)/c^2 - 1/2*(b^2*c^2*d^4 - 4*a*c^3*d^4 + 4*a*b*c^2*d^3*e - 2
4*a^2*c^2*d^2*e^2 + 12*a^2*b*c*d*e^3 - 5*a^2*b^2*e^4 + 12*a^3*c*e^4 + 2*(4*c^4*d
^3*e - 6*b*c^3*d^2*e^2 + 8*b^2*c^2*d*e^3 - 20*a*c^3*d*e^3 - 3*b^3*c*e^4 + 10*a*b
*c^2*e^4)*x^3 + (12*b*c^3*d^3*e - 6*b^2*c^2*d^2*e^2 - 48*a*c^3*d^2*e^2 + 12*b^3*
c*d*e^3 - 12*a*b*c^2*d*e^3 - 5*b^4*e^4 + 10*a*b^2*c*e^4 + 16*a^2*c^2*e^4)*x^2 +
2*(4*b^2*c^2*d^3*e - 4*a*c^3*d^3*e - 18*a*b*c^2*d^2*e^2 + 12*a*b^2*c*d*e^3 - 12*
a^2*c^2*d*e^3 - 5*a*b^3*e^4 + 14*a^2*b*c*e^4)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a
*c)*c^2)