3.1532 \(\int \frac{(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx\)

Optimal. Leaf size=172 \[ \frac{2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{c^3}-\frac{4 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}-\frac{(d+e x)^4}{a+b x+c x^2}+\frac{4 e^3 x (3 c d-b e)}{c^2}+\frac{2 e^4 x^2}{c} \]

[Out]

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

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Rubi [A]  time = 0.467408, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{c^3}-\frac{4 e (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \sqrt{b^2-4 a c}}-\frac{(d+e x)^4}{a+b x+c x^2}+\frac{4 e^3 x (3 c d-b e)}{c^2}+\frac{2 e^4 x^2}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.504513, size = 241, normalized size = 1.4 \[ \frac{\frac{-c e^3 \left (a^2 e+2 a b (2 d+e x)+4 b^2 d x\right )+b^2 e^4 (a+b x)+2 c^2 d e^2 (3 a d+2 a e x+3 b d x)-c^3 d^3 (d+4 e x)}{a+x (b+c x)}+2 e^2 \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \log (a+x (b+c x))+\frac{4 e (b e-2 c d) \left (c e (3 a e+b d)-b^2 e^2-c^2 d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+c e^3 x (8 c d-3 b e)+c^2 e^4 x^2}{c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 618, normalized size = 3.6 \[{\frac{{e}^{4}{x}^{2}}{c}}-3\,{\frac{b{e}^{4}x}{{c}^{2}}}+8\,{\frac{d{e}^{3}x}{c}}-2\,{\frac{abx{e}^{4}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) }}+4\,{\frac{axd{e}^{3}}{c \left ( c{x}^{2}+bx+a \right ) }}+{\frac{{e}^{4}x{b}^{3}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) }}-4\,{\frac{{b}^{2}xd{e}^{3}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) }}+6\,{\frac{bx{d}^{2}{e}^{2}}{c \left ( c{x}^{2}+bx+a \right ) }}-4\,{\frac{e{d}^{3}x}{c{x}^{2}+bx+a}}-{\frac{{a}^{2}{e}^{4}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) }}+{\frac{a{b}^{2}{e}^{4}}{{c}^{3} \left ( c{x}^{2}+bx+a \right ) }}-4\,{\frac{abd{e}^{3}}{{c}^{2} \left ( c{x}^{2}+bx+a \right ) }}+6\,{\frac{a{d}^{2}{e}^{2}}{c \left ( c{x}^{2}+bx+a \right ) }}-{\frac{{d}^{4}}{c{x}^{2}+bx+a}}-2\,{\frac{{e}^{4}\ln \left ( c{x}^{2}+bx+a \right ) a}{{c}^{2}}}+2\,{\frac{{e}^{4}\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}}{{c}^{3}}}-6\,{\frac{{e}^{3}\ln \left ( c{x}^{2}+bx+a \right ) db}{{c}^{2}}}+6\,{\frac{{e}^{2}\ln \left ( c{x}^{2}+bx+a \right ){d}^{2}}{c}}+12\,{\frac{ab{e}^{4}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-24\,{\frac{ad{e}^{3}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+8\,{\frac{e{d}^{3}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-4\,{\frac{{b}^{3}{e}^{4}}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+12\,{\frac{{b}^{2}d{e}^{3}}{{c}^{2}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-12\,{\frac{b{d}^{2}{e}^{2}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^4*x^2/c-3*e^4/c^2*b*x+8*e^3/c*d*x-2/c^2/(c*x^2+b*x+a)*x*a*b*e^4+4/c/(c*x^2+b*x
+a)*x*a*d*e^3+1/c^3/(c*x^2+b*x+a)*e^4*x*b^3-4/c^2/(c*x^2+b*x+a)*x*b^2*d*e^3+6/c/
(c*x^2+b*x+a)*x*b*d^2*e^2-4/(c*x^2+b*x+a)*x*d^3*e-1/c^2/(c*x^2+b*x+a)*a^2*e^4+1/
c^3/(c*x^2+b*x+a)*a*b^2*e^4-4/c^2/(c*x^2+b*x+a)*a*b*d*e^3+6/c/(c*x^2+b*x+a)*a*d^
2*e^2-1/(c*x^2+b*x+a)*d^4-2/c^2*e^4*ln(c*x^2+b*x+a)*a+2/c^3*e^4*ln(c*x^2+b*x+a)*
b^2-6/c^2*e^3*ln(c*x^2+b*x+a)*d*b+6/c*e^2*ln(c*x^2+b*x+a)*d^2+12/c^2/(4*a*c-b^2)
^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*e^4-24/c/(4*a*c-b^2)^(1/2)*arctan
((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d*e^3+8/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*
c-b^2)^(1/2))*d^3*e-4/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*
b^3*e^4+12/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d*e^3-1
2/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d^2*e^2

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.312807, 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)^2,x, algorithm="fricas")

[Out]

[(2*(2*a*c^3*d^3*e - 3*a*b*c^2*d^2*e^2 + 3*(a*b^2*c - 2*a^2*c^2)*d*e^3 - (a*b^3
- 3*a^2*b*c)*e^4 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 + 3*(b^2*c^2 - 2*a*c^3)*d*e^3
- (b^3*c - 3*a*b*c^2)*e^4)*x^2 + (2*b*c^3*d^3*e - 3*b^2*c^2*d^2*e^2 + 3*(b^3*c -
2*a*b*c^2)*d*e^3 - (b^4 - 3*a*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)) + (c^3*e^4*x^4 - c^3*d^4 + 6*a*c^2*d^2*e^2 - 4*a*b*c*d*e^3 + (a*b^2 - a^2*
c)*e^4 + 2*(4*c^3*d*e^3 - b*c^2*e^4)*x^3 + (8*b*c^2*d*e^3 - (3*b^2*c - a*c^2)*e^
4)*x^2 - (4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 4*(b^2*c - 3*a*c^2)*d*e^3 - (b^3 - 5*a
*b*c)*e^4)*x + 2*(3*a*c^2*d^2*e^2 - 3*a*b*c*d*e^3 + (a*b^2 - a^2*c)*e^4 + (3*c^3
*d^2*e^2 - 3*b*c^2*d*e^3 + (b^2*c - a*c^2)*e^4)*x^2 + (3*b*c^2*d^2*e^2 - 3*b^2*c
*d*e^3 + (b^3 - a*b*c)*e^4)*x)*log(c*x^2 + b*x + a))*sqrt(b^2 - 4*a*c))/((c^4*x^
2 + b*c^3*x + a*c^3)*sqrt(b^2 - 4*a*c)), (4*(2*a*c^3*d^3*e - 3*a*b*c^2*d^2*e^2 +
3*(a*b^2*c - 2*a^2*c^2)*d*e^3 - (a*b^3 - 3*a^2*b*c)*e^4 + (2*c^4*d^3*e - 3*b*c^
3*d^2*e^2 + 3*(b^2*c^2 - 2*a*c^3)*d*e^3 - (b^3*c - 3*a*b*c^2)*e^4)*x^2 + (2*b*c^
3*d^3*e - 3*b^2*c^2*d^2*e^2 + 3*(b^3*c - 2*a*b*c^2)*d*e^3 - (b^4 - 3*a*b^2*c)*e^
4)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (c^3*e^4*x^4 - c^3
*d^4 + 6*a*c^2*d^2*e^2 - 4*a*b*c*d*e^3 + (a*b^2 - a^2*c)*e^4 + 2*(4*c^3*d*e^3 -
b*c^2*e^4)*x^3 + (8*b*c^2*d*e^3 - (3*b^2*c - a*c^2)*e^4)*x^2 - (4*c^3*d^3*e - 6*
b*c^2*d^2*e^2 + 4*(b^2*c - 3*a*c^2)*d*e^3 - (b^3 - 5*a*b*c)*e^4)*x + 2*(3*a*c^2*
d^2*e^2 - 3*a*b*c*d*e^3 + (a*b^2 - a^2*c)*e^4 + (3*c^3*d^2*e^2 - 3*b*c^2*d*e^3 +
(b^2*c - a*c^2)*e^4)*x^2 + (3*b*c^2*d^2*e^2 - 3*b^2*c*d*e^3 + (b^3 - a*b*c)*e^4
)*x)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/((c^4*x^2 + b*c^3*x + a*c^3)*sqrt
(-b^2 + 4*a*c))]

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Sympy [A]  time = 80.9618, size = 1071, normalized size = 6.23 \[ \text{result too large to display} \]

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)**2,x)

[Out]

(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*c
+ b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a
*c - b**2)))*log(x + (8*a**2*c*e**4 - 4*a*b**2*e**4 + 12*a*b*c*d*e**3 + 4*a*c**3
*(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*
c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*
a*c - b**2))) - 24*a*c**2*d**2*e**2 - b**2*c**2*(-2*e**2*(a*c*e**2 - b**2*e**2 +
3*b*c*d*e - 3*c**2*d**2)/c**3 - 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e*
*2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2))) + 4*b*c**2*d**3*e)/
(12*a*b*c*e**4 - 24*a*c**2*d*e**3 - 4*b**3*e**4 + 12*b**2*c*d*e**3 - 12*b*c**2*d
**2*e**2 + 8*c**3*d**3*e)) + (-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2
*d**2)/c**3 + 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*
c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2)))*log(x + (8*a**2*c*e**4 - 4*a*b**2*e**4
+ 12*a*b*c*d*e**3 + 4*a*c**3*(-2*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**
2*d**2)/c**3 + 2*e*sqrt(-4*a*c + b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b
*c*d*e - c**2*d**2)/(c**3*(4*a*c - b**2))) - 24*a*c**2*d**2*e**2 - b**2*c**2*(-2
*e**2*(a*c*e**2 - b**2*e**2 + 3*b*c*d*e - 3*c**2*d**2)/c**3 + 2*e*sqrt(-4*a*c +
b**2)*(b*e - 2*c*d)*(3*a*c*e**2 - b**2*e**2 + b*c*d*e - c**2*d**2)/(c**3*(4*a*c
- b**2))) + 4*b*c**2*d**3*e)/(12*a*b*c*e**4 - 24*a*c**2*d*e**3 - 4*b**3*e**4 + 1
2*b**2*c*d*e**3 - 12*b*c**2*d**2*e**2 + 8*c**3*d**3*e)) - (a**2*c*e**4 - a*b**2*
e**4 + 4*a*b*c*d*e**3 - 6*a*c**2*d**2*e**2 + c**3*d**4 + x*(2*a*b*c*e**4 - 4*a*c
**2*d*e**3 - b**3*e**4 + 4*b**2*c*d*e**3 - 6*b*c**2*d**2*e**2 + 4*c**3*d**3*e))/
(a*c**3 + b*c**3*x + c**4*x**2) + e**4*x**2/c - x*(3*b*e**4 - 8*c*d*e**3)/c**2

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GIAC/XCAS [A]  time = 0.27463, size = 385, normalized size = 2.24 \[ \frac{2 \,{\left (3 \, c^{2} d^{2} e^{2} - 3 \, b c d e^{3} + b^{2} e^{4} - a c e^{4}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{c^{3}} + \frac{4 \,{\left (2 \, c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} + 3 \, b^{2} c d e^{3} - 6 \, a c^{2} d e^{3} - b^{3} e^{4} + 3 \, a b c e^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{3}} + \frac{c^{3} x^{2} e^{4} + 8 \, c^{3} d x e^{3} - 3 \, b c^{2} x e^{4}}{c^{4}} - \frac{c^{3} d^{4} - 6 \, a c^{2} d^{2} e^{2} + 4 \, a b c d e^{3} - a b^{2} e^{4} + a^{2} c e^{4} +{\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \, b^{2} c d e^{3} - 4 \, a c^{2} d e^{3} - b^{3} e^{4} + 2 \, a b c e^{4}\right )} x}{{\left (c x^{2} + b x + a\right )} c^{3}} \]

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)^2,x, algorithm="giac")

[Out]

2*(3*c^2*d^2*e^2 - 3*b*c*d*e^3 + b^2*e^4 - a*c*e^4)*ln(c*x^2 + b*x + a)/c^3 + 4*
(2*c^3*d^3*e - 3*b*c^2*d^2*e^2 + 3*b^2*c*d*e^3 - 6*a*c^2*d*e^3 - b^3*e^4 + 3*a*b
*c*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^3) + (c^3*x
^2*e^4 + 8*c^3*d*x*e^3 - 3*b*c^2*x*e^4)/c^4 - (c^3*d^4 - 6*a*c^2*d^2*e^2 + 4*a*b
*c*d*e^3 - a*b^2*e^4 + a^2*c*e^4 + (4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 4*b^2*c*d*e^
3 - 4*a*c^2*d*e^3 - b^3*e^4 + 2*a*b*c*e^4)*x)/((c*x^2 + b*x + a)*c^3)