3.1541 \(\int \frac{(b+2 c x) (d+e x)^3}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=126 \[ \frac{6 e \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{3 e (d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(d+e x)^3}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.220894, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{6 e \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{3 e (d+e x) (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(d+e x)^3}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.059, size = 117, normalized size = 0.93 \[ \frac{3 e \left (d + e x\right ) \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} + \frac{6 e \left (a e^{2} - b d e + c d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{\left (d + e x\right )^{3}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.51822, size = 216, normalized size = 1.71 \[ \frac{1}{2} \left (\frac{12 e \left (e (a e-b d)+c d^2\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{e \left (b c \left (7 a e^2+3 c d (d-2 e x)\right )+2 c^2 \left (3 c d^2 x-a e (12 d+5 e x)\right )+b^3 \left (-e^2\right )+b^2 c e (3 d+4 e x)\right )}{c^2 \left (4 a c-b^2\right ) (a+x (b+c x))}+\frac{c e^2 (3 a d+a e x+3 b d x)-b e^3 (a+b x)-c^2 d^2 (d+3 e x)}{c^2 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-(b*e^3*(a + b*x)) - c^2*d^2*(d + 3*e*x) + c*e^2*(3*a*d + 3*b*d*x + a*e*x))/(c
^2*(a + x*(b + c*x))^2) + (e*(-(b^3*e^2) + b^2*c*e*(3*d + 4*e*x) + b*c*(7*a*e^2
+ 3*c*d*(d - 2*e*x)) + 2*c^2*(3*c*d^2*x - a*e*(12*d + 5*e*x))))/(c^2*(-b^2 + 4*a
*c)*(a + x*(b + c*x))) + (12*e*(c*d^2 + e*(-(b*d) + a*e))*ArcTan[(b + 2*c*x)/Sqr
t[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2))/2

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Maple [B]  time = 0.016, size = 365, normalized size = 2.9 \[{\frac{1}{ \left ( c{x}^{2}+bx+a \right ) ^{2}} \left ( -{\frac{e \left ( 5\,ac{e}^{2}-2\,{b}^{2}{e}^{2}+3\,bcde-3\,{c}^{2}{d}^{2} \right ){x}^{3}}{4\,ac-{b}^{2}}}-{\frac{3\,e \left ( c{e}^{2}ab+8\,a{c}^{2}de-{b}^{3}{e}^{2}+{b}^{2}cde-3\,b{c}^{2}{d}^{2} \right ){x}^{2}}{2\,c \left ( 4\,ac-{b}^{2} \right ) }}-3\,{\frac{e \left ( c{e}^{2}{a}^{2}-a{b}^{2}{e}^{2}+3\,abcde+a{c}^{2}{d}^{2}-c{b}^{2}{d}^{2} \right ) x}{c \left ( 4\,ac-{b}^{2} \right ) }}+{\frac{3\,{a}^{2}b{e}^{3}-12\,{a}^{2}cd{e}^{2}+3\,abc{d}^{2}e-4\,a{c}^{2}{d}^{3}+{b}^{2}{d}^{3}c}{2\,c \left ( 4\,ac-{b}^{2} \right ) }} \right ) }+6\,{\frac{a{e}^{3}}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-6\,{\frac{bd{e}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+6\,{\frac{{d}^{2}ec}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-e*(5*a*c*e^2-2*b^2*e^2+3*b*c*d*e-3*c^2*d^2)/(4*a*c-b^2)*x^3-3/2*e*(a*b*c*e^2+8
*a*c^2*d*e-b^3*e^2+b^2*c*d*e-3*b*c^2*d^2)/c/(4*a*c-b^2)*x^2-3/c*e*(a^2*c*e^2-a*b
^2*e^2+3*a*b*c*d*e+a*c^2*d^2-b^2*c*d^2)/(4*a*c-b^2)*x+1/2*(3*a^2*b*e^3-12*a^2*c*
d*e^2+3*a*b*c*d^2*e-4*a*c^2*d^3+b^2*c*d^3)/c/(4*a*c-b^2))/(c*x^2+b*x+a)^2+6*e^3/
(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a-6*e^2/(4*a*c-b^2)^(3/2)*
arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d+6*e/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(
4*a*c-b^2)^(1/2))*c*d^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)^3/(c*x^2 + b*x + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.316978, 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)^3/(c*x^2 + b*x + a)^3,x, algorithm="fricas")

[Out]

[-1/2*(6*(a^2*c^2*d^2*e - a^2*b*c*d*e^2 + a^3*c*e^3 + (c^4*d^2*e - b*c^3*d*e^2 +
a*c^3*e^3)*x^4 + 2*(b*c^3*d^2*e - b^2*c^2*d*e^2 + a*b*c^2*e^3)*x^3 + ((b^2*c^2
+ 2*a*c^3)*d^2*e - (b^3*c + 2*a*b*c^2)*d*e^2 + (a*b^2*c + 2*a^2*c^2)*e^3)*x^2 +
2*(a*b*c^2*d^2*e - a*b^2*c*d*e^2 + a^2*b*c*e^3)*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)) + (3*a*b*c*d^2*e - 12*a^2*c*d*e^2 + 3*a^2*b*e^3 + (b^2*c - 4*a*c^2)*d
^3 + 2*(3*c^3*d^2*e - 3*b*c^2*d*e^2 + (2*b^2*c - 5*a*c^2)*e^3)*x^3 + 3*(3*b*c^2*
d^2*e - (b^2*c + 8*a*c^2)*d*e^2 + (b^3 - a*b*c)*e^3)*x^2 - 6*(3*a*b*c*d*e^2 - (b
^2*c - a*c^2)*d^2*e - (a*b^2 - a^2*c)*e^3)*x)*sqrt(b^2 - 4*a*c))/((a^2*b^2*c - 4
*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*x^3 + (b^4*c - 2*a*
b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)*sqrt(b^2 - 4*a*c)), -1/2
*(12*(a^2*c^2*d^2*e - a^2*b*c*d*e^2 + a^3*c*e^3 + (c^4*d^2*e - b*c^3*d*e^2 + a*c
^3*e^3)*x^4 + 2*(b*c^3*d^2*e - b^2*c^2*d*e^2 + a*b*c^2*e^3)*x^3 + ((b^2*c^2 + 2*
a*c^3)*d^2*e - (b^3*c + 2*a*b*c^2)*d*e^2 + (a*b^2*c + 2*a^2*c^2)*e^3)*x^2 + 2*(a
*b*c^2*d^2*e - a*b^2*c*d*e^2 + a^2*b*c*e^3)*x)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x
+ b)/(b^2 - 4*a*c)) + (3*a*b*c*d^2*e - 12*a^2*c*d*e^2 + 3*a^2*b*e^3 + (b^2*c -
4*a*c^2)*d^3 + 2*(3*c^3*d^2*e - 3*b*c^2*d*e^2 + (2*b^2*c - 5*a*c^2)*e^3)*x^3 + 3
*(3*b*c^2*d^2*e - (b^2*c + 8*a*c^2)*d*e^2 + (b^3 - a*b*c)*e^3)*x^2 - 6*(3*a*b*c*
d*e^2 - (b^2*c - a*c^2)*d^2*e - (a*b^2 - a^2*c)*e^3)*x)*sqrt(-b^2 + 4*a*c))/((a^
2*b^2*c - 4*a^3*c^2 + (b^2*c^3 - 4*a*c^4)*x^4 + 2*(b^3*c^2 - 4*a*b*c^3)*x^3 + (b
^4*c - 2*a*b^2*c^2 - 8*a^2*c^3)*x^2 + 2*(a*b^3*c - 4*a^2*b*c^2)*x)*sqrt(-b^2 + 4
*a*c))]

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Sympy [A]  time = 140.929, size = 762, normalized size = 6.05 \[ - 3 e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (x + \frac{- 48 a^{2} c^{2} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 24 a b^{2} c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b e^{3} - 3 b^{4} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{2} d e^{2} + 3 b c d^{2} e}{6 a c e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e} \right )} + 3 e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) \log{\left (x + \frac{48 a^{2} c^{2} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 24 a b^{2} c e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) + 3 a b e^{3} + 3 b^{4} e \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (a e^{2} - b d e + c d^{2}\right ) - 3 b^{2} d e^{2} + 3 b c d^{2} e}{6 a c e^{3} - 6 b c d e^{2} + 6 c^{2} d^{2} e} \right )} - \frac{- 3 a^{2} b e^{3} + 12 a^{2} c d e^{2} - 3 a b c d^{2} e + 4 a c^{2} d^{3} - b^{2} c d^{3} + x^{3} \left (10 a c^{2} e^{3} - 4 b^{2} c e^{3} + 6 b c^{2} d e^{2} - 6 c^{3} d^{2} e\right ) + x^{2} \left (3 a b c e^{3} + 24 a c^{2} d e^{2} - 3 b^{3} e^{3} + 3 b^{2} c d e^{2} - 9 b c^{2} d^{2} e\right ) + x \left (6 a^{2} c e^{3} - 6 a b^{2} e^{3} + 18 a b c d e^{2} + 6 a c^{2} d^{2} e - 6 b^{2} c d^{2} e\right )}{8 a^{3} c^{2} - 2 a^{2} b^{2} c + x^{4} \left (8 a c^{4} - 2 b^{2} c^{3}\right ) + x^{3} \left (16 a b c^{3} - 4 b^{3} c^{2}\right ) + x^{2} \left (16 a^{2} c^{3} + 4 a b^{2} c^{2} - 2 b^{4} c\right ) + x \left (16 a^{2} b c^{2} - 4 a b^{3} c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2)*log(x + (-48*a**2*c**2
*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) + 24*a*b**2*c*e*sqrt(-1/
(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) + 3*a*b*e**3 - 3*b**4*e*sqrt(-1/(4*
a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2) - 3*b**2*d*e**2 + 3*b*c*d**2*e)/(6*a*c
*e**3 - 6*b*c*d*e**2 + 6*c**2*d**2*e)) + 3*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2
- b*d*e + c*d**2)*log(x + (48*a**2*c**2*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b
*d*e + c*d**2) - 24*a*b**2*c*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d*
*2) + 3*a*b*e**3 + 3*b**4*e*sqrt(-1/(4*a*c - b**2)**3)*(a*e**2 - b*d*e + c*d**2)
- 3*b**2*d*e**2 + 3*b*c*d**2*e)/(6*a*c*e**3 - 6*b*c*d*e**2 + 6*c**2*d**2*e)) -
(-3*a**2*b*e**3 + 12*a**2*c*d*e**2 - 3*a*b*c*d**2*e + 4*a*c**2*d**3 - b**2*c*d**
3 + x**3*(10*a*c**2*e**3 - 4*b**2*c*e**3 + 6*b*c**2*d*e**2 - 6*c**3*d**2*e) + x*
*2*(3*a*b*c*e**3 + 24*a*c**2*d*e**2 - 3*b**3*e**3 + 3*b**2*c*d*e**2 - 9*b*c**2*d
**2*e) + x*(6*a**2*c*e**3 - 6*a*b**2*e**3 + 18*a*b*c*d*e**2 + 6*a*c**2*d**2*e -
6*b**2*c*d**2*e))/(8*a**3*c**2 - 2*a**2*b**2*c + x**4*(8*a*c**4 - 2*b**2*c**3) +
x**3*(16*a*b*c**3 - 4*b**3*c**2) + x**2*(16*a**2*c**3 + 4*a*b**2*c**2 - 2*b**4*
c) + x*(16*a**2*b*c**2 - 4*a*b**3*c))

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GIAC/XCAS [A]  time = 0.273895, size = 394, normalized size = 3.13 \[ -\frac{6 \,{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{6 \, c^{3} d^{2} x^{3} e - 6 \, b c^{2} d x^{3} e^{2} + 9 \, b c^{2} d^{2} x^{2} e + 4 \, b^{2} c x^{3} e^{3} - 10 \, a c^{2} x^{3} e^{3} - 3 \, b^{2} c d x^{2} e^{2} - 24 \, a c^{2} d x^{2} e^{2} + 6 \, b^{2} c d^{2} x e - 6 \, a c^{2} d^{2} x e + b^{2} c d^{3} - 4 \, a c^{2} d^{3} + 3 \, b^{3} x^{2} e^{3} - 3 \, a b c x^{2} e^{3} - 18 \, a b c d x e^{2} + 3 \, a b c d^{2} e + 6 \, a b^{2} x e^{3} - 6 \, a^{2} c x e^{3} - 12 \, a^{2} c d e^{2} + 3 \, a^{2} b e^{3}}{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )}{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*(c*d^2*e - b*d*e^2 + a*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2 - 4*
a*c)*sqrt(-b^2 + 4*a*c)) - 1/2*(6*c^3*d^2*x^3*e - 6*b*c^2*d*x^3*e^2 + 9*b*c^2*d^
2*x^2*e + 4*b^2*c*x^3*e^3 - 10*a*c^2*x^3*e^3 - 3*b^2*c*d*x^2*e^2 - 24*a*c^2*d*x^
2*e^2 + 6*b^2*c*d^2*x*e - 6*a*c^2*d^2*x*e + b^2*c*d^3 - 4*a*c^2*d^3 + 3*b^3*x^2*
e^3 - 3*a*b*c*x^2*e^3 - 18*a*b*c*d*x*e^2 + 3*a*b*c*d^2*e + 6*a*b^2*x*e^3 - 6*a^2
*c*x*e^3 - 12*a^2*c*d*e^2 + 3*a^2*b*e^3)/((b^2*c - 4*a*c^2)*(c*x^2 + b*x + a)^2)