3.1172 \(\int \frac{(b d+2 c d x)^4}{\left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=92 \[ -\frac{12 c^2 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

-(d^4*(b + 2*c*x)^3)/(2*(a + b*x + c*x^2)^2) - (3*c*d^4*(b + 2*c*x))/(a + b*x +
c*x^2) - (12*c^2*d^4*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c]

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Rubi [A]  time = 0.146198, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{12 c^2 d^4 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\sqrt{b^2-4 a c}}-\frac{3 c d^4 (b+2 c x)}{a+b x+c x^2}-\frac{d^4 (b+2 c x)^3}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(d^4*(b + 2*c*x)^3)/(2*(a + b*x + c*x^2)^2) - (3*c*d^4*(b + 2*c*x))/(a + b*x +
c*x^2) - (12*c^2*d^4*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/Sqrt[b^2 - 4*a*c]

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Rubi in Sympy [A]  time = 34.5833, size = 90, normalized size = 0.98 \[ - \frac{12 c^{2} d^{4} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}}} - \frac{3 c d^{4} \left (b + 2 c x\right )}{a + b x + c x^{2}} - \frac{d^{4} \left (b + 2 c x\right )^{3}}{2 \left (a + b x + c x^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-12*c**2*d**4*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/sqrt(-4*a*c + b**2) - 3*c*d
**4*(b + 2*c*x)/(a + b*x + c*x**2) - d**4*(b + 2*c*x)**3/(2*(a + b*x + c*x**2)**
2)

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Mathematica [A]  time = 0.127022, size = 89, normalized size = 0.97 \[ d^4 \left (\frac{12 c^2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}-\frac{(b+2 c x) \left (2 c \left (3 a+5 c x^2\right )+b^2+10 b c x\right )}{2 (a+x (b+c x))^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

d^4*(-((b + 2*c*x)*(b^2 + 10*b*c*x + 2*c*(3*a + 5*c*x^2)))/(2*(a + x*(b + c*x))^
2) + (12*c^2*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])

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Maple [A]  time = 0.015, size = 173, normalized size = 1.9 \[ -10\,{\frac{{d}^{4}{c}^{3}{x}^{3}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-15\,{\frac{{d}^{4}b{c}^{2}{x}^{2}}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{d}^{4}a{c}^{2}x}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-6\,{\frac{{d}^{4}x{b}^{2}c}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-3\,{\frac{{d}^{4}abc}{ \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{d}^{4}{b}^{3}}{2\, \left ( c{x}^{2}+bx+a \right ) ^{2}}}+12\,{\frac{{c}^{2}{d}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-10*d^4/(c*x^2+b*x+a)^2*c^3*x^3-15*d^4/(c*x^2+b*x+a)^2*b*c^2*x^2-6*d^4/(c*x^2+b*
x+a)^2*a*c^2*x-6*d^4/(c*x^2+b*x+a)^2*x*b^2*c-3*d^4/(c*x^2+b*x+a)^2*a*b*c-1/2*d^4
/(c*x^2+b*x+a)^2*b^3+12*d^4*c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(
1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223307, size = 1, normalized size = 0.01 \[ \left [\frac{12 \,{\left (c^{4} d^{4} x^{4} + 2 \, b c^{3} d^{4} x^{3} + 2 \, a b c^{2} d^{4} x + a^{2} c^{2} d^{4} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) -{\left (20 \, c^{3} d^{4} x^{3} + 30 \, b c^{2} d^{4} x^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} x +{\left (b^{3} + 6 \, a b c\right )} d^{4}\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{24 \,{\left (c^{4} d^{4} x^{4} + 2 \, b c^{3} d^{4} x^{3} + 2 \, a b c^{2} d^{4} x + a^{2} c^{2} d^{4} +{\left (b^{2} c^{2} + 2 \, a c^{3}\right )} d^{4} x^{2}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (20 \, c^{3} d^{4} x^{3} + 30 \, b c^{2} d^{4} x^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{4} x +{\left (b^{3} + 6 \, a b c\right )} d^{4}\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(12*(c^4*d^4*x^4 + 2*b*c^3*d^4*x^3 + 2*a*b*c^2*d^4*x + a^2*c^2*d^4 + (b^2*c
^2 + 2*a*c^3)*d^4*x^2)*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)) - (20*c^3*d^4*x^3
+ 30*b*c^2*d^4*x^2 + 12*(b^2*c + a*c^2)*d^4*x + (b^3 + 6*a*b*c)*d^4)*sqrt(b^2 -
4*a*c))/((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(b^2 - 4
*a*c)), 1/2*(24*(c^4*d^4*x^4 + 2*b*c^3*d^4*x^3 + 2*a*b*c^2*d^4*x + a^2*c^2*d^4 +
(b^2*c^2 + 2*a*c^3)*d^4*x^2)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*
c)) - (20*c^3*d^4*x^3 + 30*b*c^2*d^4*x^2 + 12*(b^2*c + a*c^2)*d^4*x + (b^3 + 6*a
*b*c)*d^4)*sqrt(-b^2 + 4*a*c))/((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x
^2 + a^2)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 9.06161, size = 303, normalized size = 3.29 \[ - 6 c^{2} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{- 24 a c^{3} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} + 6 b^{2} c^{2} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} + 6 b c^{2} d^{4}}{12 c^{3} d^{4}} \right )} + 6 c^{2} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} \log{\left (x + \frac{24 a c^{3} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} - 6 b^{2} c^{2} d^{4} \sqrt{- \frac{1}{4 a c - b^{2}}} + 6 b c^{2} d^{4}}{12 c^{3} d^{4}} \right )} - \frac{6 a b c d^{4} + b^{3} d^{4} + 30 b c^{2} d^{4} x^{2} + 20 c^{3} d^{4} x^{3} + x \left (12 a c^{2} d^{4} + 12 b^{2} c d^{4}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-6*c**2*d**4*sqrt(-1/(4*a*c - b**2))*log(x + (-24*a*c**3*d**4*sqrt(-1/(4*a*c - b
**2)) + 6*b**2*c**2*d**4*sqrt(-1/(4*a*c - b**2)) + 6*b*c**2*d**4)/(12*c**3*d**4)
) + 6*c**2*d**4*sqrt(-1/(4*a*c - b**2))*log(x + (24*a*c**3*d**4*sqrt(-1/(4*a*c -
b**2)) - 6*b**2*c**2*d**4*sqrt(-1/(4*a*c - b**2)) + 6*b*c**2*d**4)/(12*c**3*d**
4)) - (6*a*b*c*d**4 + b**3*d**4 + 30*b*c**2*d**4*x**2 + 20*c**3*d**4*x**3 + x*(1
2*a*c**2*d**4 + 12*b**2*c*d**4))/(2*a**2 + 4*a*b*x + 4*b*c*x**3 + 2*c**2*x**4 +
x**2*(4*a*c + 2*b**2))

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GIAC/XCAS [A]  time = 0.217635, size = 154, normalized size = 1.67 \[ \frac{12 \, c^{2} d^{4} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} - \frac{20 \, c^{3} d^{4} x^{3} + 30 \, b c^{2} d^{4} x^{2} + 12 \, b^{2} c d^{4} x + 12 \, a c^{2} d^{4} x + b^{3} d^{4} + 6 \, a b c d^{4}}{2 \,{\left (c x^{2} + b x + a\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*c^2*d^4*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/sqrt(-b^2 + 4*a*c) - 1/2*(20*c
^3*d^4*x^3 + 30*b*c^2*d^4*x^2 + 12*b^2*c*d^4*x + 12*a*c^2*d^4*x + b^3*d^4 + 6*a*
b*c*d^4)/(c*x^2 + b*x + a)^2