3.1147 \(\int \frac{(b d+2 c d x)^4}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=72 \[ 2 d^4 \left (b^2-4 a c\right ) (b+2 c x)-2 d^4 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+\frac{2}{3} d^4 (b+2 c x)^3 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.134653, antiderivative size = 72, 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 \[ 2 d^4 \left (b^2-4 a c\right ) (b+2 c x)-2 d^4 \left (b^2-4 a c\right )^{3/2} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+\frac{2}{3} d^4 (b+2 c x)^3 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 34.6187, size = 83, normalized size = 1.15 \[ 2 b d^{4} \left (- 4 a c + b^{2}\right ) + 4 c d^{4} x \left (- 4 a c + b^{2}\right ) + \frac{2 d^{4} \left (b + 2 c x\right )^{3}}{3} - 2 d^{4} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )} \]

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),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0834551, size = 72, normalized size = 1. \[ d^4 \left (\frac{8}{3} c x \left (2 c \left (c x^2-3 a\right )+3 b^2+3 b c x\right )+2 \left (4 a c-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

d^4*((8*c*x*(3*b^2 + 3*b*c*x + 2*c*(-3*a + c*x^2)))/3 + 2*(-b^2 + 4*a*c)^(3/2)*A
rcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])

_______________________________________________________________________________________

Maple [B]  time = 0.004, size = 170, normalized size = 2.4 \[{\frac{16\,{d}^{4}{c}^{3}{x}^{3}}{3}}+8\,{d}^{4}b{c}^{2}{x}^{2}-16\,{d}^{4}a{c}^{2}x+8\,{d}^{4}x{b}^{2}c+32\,{\frac{{d}^{4}{a}^{2}{c}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-16\,{\frac{{d}^{4}ac{b}^{2}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{{d}^{4}{b}^{4}}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

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),x)

[Out]

16/3*d^4*c^3*x^3+8*d^4*b*c^2*x^2-16*d^4*a*c^2*x+8*d^4*x*b^2*c+32*d^4/(4*a*c-b^2)
^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*c^2-16*d^4/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*c*b^2+2*d^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)
/(4*a*c-b^2)^(1/2))*b^4

_______________________________________________________________________________________

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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.212042, size = 1, normalized size = 0.01 \[ \left [\frac{16}{3} \, c^{3} d^{4} x^{3} + 8 \, b c^{2} d^{4} x^{2} -{\left (b^{2} - 4 \, a c\right )}^{\frac{3}{2}} d^{4} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 8 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d^{4} x, \frac{16}{3} \, c^{3} d^{4} x^{3} + 8 \, b c^{2} d^{4} x^{2} - 2 \,{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c} d^{4} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right ) + 8 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d^{4} x\right ] \]

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),x, algorithm="fricas")

[Out]

[16/3*c^3*d^4*x^3 + 8*b*c^2*d^4*x^2 - (b^2 - 4*a*c)^(3/2)*d^4*log((2*c^2*x^2 + 2
*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 8*(b^
2*c - 2*a*c^2)*d^4*x, 16/3*c^3*d^4*x^3 + 8*b*c^2*d^4*x^2 - 2*(b^2 - 4*a*c)*sqrt(
-b^2 + 4*a*c)*d^4*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c)) + 8*(b^2*c - 2*a*c^2)*d
^4*x]

_______________________________________________________________________________________

Sympy [A]  time = 121.962, size = 204, normalized size = 2.83 \[ 8 b c^{2} d^{4} x^{2} + \frac{16 c^{3} d^{4} x^{3}}{3} - d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{4 a b c d^{4} - b^{3} d^{4} - d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{8 a c^{2} d^{4} - 2 b^{2} c d^{4}} \right )} + d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}} \log{\left (x + \frac{4 a b c d^{4} - b^{3} d^{4} + d^{4} \sqrt{- \left (4 a c - b^{2}\right )^{3}}}{8 a c^{2} d^{4} - 2 b^{2} c d^{4}} \right )} + x \left (- 16 a c^{2} d^{4} + 8 b^{2} c d^{4}\right ) \]

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),x)

[Out]

8*b*c**2*d**4*x**2 + 16*c**3*d**4*x**3/3 - d**4*sqrt(-(4*a*c - b**2)**3)*log(x +
(4*a*b*c*d**4 - b**3*d**4 - d**4*sqrt(-(4*a*c - b**2)**3))/(8*a*c**2*d**4 - 2*b
**2*c*d**4)) + d**4*sqrt(-(4*a*c - b**2)**3)*log(x + (4*a*b*c*d**4 - b**3*d**4 +
d**4*sqrt(-(4*a*c - b**2)**3))/(8*a*c**2*d**4 - 2*b**2*c*d**4)) + x*(-16*a*c**2
*d**4 + 8*b**2*c*d**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214704, size = 155, normalized size = 2.15 \[ \frac{2 \,{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c}} + \frac{8 \,{\left (2 \, c^{6} d^{4} x^{3} + 3 \, b c^{5} d^{4} x^{2} + 3 \, b^{2} c^{4} d^{4} x - 6 \, a c^{5} d^{4} x\right )}}{3 \, c^{3}} \]

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),x, algorithm="giac")

[Out]

2*(b^4*d^4 - 8*a*b^2*c*d^4 + 16*a^2*c^2*d^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*
c))/sqrt(-b^2 + 4*a*c) + 8/3*(2*c^6*d^4*x^3 + 3*b*c^5*d^4*x^2 + 3*b^2*c^4*d^4*x
- 6*a*c^5*d^4*x)/c^3