3.1154 \(\int \frac{1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=86 \[ \frac{2}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.17215, antiderivative size = 86, 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 \[ \frac{2}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac{2}{3 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3}-\frac{2 \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 37.4099, size = 82, normalized size = 0.95 \[ - \frac{2 \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{d^{4} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{2}{d^{4} \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right )^{2}} + \frac{2}{3 d^{4} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a),x)

[Out]

-2*atanh((b + 2*c*x)/sqrt(-4*a*c + b**2))/(d**4*(-4*a*c + b**2)**(5/2)) + 2/(d**
4*(b + 2*c*x)*(-4*a*c + b**2)**2) + 2/(3*d**4*(b + 2*c*x)**3*(-4*a*c + b**2))

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Mathematica [A]  time = 0.0970067, size = 83, normalized size = 0.97 \[ \frac{2 \left (\frac{b^2-4 a c}{(b+2 c x)^3}+\frac{3 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{3}{b+2 c x}\right )}{3 d^4 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^4*(a + b*x + c*x^2)),x]

[Out]

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

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Maple [A]  time = 0.01, size = 89, normalized size = 1. \[ 2\,{\frac{1}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( 2\,cx+b \right ) }}-{\frac{2}{3\,{d}^{4} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) ^{3}}}+2\,{\frac{1}{{d}^{4} \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^4/(c*x^2+b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.223805, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) + 8 \,{\left (3 \, c^{2} x^{2} + 3 \, b c x + b^{2} - a c\right )} \sqrt{b^{2} - 4 \, a c}}{3 \,{\left (8 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \,{\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x +{\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (3 \,{\left (8 \, c^{3} x^{3} + 12 \, b c^{2} x^{2} + 6 \, b^{2} c x + b^{3}\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 4 \,{\left (3 \, c^{2} x^{2} + 3 \, b c x + b^{2} - a c\right )} \sqrt{-b^{2} + 4 \, a c}\right )}}{3 \,{\left (8 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{4} x^{3} + 12 \,{\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} d^{4} x^{2} + 6 \,{\left (b^{6} c - 8 \, a b^{4} c^{2} + 16 \, a^{2} b^{2} c^{3}\right )} d^{4} x +{\left (b^{7} - 8 \, a b^{5} c + 16 \, a^{2} b^{3} c^{2}\right )} d^{4}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)),x, algorithm="fricas")

[Out]

[1/3*(3*(8*c^3*x^3 + 12*b*c^2*x^2 + 6*b^2*c*x + b^3)*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)) + 8*(3*c^2*x^2 + 3*b*c*x + b^2 - a*c)*sqrt(b^2 - 4*a*c))/((8*(b^4*c
^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^4*x^3 + 12*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^
4)*d^4*x^2 + 6*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^4*x + (b^7 - 8*a*b^5*c +
16*a^2*b^3*c^2)*d^4)*sqrt(b^2 - 4*a*c)), 2/3*(3*(8*c^3*x^3 + 12*b*c^2*x^2 + 6*b
^2*c*x + b^3)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 4*(3*c^2*x
^2 + 3*b*c*x + b^2 - a*c)*sqrt(-b^2 + 4*a*c))/((8*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^
2*c^5)*d^4*x^3 + 12*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d^4*x^2 + 6*(b^6*c -
8*a*b^4*c^2 + 16*a^2*b^2*c^3)*d^4*x + (b^7 - 8*a*b^5*c + 16*a^2*b^3*c^2)*d^4)*sq
rt(-b^2 + 4*a*c))]

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Sympy [A]  time = 8.34749, size = 442, normalized size = 5.14 \[ \frac{- 8 a c + 8 b^{2} + 24 b c x + 24 c^{2} x^{2}}{48 a^{2} b^{3} c^{2} d^{4} - 24 a b^{5} c d^{4} + 3 b^{7} d^{4} + x^{3} \left (384 a^{2} c^{5} d^{4} - 192 a b^{2} c^{4} d^{4} + 24 b^{4} c^{3} d^{4}\right ) + x^{2} \left (576 a^{2} b c^{4} d^{4} - 288 a b^{3} c^{3} d^{4} + 36 b^{5} c^{2} d^{4}\right ) + x \left (288 a^{2} b^{2} c^{3} d^{4} - 144 a b^{4} c^{2} d^{4} + 18 b^{6} c d^{4}\right )} - \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{- 64 a^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 48 a^{2} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 12 a b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + b}{2 c} \right )}}{d^{4}} + \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \log{\left (x + \frac{64 a^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - 48 a^{2} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + 12 a b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} - b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} + b}{2 c} \right )}}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**4/(c*x**2+b*x+a),x)

[Out]

(-8*a*c + 8*b**2 + 24*b*c*x + 24*c**2*x**2)/(48*a**2*b**3*c**2*d**4 - 24*a*b**5*
c*d**4 + 3*b**7*d**4 + x**3*(384*a**2*c**5*d**4 - 192*a*b**2*c**4*d**4 + 24*b**4
*c**3*d**4) + x**2*(576*a**2*b*c**4*d**4 - 288*a*b**3*c**3*d**4 + 36*b**5*c**2*d
**4) + x*(288*a**2*b**2*c**3*d**4 - 144*a*b**4*c**2*d**4 + 18*b**6*c*d**4)) - sq
rt(-1/(4*a*c - b**2)**5)*log(x + (-64*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5) + 48*
a**2*b**2*c**2*sqrt(-1/(4*a*c - b**2)**5) - 12*a*b**4*c*sqrt(-1/(4*a*c - b**2)**
5) + b**6*sqrt(-1/(4*a*c - b**2)**5) + b)/(2*c))/d**4 + sqrt(-1/(4*a*c - b**2)**
5)*log(x + (64*a**3*c**3*sqrt(-1/(4*a*c - b**2)**5) - 48*a**2*b**2*c**2*sqrt(-1/
(4*a*c - b**2)**5) + 12*a*b**4*c*sqrt(-1/(4*a*c - b**2)**5) - b**6*sqrt(-1/(4*a*
c - b**2)**5) + b)/(2*c))/d**4

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GIAC/XCAS [A]  time = 0.215957, size = 173, normalized size = 2.01 \[ \frac{2 \, \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{8 \,{\left (3 \, c^{2} x^{2} + 3 \, b c x + b^{2} - a c\right )}}{3 \,{\left (b^{4} d^{4} - 8 \, a b^{2} c d^{4} + 16 \, a^{2} c^{2} d^{4}\right )}{\left (2 \, c x + b\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^4*(c*x^2 + b*x + a)),x, algorithm="giac")

[Out]

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