3.1153 \(\int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=70 \[ \frac{\log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^2}+\frac{1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.107803, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^2}+\frac{1}{d^3 \left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.7638, size = 70, normalized size = 1. \[ - \frac{2 \log{\left (b + 2 c x \right )}}{d^{3} \left (- 4 a c + b^{2}\right )^{2}} + \frac{\log{\left (a + b x + c x^{2} \right )}}{d^{3} \left (- 4 a c + b^{2}\right )^{2}} + \frac{1}{d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a),x)

[Out]

-2*log(b + 2*c*x)/(d**3*(-4*a*c + b**2)**2) + log(a + b*x + c*x**2)/(d**3*(-4*a*
c + b**2)**2) + 1/(d**3*(b + 2*c*x)**2*(-4*a*c + b**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0774938, size = 65, normalized size = 0.93 \[ \frac{\frac{\log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^2}+\frac{1}{\left (b^2-4 a c\right ) (b+2 c x)^2}-\frac{2 \log (b+2 c x)}{\left (b^2-4 a c\right )^2}}{d^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((b*d + 2*c*d*x)^3*(a + b*x + c*x^2)),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 78, normalized size = 1.1 \[ -2\,{\frac{\ln \left ( 2\,cx+b \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}}-{\frac{1}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) \left ( 2\,cx+b \right ) ^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2*c*d*x+b*d)^3/(c*x^2+b*x+a),x)

[Out]

-2/d^3/(4*a*c-b^2)^2*ln(2*c*x+b)-1/d^3/(4*a*c-b^2)/(2*c*x+b)^2+1/d^3/(4*a*c-b^2)
^2*ln(c*x^2+b*x+a)

_______________________________________________________________________________________

Maxima [A]  time = 0.685791, size = 174, normalized size = 2.49 \[ \frac{1}{4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{3} x^{2} + 4 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d^{3} x +{\left (b^{4} - 4 \, a b^{2} c\right )} d^{3}} + \frac{\log \left (c x^{2} + b x + a\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3}} - \frac{2 \, \log \left (2 \, c x + b\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)),x, algorithm="maxima")

[Out]

1/(4*(b^2*c^2 - 4*a*c^3)*d^3*x^2 + 4*(b^3*c - 4*a*b*c^2)*d^3*x + (b^4 - 4*a*b^2*
c)*d^3) + log(c*x^2 + b*x + a)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^3) - 2*log(2*c*
x + b)/((b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^3)

_______________________________________________________________________________________

Fricas [A]  time = 0.21508, size = 211, normalized size = 3.01 \[ \frac{b^{2} - 4 \, a c +{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (4 \, c^{2} x^{2} + 4 \, b c x + b^{2}\right )} \log \left (2 \, c x + b\right )}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{3} x^{2} + 4 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{3} x +{\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2}\right )} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 7.3073, size = 119, normalized size = 1.7 \[ - \frac{1}{4 a b^{2} c d^{3} - b^{4} d^{3} + x^{2} \left (16 a c^{3} d^{3} - 4 b^{2} c^{2} d^{3}\right ) + x \left (16 a b c^{2} d^{3} - 4 b^{3} c d^{3}\right )} - \frac{2 \log{\left (\frac{b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{2}} + \frac{\log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2*c*d*x+b*d)**3/(c*x**2+b*x+a),x)

[Out]

-1/(4*a*b**2*c*d**3 - b**4*d**3 + x**2*(16*a*c**3*d**3 - 4*b**2*c**2*d**3) + x*(
16*a*b*c**2*d**3 - 4*b**3*c*d**3)) - 2*log(b/(2*c) + x)/(d**3*(4*a*c - b**2)**2)
+ log(a/c + b*x/c + x**2)/(d**3*(4*a*c - b**2)**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217941, size = 150, normalized size = 2.14 \[ -\frac{2 \, c{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{b^{4} c d^{3} - 8 \, a b^{2} c^{2} d^{3} + 16 \, a^{2} c^{3} d^{3}} + \frac{{\rm ln}\left (c x^{2} + b x + a\right )}{b^{4} d^{3} - 8 \, a b^{2} c d^{3} + 16 \, a^{2} c^{2} d^{3}} + \frac{1}{{\left (b^{2} - 4 \, a c\right )}{\left (2 \, c x + b\right )}^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*c*d*x + b*d)^3*(c*x^2 + b*x + a)),x, algorithm="giac")

[Out]

-2*c*ln(abs(2*c*x + b))/(b^4*c*d^3 - 8*a*b^2*c^2*d^3 + 16*a^2*c^3*d^3) + ln(c*x^
2 + b*x + a)/(b^4*d^3 - 8*a*b^2*c*d^3 + 16*a^2*c^2*d^3) + 1/((b^2 - 4*a*c)*(2*c*
x + b)^2*d^3)