3.1178 \(\int \frac{1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx\)

Optimal. Leaf size=154 \[ \frac{48 c^2 \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^4}+\frac{48 c^2}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}-\frac{96 c^2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^4}+\frac{6 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2} \]

[Out]

(48*c^2)/((b^2 - 4*a*c)^3*d^3*(b + 2*c*x)^2) - 1/(2*(b^2 - 4*a*c)*d^3*(b + 2*c*x
)^2*(a + b*x + c*x^2)^2) + (6*c)/((b^2 - 4*a*c)^2*d^3*(b + 2*c*x)^2*(a + b*x + c
*x^2)) - (96*c^2*Log[b + 2*c*x])/((b^2 - 4*a*c)^4*d^3) + (48*c^2*Log[a + b*x + c
*x^2])/((b^2 - 4*a*c)^4*d^3)

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Rubi [A]  time = 0.232916, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{48 c^2 \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^4}+\frac{48 c^2}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}-\frac{96 c^2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^4}+\frac{6 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac{1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

(48*c^2)/((b^2 - 4*a*c)^3*d^3*(b + 2*c*x)^2) - 1/(2*(b^2 - 4*a*c)*d^3*(b + 2*c*x
)^2*(a + b*x + c*x^2)^2) + (6*c)/((b^2 - 4*a*c)^2*d^3*(b + 2*c*x)^2*(a + b*x + c
*x^2)) - (96*c^2*Log[b + 2*c*x])/((b^2 - 4*a*c)^4*d^3) + (48*c^2*Log[a + b*x + c
*x^2])/((b^2 - 4*a*c)^4*d^3)

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Rubi in Sympy [A]  time = 76.915, size = 155, normalized size = 1.01 \[ - \frac{96 c^{2} \log{\left (b + 2 c x \right )}}{d^{3} \left (- 4 a c + b^{2}\right )^{4}} + \frac{48 c^{2} \log{\left (a + b x + c x^{2} \right )}}{d^{3} \left (- 4 a c + b^{2}\right )^{4}} + \frac{48 c^{2}}{d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )^{3}} + \frac{6 c}{d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x + c x^{2}\right )} - \frac{1}{2 d^{3} \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )^{2}} \]

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)**3,x)

[Out]

-96*c**2*log(b + 2*c*x)/(d**3*(-4*a*c + b**2)**4) + 48*c**2*log(a + b*x + c*x**2
)/(d**3*(-4*a*c + b**2)**4) + 48*c**2/(d**3*(b + 2*c*x)**2*(-4*a*c + b**2)**3) +
6*c/(d**3*(b + 2*c*x)**2*(-4*a*c + b**2)**2*(a + b*x + c*x**2)) - 1/(2*d**3*(b
+ 2*c*x)**2*(-4*a*c + b**2)*(a + b*x + c*x**2)**2)

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Mathematica [A]  time = 0.274975, size = 111, normalized size = 0.72 \[ \frac{\frac{32 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^2}+\frac{16 c \left (b^2-4 a c\right )}{a+x (b+c x)}-\frac{\left (b^2-4 a c\right )^2}{(a+x (b+c x))^2}+96 c^2 \log (a+x (b+c x))-192 c^2 \log (b+2 c x)}{2 d^3 \left (b^2-4 a c\right )^4} \]

Antiderivative was successfully verified.

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

[Out]

((32*c^2*(b^2 - 4*a*c))/(b + 2*c*x)^2 - (b^2 - 4*a*c)^2/(a + x*(b + c*x))^2 + (1
6*c*(b^2 - 4*a*c))/(a + x*(b + c*x)) - 192*c^2*Log[b + 2*c*x] + 96*c^2*Log[a + x
*(b + c*x)])/(2*(b^2 - 4*a*c)^4*d^3)

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Maple [B]  time = 0.026, size = 332, normalized size = 2.2 \[ -96\,{\frac{{c}^{2}\ln \left ( 2\,cx+b \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}}-16\,{\frac{{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{3} \left ( 2\,cx+b \right ) ^{2}}}-32\,{\frac{a{x}^{2}{c}^{3}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+8\,{\frac{{b}^{2}{x}^{2}{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-32\,{\frac{abx{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+8\,{\frac{{b}^{3}cx}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-40\,{\frac{{a}^{2}{c}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+12\,{\frac{ac{b}^{2}}{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}-{\frac{{b}^{4}}{2\,{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4} \left ( c{x}^{2}+bx+a \right ) ^{2}}}+48\,{\frac{{c}^{2}\ln \left ( c{x}^{2}+bx+a \right ) }{{d}^{3} \left ( 4\,ac-{b}^{2} \right ) ^{4}}} \]

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)^3,x)

[Out]

-96/d^3*c^2/(4*a*c-b^2)^4*ln(2*c*x+b)-16/d^3*c^2/(4*a*c-b^2)^3/(2*c*x+b)^2-32/d^
3/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*x^2*a*c^3+8/d^3/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*x^
2*b^2*c^2-32/d^3/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*x*a*b*c^2+8/d^3/(4*a*c-b^2)^4/(c*
x^2+b*x+a)^2*b^3*c*x-40/d^3/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*a^2*c^2+12/d^3/(4*a*c-
b^2)^4/(c*x^2+b*x+a)^2*a*c*b^2-1/2/d^3/(4*a*c-b^2)^4/(c*x^2+b*x+a)^2*b^4+48/d^3/
(4*a*c-b^2)^4*c^2*ln(c*x^2+b*x+a)

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Maxima [A]  time = 0.700749, size = 747, normalized size = 4.85 \[ \frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2} + 36 \,{\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \,{\left (b^{3} c + 12 \, a b c^{2}\right )} x}{2 \,{\left (4 \,{\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \,{\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} +{\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \,{\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} +{\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \,{\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x +{\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\right )}} + \frac{48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} - \frac{96 \, c^{2} \log \left (2 \, c x + b\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} \]

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)^3),x, algorithm="maxima")

[Out]

1/2*(96*c^4*x^4 + 192*b*c^3*x^3 - b^4 + 20*a*b^2*c + 32*a^2*c^2 + 36*(3*b^2*c^2
+ 4*a*c^3)*x^2 + 12*(b^3*c + 12*a*b*c^2)*x)/(4*(b^6*c^4 - 12*a*b^4*c^5 + 48*a^2*
b^2*c^6 - 64*a^3*c^7)*d^3*x^6 + 12*(b^7*c^3 - 12*a*b^5*c^4 + 48*a^2*b^3*c^5 - 64
*a^3*b*c^6)*d^3*x^5 + (13*b^8*c^2 - 148*a*b^6*c^3 + 528*a^2*b^4*c^4 - 448*a^3*b^
2*c^5 - 512*a^4*c^6)*d^3*x^4 + 2*(3*b^9*c - 28*a*b^7*c^2 + 48*a^2*b^5*c^3 + 192*
a^3*b^3*c^4 - 512*a^4*b*c^5)*d^3*x^3 + (b^10 - 2*a*b^8*c - 68*a^2*b^6*c^2 + 368*
a^3*b^4*c^3 - 448*a^4*b^2*c^4 - 256*a^5*c^5)*d^3*x^2 + 2*(a*b^9 - 10*a^2*b^7*c +
24*a^3*b^5*c^2 + 32*a^4*b^3*c^3 - 128*a^5*b*c^4)*d^3*x + (a^2*b^8 - 12*a^3*b^6*
c + 48*a^4*b^4*c^2 - 64*a^5*b^2*c^3)*d^3) + 48*c^2*log(c*x^2 + b*x + a)/((b^8 -
16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*d^3) - 96*c^2*log(2
*c*x + b)/((b^8 - 16*a*b^6*c + 96*a^2*b^4*c^2 - 256*a^3*b^2*c^3 + 256*a^4*c^4)*d
^3)

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Fricas [A]  time = 0.235236, size = 1092, normalized size = 7.09 \[ \text{result too large to display} \]

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)^3),x, algorithm="fricas")

[Out]

-1/2*(b^6 - 24*a*b^4*c + 48*a^2*b^2*c^2 + 128*a^3*c^3 - 96*(b^2*c^4 - 4*a*c^5)*x
^4 - 192*(b^3*c^3 - 4*a*b*c^4)*x^3 - 36*(3*b^4*c^2 - 8*a*b^2*c^3 - 16*a^2*c^4)*x
^2 - 12*(b^5*c + 8*a*b^3*c^2 - 48*a^2*b*c^3)*x - 96*(4*c^6*x^6 + 12*b*c^5*x^5 +
a^2*b^2*c^2 + (13*b^2*c^4 + 8*a*c^5)*x^4 + 2*(3*b^3*c^3 + 8*a*b*c^4)*x^3 + (b^4*
c^2 + 10*a*b^2*c^3 + 4*a^2*c^4)*x^2 + 2*(a*b^3*c^2 + 2*a^2*b*c^3)*x)*log(c*x^2 +
b*x + a) + 192*(4*c^6*x^6 + 12*b*c^5*x^5 + a^2*b^2*c^2 + (13*b^2*c^4 + 8*a*c^5)
*x^4 + 2*(3*b^3*c^3 + 8*a*b*c^4)*x^3 + (b^4*c^2 + 10*a*b^2*c^3 + 4*a^2*c^4)*x^2
+ 2*(a*b^3*c^2 + 2*a^2*b*c^3)*x)*log(2*c*x + b))/(4*(b^8*c^4 - 16*a*b^6*c^5 + 96
*a^2*b^4*c^6 - 256*a^3*b^2*c^7 + 256*a^4*c^8)*d^3*x^6 + 12*(b^9*c^3 - 16*a*b^7*c
^4 + 96*a^2*b^5*c^5 - 256*a^3*b^3*c^6 + 256*a^4*b*c^7)*d^3*x^5 + (13*b^10*c^2 -
200*a*b^8*c^3 + 1120*a^2*b^6*c^4 - 2560*a^3*b^4*c^5 + 1280*a^4*b^2*c^6 + 2048*a^
5*c^7)*d^3*x^4 + 2*(3*b^11*c - 40*a*b^9*c^2 + 160*a^2*b^7*c^3 - 1280*a^4*b^3*c^5
+ 2048*a^5*b*c^6)*d^3*x^3 + (b^12 - 6*a*b^10*c - 60*a^2*b^8*c^2 + 640*a^3*b^6*c
^3 - 1920*a^4*b^4*c^4 + 1536*a^5*b^2*c^5 + 1024*a^6*c^6)*d^3*x^2 + 2*(a*b^11 - 1
4*a^2*b^9*c + 64*a^3*b^7*c^2 - 64*a^4*b^5*c^3 - 256*a^5*b^3*c^4 + 512*a^6*b*c^5)
*d^3*x + (a^2*b^10 - 16*a^3*b^8*c + 96*a^4*b^6*c^2 - 256*a^5*b^4*c^3 + 256*a^6*b
^2*c^4)*d^3)

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Sympy [A]  time = 87.719, size = 597, normalized size = 3.88 \[ - \frac{96 c^{2} \log{\left (\frac{b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} + \frac{48 c^{2} \log{\left (\frac{a}{c} + \frac{b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} - \frac{32 a^{2} c^{2} + 20 a b^{2} c - b^{4} + 192 b c^{3} x^{3} + 96 c^{4} x^{4} + x^{2} \left (144 a c^{3} + 108 b^{2} c^{2}\right ) + x \left (144 a b c^{2} + 12 b^{3} c\right )}{128 a^{5} b^{2} c^{3} d^{3} - 96 a^{4} b^{4} c^{2} d^{3} + 24 a^{3} b^{6} c d^{3} - 2 a^{2} b^{8} d^{3} + x^{6} \left (512 a^{3} c^{7} d^{3} - 384 a^{2} b^{2} c^{6} d^{3} + 96 a b^{4} c^{5} d^{3} - 8 b^{6} c^{4} d^{3}\right ) + x^{5} \left (1536 a^{3} b c^{6} d^{3} - 1152 a^{2} b^{3} c^{5} d^{3} + 288 a b^{5} c^{4} d^{3} - 24 b^{7} c^{3} d^{3}\right ) + x^{4} \left (1024 a^{4} c^{6} d^{3} + 896 a^{3} b^{2} c^{5} d^{3} - 1056 a^{2} b^{4} c^{4} d^{3} + 296 a b^{6} c^{3} d^{3} - 26 b^{8} c^{2} d^{3}\right ) + x^{3} \left (2048 a^{4} b c^{5} d^{3} - 768 a^{3} b^{3} c^{4} d^{3} - 192 a^{2} b^{5} c^{3} d^{3} + 112 a b^{7} c^{2} d^{3} - 12 b^{9} c d^{3}\right ) + x^{2} \left (512 a^{5} c^{5} d^{3} + 896 a^{4} b^{2} c^{4} d^{3} - 736 a^{3} b^{4} c^{3} d^{3} + 136 a^{2} b^{6} c^{2} d^{3} + 4 a b^{8} c d^{3} - 2 b^{10} d^{3}\right ) + x \left (512 a^{5} b c^{4} d^{3} - 128 a^{4} b^{3} c^{3} d^{3} - 96 a^{3} b^{5} c^{2} d^{3} + 40 a^{2} b^{7} c d^{3} - 4 a b^{9} d^{3}\right )} \]

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)**3,x)

[Out]

-96*c**2*log(b/(2*c) + x)/(d**3*(4*a*c - b**2)**4) + 48*c**2*log(a/c + b*x/c + x
**2)/(d**3*(4*a*c - b**2)**4) - (32*a**2*c**2 + 20*a*b**2*c - b**4 + 192*b*c**3*
x**3 + 96*c**4*x**4 + x**2*(144*a*c**3 + 108*b**2*c**2) + x*(144*a*b*c**2 + 12*b
**3*c))/(128*a**5*b**2*c**3*d**3 - 96*a**4*b**4*c**2*d**3 + 24*a**3*b**6*c*d**3
- 2*a**2*b**8*d**3 + x**6*(512*a**3*c**7*d**3 - 384*a**2*b**2*c**6*d**3 + 96*a*b
**4*c**5*d**3 - 8*b**6*c**4*d**3) + x**5*(1536*a**3*b*c**6*d**3 - 1152*a**2*b**3
*c**5*d**3 + 288*a*b**5*c**4*d**3 - 24*b**7*c**3*d**3) + x**4*(1024*a**4*c**6*d*
*3 + 896*a**3*b**2*c**5*d**3 - 1056*a**2*b**4*c**4*d**3 + 296*a*b**6*c**3*d**3 -
26*b**8*c**2*d**3) + x**3*(2048*a**4*b*c**5*d**3 - 768*a**3*b**3*c**4*d**3 - 19
2*a**2*b**5*c**3*d**3 + 112*a*b**7*c**2*d**3 - 12*b**9*c*d**3) + x**2*(512*a**5*
c**5*d**3 + 896*a**4*b**2*c**4*d**3 - 736*a**3*b**4*c**3*d**3 + 136*a**2*b**6*c*
*2*d**3 + 4*a*b**8*c*d**3 - 2*b**10*d**3) + x*(512*a**5*b*c**4*d**3 - 128*a**4*b
**3*c**3*d**3 - 96*a**3*b**5*c**2*d**3 + 40*a**2*b**7*c*d**3 - 4*a*b**9*d**3))

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GIAC/XCAS [A]  time = 0.221425, size = 408, normalized size = 2.65 \[ -\frac{96 \, c^{3}{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{b^{8} c d^{3} - 16 \, a b^{6} c^{2} d^{3} + 96 \, a^{2} b^{4} c^{3} d^{3} - 256 \, a^{3} b^{2} c^{4} d^{3} + 256 \, a^{4} c^{5} d^{3}} + \frac{48 \, c^{2}{\rm ln}\left (c x^{2} + b x + a\right )}{b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}} + \frac{96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 108 \, b^{2} c^{2} x^{2} + 144 \, a c^{3} x^{2} + 12 \, b^{3} c x + 144 \, a b c^{2} x - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2}}{2 \,{\left (b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}\right )}{\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + b^{2} x + 2 \, a c x + a b\right )}^{2}} \]

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)^3),x, algorithm="giac")

[Out]

-96*c^3*ln(abs(2*c*x + b))/(b^8*c*d^3 - 16*a*b^6*c^2*d^3 + 96*a^2*b^4*c^3*d^3 -
256*a^3*b^2*c^4*d^3 + 256*a^4*c^5*d^3) + 48*c^2*ln(c*x^2 + b*x + a)/(b^8*d^3 - 1
6*a*b^6*c*d^3 + 96*a^2*b^4*c^2*d^3 - 256*a^3*b^2*c^3*d^3 + 256*a^4*c^4*d^3) + 1/
2*(96*c^4*x^4 + 192*b*c^3*x^3 + 108*b^2*c^2*x^2 + 144*a*c^3*x^2 + 12*b^3*c*x + 1
44*a*b*c^2*x - b^4 + 20*a*b^2*c + 32*a^2*c^2)/((b^6*d^3 - 12*a*b^4*c*d^3 + 48*a^
2*b^2*c^2*d^3 - 64*a^3*c^3*d^3)*(2*c^2*x^3 + 3*b*c*x^2 + b^2*x + 2*a*c*x + a*b)^
2)