3.1133 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx\)

Optimal. Leaf size=100 \[ -\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^2}{256 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{256 c^4 d^3 (b+2 c x)^2}+\frac{3 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{128 c^4 d^3}+\frac{(b+2 c x)^4}{512 c^4 d^3} \]

[Out]

(b^2 - 4*a*c)^3/(256*c^4*d^3*(b + 2*c*x)^2) - (3*(b^2 - 4*a*c)*(b + 2*c*x)^2)/(2
56*c^4*d^3) + (b + 2*c*x)^4/(512*c^4*d^3) + (3*(b^2 - 4*a*c)^2*Log[b + 2*c*x])/(
128*c^4*d^3)

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Rubi [A]  time = 0.249724, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^2}{256 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{256 c^4 d^3 (b+2 c x)^2}+\frac{3 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{128 c^4 d^3}+\frac{(b+2 c x)^4}{512 c^4 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b^2 - 4*a*c)^3/(256*c^4*d^3*(b + 2*c*x)^2) - (3*(b^2 - 4*a*c)*(b + 2*c*x)^2)/(2
56*c^4*d^3) + (b + 2*c*x)^4/(512*c^4*d^3) + (3*(b^2 - 4*a*c)^2*Log[b + 2*c*x])/(
128*c^4*d^3)

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Rubi in Sympy [A]  time = 39.0247, size = 99, normalized size = 0.99 \[ \frac{\left (b + 2 c x\right )^{4}}{512 c^{4} d^{3}} - \frac{3 \left (b + 2 c x\right )^{2} \left (- 4 a c + b^{2}\right )}{256 c^{4} d^{3}} + \frac{3 \left (- 4 a c + b^{2}\right )^{2} \log{\left (b + 2 c x \right )}}{128 c^{4} d^{3}} + \frac{\left (- 4 a c + b^{2}\right )^{3}}{256 c^{4} d^{3} \left (b + 2 c x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b + 2*c*x)**4/(512*c**4*d**3) - 3*(b + 2*c*x)**2*(-4*a*c + b**2)/(256*c**4*d**3
) + 3*(-4*a*c + b**2)**2*log(b + 2*c*x)/(128*c**4*d**3) + (-4*a*c + b**2)**3/(25
6*c**4*d**3*(b + 2*c*x)**2)

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Mathematica [A]  time = 0.186329, size = 90, normalized size = 0.9 \[ \frac{\frac{\left (b^2-4 a c\right )^3}{c^4 (b+2 c x)^2}+\frac{6 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{c^4}-\frac{8 b x \left (b^2-6 a c\right )}{c^3}+\frac{48 a x^2}{c}+\frac{16 b x^3}{c}+8 x^4}{256 d^3} \]

Antiderivative was successfully verified.

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

[Out]

((-8*b*(b^2 - 6*a*c)*x)/c^3 + (48*a*x^2)/c + (16*b*x^3)/c + 8*x^4 + (b^2 - 4*a*c
)^3/(c^4*(b + 2*c*x)^2) + (6*(b^2 - 4*a*c)^2*Log[b + 2*c*x])/c^4)/(256*d^3)

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Maple [B]  time = 0.012, size = 192, normalized size = 1.9 \[{\frac{{x}^{4}}{32\,{d}^{3}}}+{\frac{b{x}^{3}}{16\,c{d}^{3}}}+{\frac{3\,a{x}^{2}}{16\,c{d}^{3}}}+{\frac{3\,abx}{16\,{c}^{2}{d}^{3}}}-{\frac{x{b}^{3}}{32\,{c}^{3}{d}^{3}}}-{\frac{{a}^{3}}{4\,c{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{3\,{a}^{2}{b}^{2}}{16\,{c}^{2}{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{3\,a{b}^{4}}{64\,{c}^{3}{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{{b}^{6}}{256\,{d}^{3}{c}^{4} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{3\,\ln \left ( 2\,cx+b \right ){a}^{2}}{8\,{c}^{2}{d}^{3}}}-{\frac{3\,\ln \left ( 2\,cx+b \right ) a{b}^{2}}{16\,{c}^{3}{d}^{3}}}+{\frac{3\,\ln \left ( 2\,cx+b \right ){b}^{4}}{128\,{d}^{3}{c}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/32/d^3*x^4+1/16/d^3/c*x^3*b+3/16/d^3/c*a*x^2+3/16/d^3/c^2*x*a*b-1/32/d^3/c^3*x
*b^3-1/4/d^3/c/(2*c*x+b)^2*a^3+3/16/d^3/c^2/(2*c*x+b)^2*a^2*b^2-3/64/d^3/c^3/(2*
c*x+b)^2*a*b^4+1/256/d^3/c^4/(2*c*x+b)^2*b^6+3/8/d^3/c^2*ln(2*c*x+b)*a^2-3/16/d^
3/c^3*ln(2*c*x+b)*a*b^2+3/128/d^3/c^4*ln(2*c*x+b)*b^4

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Maxima [A]  time = 0.70672, size = 198, normalized size = 1.98 \[ \frac{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{256 \,{\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} + \frac{c^{3} x^{4} + 2 \, b c^{2} x^{3} + 6 \, a c^{2} x^{2} -{\left (b^{3} - 6 \, a b c\right )} x}{32 \, c^{3} d^{3}} + \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/(4*c^6*d^3*x^2 + 4*b*c^5*
d^3*x + b^2*c^4*d^3) + 1/32*(c^3*x^4 + 2*b*c^2*x^3 + 6*a*c^2*x^2 - (b^3 - 6*a*b*
c)*x)/(c^3*d^3) + 3/128*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*log(2*c*x + b)/(c^4*d^3)

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Fricas [A]  time = 0.205489, size = 340, normalized size = 3.4 \[ \frac{32 \, c^{6} x^{6} + 96 \, b c^{5} x^{5} + b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 24 \,{\left (3 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} - 16 \,{\left (b^{3} c^{3} - 24 \, a b c^{4}\right )} x^{3} - 16 \,{\left (2 \, b^{4} c^{2} - 15 \, a b^{2} c^{3}\right )} x^{2} - 8 \,{\left (b^{5} c - 6 \, a b^{3} c^{2}\right )} x + 6 \,{\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 4 \,{\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )} \log \left (2 \, c x + b\right )}{256 \,{\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/256*(32*c^6*x^6 + 96*b*c^5*x^5 + b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^
3 + 24*(3*b^2*c^4 + 8*a*c^5)*x^4 - 16*(b^3*c^3 - 24*a*b*c^4)*x^3 - 16*(2*b^4*c^2
- 15*a*b^2*c^3)*x^2 - 8*(b^5*c - 6*a*b^3*c^2)*x + 6*(b^6 - 8*a*b^4*c + 16*a^2*b
^2*c^2 + 4*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 4*(b^5*c - 8*a*b^3*c^2 + 1
6*a^2*b*c^3)*x)*log(2*c*x + b))/(4*c^6*d^3*x^2 + 4*b*c^5*d^3*x + b^2*c^4*d^3)

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Sympy [A]  time = 5.54392, size = 150, normalized size = 1.5 \[ \frac{3 a x^{2}}{16 c d^{3}} + \frac{b x^{3}}{16 c d^{3}} - \frac{64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}}{256 b^{2} c^{4} d^{3} + 1024 b c^{5} d^{3} x + 1024 c^{6} d^{3} x^{2}} + \frac{x^{4}}{32 d^{3}} + \frac{x \left (6 a b c - b^{3}\right )}{32 c^{3} d^{3}} + \frac{3 \left (4 a c - b^{2}\right )^{2} \log{\left (b + 2 c x \right )}}{128 c^{4} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*a*x**2/(16*c*d**3) + b*x**3/(16*c*d**3) - (64*a**3*c**3 - 48*a**2*b**2*c**2 +
12*a*b**4*c - b**6)/(256*b**2*c**4*d**3 + 1024*b*c**5*d**3*x + 1024*c**6*d**3*x*
*2) + x**4/(32*d**3) + x*(6*a*b*c - b**3)/(32*c**3*d**3) + 3*(4*a*c - b**2)**2*l
og(b + 2*c*x)/(128*c**4*d**3)

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GIAC/XCAS [A]  time = 0.216081, size = 200, normalized size = 2. \[ \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d^{3}} + \frac{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{256 \,{\left (2 \, c x + b\right )}^{2} c^{4} d^{3}} + \frac{c^{12} d^{9} x^{4} + 2 \, b c^{11} d^{9} x^{3} + 6 \, a c^{11} d^{9} x^{2} - b^{3} c^{9} d^{9} x + 6 \, a b c^{10} d^{9} x}{32 \, c^{12} d^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/128*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*ln(abs(2*c*x + b))/(c^4*d^3) + 1/256*(b^6 -
12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/((2*c*x + b)^2*c^4*d^3) + 1/32*(c^12*
d^9*x^4 + 2*b*c^11*d^9*x^3 + 6*a*c^11*d^9*x^2 - b^3*c^9*d^9*x + 6*a*b*c^10*d^9*x
)/(c^12*d^12)