3.1137 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx\)

Optimal. Leaf size=100 \[ \frac{\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac{3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac{3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac{\log (b+2 c x)}{128 c^4 d^7} \]

[Out]

(b^2 - 4*a*c)^3/(768*c^4*d^7*(b + 2*c*x)^6) - (3*(b^2 - 4*a*c)^2)/(512*c^4*d^7*(
b + 2*c*x)^4) + (3*(b^2 - 4*a*c))/(256*c^4*d^7*(b + 2*c*x)^2) + Log[b + 2*c*x]/(
128*c^4*d^7)

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Rubi [A]  time = 0.209038, 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 \[ \frac{\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac{3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac{3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac{\log (b+2 c x)}{128 c^4 d^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^7,x]

[Out]

(b^2 - 4*a*c)^3/(768*c^4*d^7*(b + 2*c*x)^6) - (3*(b^2 - 4*a*c)^2)/(512*c^4*d^7*(
b + 2*c*x)^4) + (3*(b^2 - 4*a*c))/(256*c^4*d^7*(b + 2*c*x)^2) + Log[b + 2*c*x]/(
128*c^4*d^7)

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Rubi in Sympy [A]  time = 42.0429, size = 99, normalized size = 0.99 \[ \frac{\log{\left (b + 2 c x \right )}}{128 c^{4} d^{7}} + \frac{3 \left (- 4 a c + b^{2}\right )}{256 c^{4} d^{7} \left (b + 2 c x\right )^{2}} - \frac{3 \left (- 4 a c + b^{2}\right )^{2}}{512 c^{4} d^{7} \left (b + 2 c x\right )^{4}} + \frac{\left (- 4 a c + b^{2}\right )^{3}}{768 c^{4} d^{7} \left (b + 2 c x\right )^{6}} \]

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)**7,x)

[Out]

log(b + 2*c*x)/(128*c**4*d**7) + 3*(-4*a*c + b**2)/(256*c**4*d**7*(b + 2*c*x)**2
) - 3*(-4*a*c + b**2)**2/(512*c**4*d**7*(b + 2*c*x)**4) + (-4*a*c + b**2)**3/(76
8*c**4*d**7*(b + 2*c*x)**6)

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Mathematica [A]  time = 0.083074, size = 78, normalized size = 0.78 \[ \frac{\frac{2 \left (b^2-4 a c\right )^3}{(b+2 c x)^6}-\frac{9 \left (b^2-4 a c\right )^2}{(b+2 c x)^4}+\frac{18 \left (b^2-4 a c\right )}{(b+2 c x)^2}+12 \log (b+2 c x)}{1536 c^4 d^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^7,x]

[Out]

((2*(b^2 - 4*a*c)^3)/(b + 2*c*x)^6 - (9*(b^2 - 4*a*c)^2)/(b + 2*c*x)^4 + (18*(b^
2 - 4*a*c))/(b + 2*c*x)^2 + 12*Log[b + 2*c*x])/(1536*c^4*d^7)

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Maple [B]  time = 0.015, size = 191, normalized size = 1.9 \[ -{\frac{3\,a}{64\,{d}^{7}{c}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{3\,{b}^{2}}{256\,{c}^{4}{d}^{7} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{\ln \left ( 2\,cx+b \right ) }{128\,{c}^{4}{d}^{7}}}-{\frac{3\,{a}^{2}}{32\,{d}^{7}{c}^{2} \left ( 2\,cx+b \right ) ^{4}}}+{\frac{3\,a{b}^{2}}{64\,{d}^{7}{c}^{3} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{3\,{b}^{4}}{512\,{c}^{4}{d}^{7} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{{a}^{3}}{12\,{d}^{7}c \left ( 2\,cx+b \right ) ^{6}}}+{\frac{{a}^{2}{b}^{2}}{16\,{d}^{7}{c}^{2} \left ( 2\,cx+b \right ) ^{6}}}-{\frac{a{b}^{4}}{64\,{d}^{7}{c}^{3} \left ( 2\,cx+b \right ) ^{6}}}+{\frac{{b}^{6}}{768\,{c}^{4}{d}^{7} \left ( 2\,cx+b \right ) ^{6}}} \]

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)^7,x)

[Out]

-3/64/d^7/c^3/(2*c*x+b)^2*a+3/256/d^7/c^4/(2*c*x+b)^2*b^2+1/128*ln(2*c*x+b)/c^4/
d^7-3/32/d^7/c^2/(2*c*x+b)^4*a^2+3/64/d^7/c^3/(2*c*x+b)^4*a*b^2-3/512/d^7/c^4/(2
*c*x+b)^4*b^4-1/12/d^7/c/(2*c*x+b)^6*a^3+1/16/d^7/c^2/(2*c*x+b)^6*a^2*b^2-1/64/d
^7/c^3/(2*c*x+b)^6*a*b^4+1/768/d^7/c^4/(2*c*x+b)^6*b^6

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Maxima [A]  time = 0.687611, size = 321, normalized size = 3.21 \[ \frac{11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \,{\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{1536 \,{\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} + \frac{\log \left (2 \, c x + b\right )}{128 \, c^{4} d^{7}} \]

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)^7,x, algorithm="maxima")

[Out]

1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^2 - 128*a^3*c^3 + 288*(b^2*c^4 - 4*a*
c^5)*x^4 + 576*(b^3*c^3 - 4*a*b*c^4)*x^3 + 36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^
2*c^4)*x^2 + 36*(3*b^5*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*x)/(64*c^10*d^7*x^6 + 192
*b*c^9*d^7*x^5 + 240*b^2*c^8*d^7*x^4 + 160*b^3*c^7*d^7*x^3 + 60*b^4*c^6*d^7*x^2
+ 12*b^5*c^5*d^7*x + b^6*c^4*d^7) + 1/128*log(2*c*x + b)/(c^4*d^7)

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Fricas [A]  time = 0.220656, size = 394, normalized size = 3.94 \[ \frac{11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \,{\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x + 12 \,{\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \log \left (2 \, c x + b\right )}{1536 \,{\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} \]

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)^7,x, algorithm="fricas")

[Out]

1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^2 - 128*a^3*c^3 + 288*(b^2*c^4 - 4*a*
c^5)*x^4 + 576*(b^3*c^3 - 4*a*b*c^4)*x^3 + 36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^
2*c^4)*x^2 + 36*(3*b^5*c - 8*a*b^3*c^2 - 16*a^2*b*c^3)*x + 12*(64*c^6*x^6 + 192*
b*c^5*x^5 + 240*b^2*c^4*x^4 + 160*b^3*c^3*x^3 + 60*b^4*c^2*x^2 + 12*b^5*c*x + b^
6)*log(2*c*x + b))/(64*c^10*d^7*x^6 + 192*b*c^9*d^7*x^5 + 240*b^2*c^8*d^7*x^4 +
160*b^3*c^7*d^7*x^3 + 60*b^4*c^6*d^7*x^2 + 12*b^5*c^5*d^7*x + b^6*c^4*d^7)

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Sympy [A]  time = 28.2292, size = 245, normalized size = 2.45 \[ - \frac{128 a^{3} c^{3} + 48 a^{2} b^{2} c^{2} + 24 a b^{4} c - 11 b^{6} + x^{4} \left (1152 a c^{5} - 288 b^{2} c^{4}\right ) + x^{3} \left (2304 a b c^{4} - 576 b^{3} c^{3}\right ) + x^{2} \left (576 a^{2} c^{4} + 1440 a b^{2} c^{3} - 396 b^{4} c^{2}\right ) + x \left (576 a^{2} b c^{3} + 288 a b^{3} c^{2} - 108 b^{5} c\right )}{1536 b^{6} c^{4} d^{7} + 18432 b^{5} c^{5} d^{7} x + 92160 b^{4} c^{6} d^{7} x^{2} + 245760 b^{3} c^{7} d^{7} x^{3} + 368640 b^{2} c^{8} d^{7} x^{4} + 294912 b c^{9} d^{7} x^{5} + 98304 c^{10} d^{7} x^{6}} + \frac{\log{\left (b + 2 c x \right )}}{128 c^{4} d^{7}} \]

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)**7,x)

[Out]

-(128*a**3*c**3 + 48*a**2*b**2*c**2 + 24*a*b**4*c - 11*b**6 + x**4*(1152*a*c**5
- 288*b**2*c**4) + x**3*(2304*a*b*c**4 - 576*b**3*c**3) + x**2*(576*a**2*c**4 +
1440*a*b**2*c**3 - 396*b**4*c**2) + x*(576*a**2*b*c**3 + 288*a*b**3*c**2 - 108*b
**5*c))/(1536*b**6*c**4*d**7 + 18432*b**5*c**5*d**7*x + 92160*b**4*c**6*d**7*x**
2 + 245760*b**3*c**7*d**7*x**3 + 368640*b**2*c**8*d**7*x**4 + 294912*b*c**9*d**7
*x**5 + 98304*c**10*d**7*x**6) + log(b + 2*c*x)/(128*c**4*d**7)

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GIAC/XCAS [A]  time = 0.216524, size = 220, normalized size = 2.2 \[ \frac{{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d^{7}} + \frac{11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \,{\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \,{\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{1536 \,{\left (2 \, c x + b\right )}^{6} c^{4} d^{7}} \]

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)^7,x, algorithm="giac")

[Out]

1/128*ln(abs(2*c*x + b))/(c^4*d^7) + 1/1536*(11*b^6 - 24*a*b^4*c - 48*a^2*b^2*c^
2 - 128*a^3*c^3 + 288*(b^2*c^4 - 4*a*c^5)*x^4 + 576*(b^3*c^3 - 4*a*b*c^4)*x^3 +
36*(11*b^4*c^2 - 40*a*b^2*c^3 - 16*a^2*c^4)*x^2 + 36*(3*b^5*c - 8*a*b^3*c^2 - 16
*a^2*b*c^3)*x)/((2*c*x + b)^6*c^4*d^7)