3.1136 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=94 \[ \frac{\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac{\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac{3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}+\frac{x}{64 c^3 d^6} \]

[Out]

x/(64*c^3*d^6) + (b^2 - 4*a*c)^3/(640*c^4*d^6*(b + 2*c*x)^5) - (b^2 - 4*a*c)^2/(
128*c^4*d^6*(b + 2*c*x)^3) + (3*(b^2 - 4*a*c))/(128*c^4*d^6*(b + 2*c*x))

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Rubi [A]  time = 0.208068, antiderivative size = 94, 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 \[ \frac{\left (b^2-4 a c\right )^3}{640 c^4 d^6 (b+2 c x)^5}-\frac{\left (b^2-4 a c\right )^2}{128 c^4 d^6 (b+2 c x)^3}+\frac{3 \left (b^2-4 a c\right )}{128 c^4 d^6 (b+2 c x)}+\frac{x}{64 c^3 d^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x]

[Out]

x/(64*c^3*d^6) + (b^2 - 4*a*c)^3/(640*c^4*d^6*(b + 2*c*x)^5) - (b^2 - 4*a*c)^2/(
128*c^4*d^6*(b + 2*c*x)^3) + (3*(b^2 - 4*a*c))/(128*c^4*d^6*(b + 2*c*x))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int \frac{1}{64}\, dx}{c^{3} d^{6}} + \frac{3 \left (- 4 a c + b^{2}\right )}{128 c^{4} d^{6} \left (b + 2 c x\right )} - \frac{\left (- 4 a c + b^{2}\right )^{2}}{128 c^{4} d^{6} \left (b + 2 c x\right )^{3}} + \frac{\left (- 4 a c + b^{2}\right )^{3}}{640 c^{4} d^{6} \left (b + 2 c x\right )^{5}} \]

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)**6,x)

[Out]

Integral(1/64, x)/(c**3*d**6) + 3*(-4*a*c + b**2)/(128*c**4*d**6*(b + 2*c*x)) -
(-4*a*c + b**2)**2/(128*c**4*d**6*(b + 2*c*x)**3) + (-4*a*c + b**2)**3/(640*c**4
*d**6*(b + 2*c*x)**5)

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Mathematica [A]  time = 0.117009, size = 72, normalized size = 0.77 \[ \frac{\frac{\left (b^2-4 a c\right )^3}{(b+2 c x)^5}-\frac{5 \left (b^2-4 a c\right )^2}{(b+2 c x)^3}+\frac{15 \left (b^2-4 a c\right )}{b+2 c x}+10 c x}{640 c^4 d^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^6,x]

[Out]

(10*c*x + (b^2 - 4*a*c)^3/(b + 2*c*x)^5 - (5*(b^2 - 4*a*c)^2)/(b + 2*c*x)^3 + (1
5*(b^2 - 4*a*c))/(b + 2*c*x))/(640*c^4*d^6)

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Maple [A]  time = 0.011, size = 114, normalized size = 1.2 \[{\frac{1}{{d}^{6}} \left ({\frac{x}{64\,{c}^{3}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{640\,{c}^{4} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{384\,{c}^{4} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{12\,ac-3\,{b}^{2}}{128\,{c}^{4} \left ( 2\,cx+b \right ) }} \right ) } \]

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)^6,x)

[Out]

1/d^6*(1/64*x/c^3-1/640/c^4*(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)/(2*c*x+b)
^5-1/384*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^3-1/128*(12*a*c-3*b^2)/c^4/
(2*c*x+b))

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Maxima [A]  time = 0.687257, size = 294, normalized size = 3.13 \[ \frac{11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 240 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 480 \,{\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 20 \,{\left (17 \, b^{4} c^{2} - 64 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 20 \,{\left (5 \, b^{5} c - 16 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{640 \,{\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} + \frac{x}{64 \, c^{3} d^{6}} \]

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)^6,x, algorithm="maxima")

[Out]

1/640*(11*b^6 - 32*a*b^4*c - 32*a^2*b^2*c^2 - 64*a^3*c^3 + 240*(b^2*c^4 - 4*a*c^
5)*x^4 + 480*(b^3*c^3 - 4*a*b*c^4)*x^3 + 20*(17*b^4*c^2 - 64*a*b^2*c^3 - 16*a^2*
c^4)*x^2 + 20*(5*b^5*c - 16*a*b^3*c^2 - 16*a^2*b*c^3)*x)/(32*c^9*d^6*x^5 + 80*b*
c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^6*d^6*x^2 + 10*b^4*c^5*d^6*x + b^5*c
^4*d^6) + 1/64*x/(c^3*d^6)

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Fricas [A]  time = 0.21063, size = 306, normalized size = 3.26 \[ \frac{320 \, c^{6} x^{6} + 800 \, b c^{5} x^{5} + 11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 80 \,{\left (13 \, b^{2} c^{4} - 12 \, a c^{5}\right )} x^{4} + 80 \,{\left (11 \, b^{3} c^{3} - 24 \, a b c^{4}\right )} x^{3} + 40 \,{\left (11 \, b^{4} c^{2} - 32 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} + 10 \,{\left (11 \, b^{5} c - 32 \, a b^{3} c^{2} - 32 \, a^{2} b c^{3}\right )} x}{640 \,{\left (32 \, c^{9} d^{6} x^{5} + 80 \, b c^{8} d^{6} x^{4} + 80 \, b^{2} c^{7} d^{6} x^{3} + 40 \, b^{3} c^{6} d^{6} x^{2} + 10 \, b^{4} c^{5} d^{6} x + b^{5} c^{4} d^{6}\right )}} \]

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)^6,x, algorithm="fricas")

[Out]

1/640*(320*c^6*x^6 + 800*b*c^5*x^5 + 11*b^6 - 32*a*b^4*c - 32*a^2*b^2*c^2 - 64*a
^3*c^3 + 80*(13*b^2*c^4 - 12*a*c^5)*x^4 + 80*(11*b^3*c^3 - 24*a*b*c^4)*x^3 + 40*
(11*b^4*c^2 - 32*a*b^2*c^3 - 8*a^2*c^4)*x^2 + 10*(11*b^5*c - 32*a*b^3*c^2 - 32*a
^2*b*c^3)*x)/(32*c^9*d^6*x^5 + 80*b*c^8*d^6*x^4 + 80*b^2*c^7*d^6*x^3 + 40*b^3*c^
6*d^6*x^2 + 10*b^4*c^5*d^6*x + b^5*c^4*d^6)

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Sympy [A]  time = 15.1009, size = 223, normalized size = 2.37 \[ - \frac{64 a^{3} c^{3} + 32 a^{2} b^{2} c^{2} + 32 a b^{4} c - 11 b^{6} + x^{4} \left (960 a c^{5} - 240 b^{2} c^{4}\right ) + x^{3} \left (1920 a b c^{4} - 480 b^{3} c^{3}\right ) + x^{2} \left (320 a^{2} c^{4} + 1280 a b^{2} c^{3} - 340 b^{4} c^{2}\right ) + x \left (320 a^{2} b c^{3} + 320 a b^{3} c^{2} - 100 b^{5} c\right )}{640 b^{5} c^{4} d^{6} + 6400 b^{4} c^{5} d^{6} x + 25600 b^{3} c^{6} d^{6} x^{2} + 51200 b^{2} c^{7} d^{6} x^{3} + 51200 b c^{8} d^{6} x^{4} + 20480 c^{9} d^{6} x^{5}} + \frac{x}{64 c^{3} d^{6}} \]

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)**6,x)

[Out]

-(64*a**3*c**3 + 32*a**2*b**2*c**2 + 32*a*b**4*c - 11*b**6 + x**4*(960*a*c**5 -
240*b**2*c**4) + x**3*(1920*a*b*c**4 - 480*b**3*c**3) + x**2*(320*a**2*c**4 + 12
80*a*b**2*c**3 - 340*b**4*c**2) + x*(320*a**2*b*c**3 + 320*a*b**3*c**2 - 100*b**
5*c))/(640*b**5*c**4*d**6 + 6400*b**4*c**5*d**6*x + 25600*b**3*c**6*d**6*x**2 +
51200*b**2*c**7*d**6*x**3 + 51200*b*c**8*d**6*x**4 + 20480*c**9*d**6*x**5) + x/(
64*c**3*d**6)

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GIAC/XCAS [A]  time = 0.214436, size = 216, normalized size = 2.3 \[ \frac{x}{64 \, c^{3} d^{6}} + \frac{240 \, b^{2} c^{4} x^{4} - 960 \, a c^{5} x^{4} + 480 \, b^{3} c^{3} x^{3} - 1920 \, a b c^{4} x^{3} + 340 \, b^{4} c^{2} x^{2} - 1280 \, a b^{2} c^{3} x^{2} - 320 \, a^{2} c^{4} x^{2} + 100 \, b^{5} c x - 320 \, a b^{3} c^{2} x - 320 \, a^{2} b c^{3} x + 11 \, b^{6} - 32 \, a b^{4} c - 32 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{640 \,{\left (2 \, c x + b\right )}^{5} c^{4} d^{6}} \]

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)^6,x, algorithm="giac")

[Out]

1/64*x/(c^3*d^6) + 1/640*(240*b^2*c^4*x^4 - 960*a*c^5*x^4 + 480*b^3*c^3*x^3 - 19
20*a*b*c^4*x^3 + 340*b^4*c^2*x^2 - 1280*a*b^2*c^3*x^2 - 320*a^2*c^4*x^2 + 100*b^
5*c*x - 320*a*b^3*c^2*x - 320*a^2*b*c^3*x + 11*b^6 - 32*a*b^4*c - 32*a^2*b^2*c^2
- 64*a^3*c^3)/((2*c*x + b)^5*c^4*d^6)