3.1132 \(\int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.241676, 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 \[ -\frac{\left (b^2-4 a c\right ) (b+2 c x)^3}{128 c^4 d^2}+\frac{\left (b^2-4 a c\right )^3}{128 c^4 d^2 (b+2 c x)}+\frac{3 x \left (b^2-4 a c\right )^2}{64 c^3 d^2}+\frac{(b+2 c x)^5}{640 c^4 d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.194843, size = 101, normalized size = 1.07 \[ \frac{\frac{10 x \left (48 a^2 c^2-12 a b^2 c+b^4\right )}{c^3}+\frac{5 \left (b^2-4 a c\right )^3}{c^4 (b+2 c x)}-\frac{20 b x^2 \left (b^2-12 a c\right )}{c^2}+\frac{40 x^3 \left (4 a c+b^2\right )}{c}+80 b x^4+32 c x^5}{640 d^2} \]

Antiderivative was successfully verified.

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

[Out]

((10*(b^4 - 12*a*b^2*c + 48*a^2*c^2)*x)/c^3 - (20*b*(b^2 - 12*a*c)*x^2)/c^2 + (4
0*(b^2 + 4*a*c)*x^3)/c + 80*b*x^4 + 32*c*x^5 + (5*(b^2 - 4*a*c)^3)/(c^4*(b + 2*c
*x)))/(640*d^2)

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Maple [A]  time = 0.007, size = 135, normalized size = 1.4 \[{\frac{1}{{d}^{2}} \left ({\frac{1}{64\,{c}^{3}} \left ({\frac{16\,{x}^{5}{c}^{4}}{5}}+8\,{x}^{4}b{c}^{3}+16\,{x}^{3}a{c}^{3}+4\,{x}^{3}{b}^{2}{c}^{2}+24\,{x}^{2}ab{c}^{2}-2\,{b}^{3}c{x}^{2}+48\,{a}^{2}{c}^{2}x-12\,xa{b}^{2}c+x{b}^{4} \right ) }-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{128\,{c}^{4} \left ( 2\,cx+b \right ) }} \right ) } \]

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)^2,x)

[Out]

1/d^2*(1/64/c^3*(16/5*x^5*c^4+8*x^4*b*c^3+16*x^3*a*c^3+4*x^3*b^2*c^2+24*x^2*a*b*
c^2-2*b^3*c*x^2+48*a^2*c^2*x-12*x*a*b^2*c+x*b^4)-1/128*(64*a^3*c^3-48*a^2*b^2*c^
2+12*a*b^4*c-b^6)/c^4/(2*c*x+b))

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Maxima [A]  time = 0.697447, size = 186, normalized size = 1.98 \[ \frac{b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{128 \,{\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}} + \frac{16 \, c^{4} x^{5} + 40 \, b c^{3} x^{4} + 20 \,{\left (b^{2} c^{2} + 4 \, a c^{3}\right )} x^{3} - 10 \,{\left (b^{3} c - 12 \, a b c^{2}\right )} x^{2} + 5 \,{\left (b^{4} - 12 \, a b^{2} c + 48 \, a^{2} c^{2}\right )} x}{320 \, c^{3} d^{2}} \]

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)^2,x, algorithm="maxima")

[Out]

1/128*(b^6 - 12*a*b^4*c + 48*a^2*b^2*c^2 - 64*a^3*c^3)/(2*c^5*d^2*x + b*c^4*d^2)
+ 1/320*(16*c^4*x^5 + 40*b*c^3*x^4 + 20*(b^2*c^2 + 4*a*c^3)*x^3 - 10*(b^3*c - 1
2*a*b*c^2)*x^2 + 5*(b^4 - 12*a*b^2*c + 48*a^2*c^2)*x)/(c^3*d^2)

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Fricas [A]  time = 0.200344, size = 186, normalized size = 1.98 \[ \frac{64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 640 \, a b c^{4} x^{3} + 960 \, a^{2} c^{4} x^{2} + 5 \, b^{6} - 60 \, a b^{4} c + 240 \, a^{2} b^{2} c^{2} - 320 \, a^{3} c^{3} + 160 \,{\left (b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 10 \,{\left (b^{5} c - 12 \, a b^{3} c^{2} + 48 \, a^{2} b c^{3}\right )} x}{640 \,{\left (2 \, c^{5} d^{2} x + b c^{4} d^{2}\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

1/640*(64*c^6*x^6 + 192*b*c^5*x^5 + 640*a*b*c^4*x^3 + 960*a^2*c^4*x^2 + 5*b^6 -
60*a*b^4*c + 240*a^2*b^2*c^2 - 320*a^3*c^3 + 160*(b^2*c^4 + 2*a*c^5)*x^4 + 10*(b
^5*c - 12*a*b^3*c^2 + 48*a^2*b*c^3)*x)/(2*c^5*d^2*x + b*c^4*d^2)

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Sympy [A]  time = 2.95258, size = 143, normalized size = 1.52 \[ \frac{b x^{4}}{8 d^{2}} + \frac{c x^{5}}{20 d^{2}} - \frac{64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}}{128 b c^{4} d^{2} + 256 c^{5} d^{2} x} + \frac{x^{3} \left (4 a c + b^{2}\right )}{16 c d^{2}} + \frac{x^{2} \left (12 a b c - b^{3}\right )}{32 c^{2} d^{2}} + \frac{x \left (48 a^{2} c^{2} - 12 a b^{2} c + b^{4}\right )}{64 c^{3} d^{2}} \]

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)**2,x)

[Out]

b*x**4/(8*d**2) + c*x**5/(20*d**2) - (64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b*
*4*c - b**6)/(128*b*c**4*d**2 + 256*c**5*d**2*x) + x**3*(4*a*c + b**2)/(16*c*d**
2) + x**2*(12*a*b*c - b**3)/(32*c**2*d**2) + x*(48*a**2*c**2 - 12*a*b**2*c + b**
4)/(64*c**3*d**2)

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GIAC/XCAS [A]  time = 0.214241, size = 298, normalized size = 3.17 \[ \frac{{\left (\frac{15 \, b^{4} d^{4}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{120 \, a b^{2} c d^{4}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac{240 \, a^{2} c^{2} d^{4}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{5 \, b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{20 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} + 1\right )}{\left (2 \, c d x + b d\right )}^{5}}{640 \, c^{4} d^{7}} + \frac{\frac{b^{6} c^{5} d^{11}}{2 \, c d x + b d} - \frac{12 \, a b^{4} c^{6} d^{11}}{2 \, c d x + b d} + \frac{48 \, a^{2} b^{2} c^{7} d^{11}}{2 \, c d x + b d} - \frac{64 \, a^{3} c^{8} d^{11}}{2 \, c d x + b d}}{128 \, c^{9} d^{12}} \]

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)^2,x, algorithm="giac")

[Out]

1/640*(15*b^4*d^4/(2*c*d*x + b*d)^4 - 120*a*b^2*c*d^4/(2*c*d*x + b*d)^4 + 240*a^
2*c^2*d^4/(2*c*d*x + b*d)^4 - 5*b^2*d^2/(2*c*d*x + b*d)^2 + 20*a*c*d^2/(2*c*d*x
+ b*d)^2 + 1)*(2*c*d*x + b*d)^5/(c^4*d^7) + 1/128*(b^6*c^5*d^11/(2*c*d*x + b*d)
- 12*a*b^4*c^6*d^11/(2*c*d*x + b*d) + 48*a^2*b^2*c^7*d^11/(2*c*d*x + b*d) - 64*a
^3*c^8*d^11/(2*c*d*x + b*d))/(c^9*d^12)