3.1144 \(\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^3} \, dx\)

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)/(256*c^4*d^3) + (b + 2*c*x)^4/(5
12*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 = 0.0982906, 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, Rules used = {683} \[ -\frac{3 \left (b^2-4 a c\right ) (b+2 c x)^2}{256 c^4 d^3}+\frac{\left (b^2-4 a c\right )^3}{256 c^4 d^3 (b+2 c x)^2}+\frac{3 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{128 c^4 d^3}+\frac{(b+2 c x)^4}{512 c^4 d^3} \]

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)/(256*c^4*d^3) + (b + 2*c*x)^4/(5
12*c^4*d^3) + (3*(b^2 - 4*a*c)^2*Log[b + 2*c*x])/(128*c^4*d^3)

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^3} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^3}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)}+\frac{3 \left (-b^2+4 a c\right ) (b d+2 c d x)}{64 c^3 d^4}+\frac{(b d+2 c d x)^3}{64 c^3 d^6}\right ) \, dx\\ &=\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 ) (b+2 c x)^2}{256 c^4 d^3}+\frac{(b+2 c x)^4}{512 c^4 d^3}+\frac{3 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{128 c^4 d^3}\\ \end{align*}

Mathematica [A]  time = 0.0946953, size = 90, normalized size = 0.9 \[ \frac{-\frac{8 b x \left (b^2-6 a c\right )}{c^3}+\frac{\left (b^2-4 a c\right )^3}{c^4 (b+2 c x)^2}+\frac{6 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{c^4}+\frac{48 a x^2}{c}+\frac{16 b x^3}{c}+8 x^4}{256 d^3} \]

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 = 0.047, size = 192, normalized size = 1.9 \begin{align*}{\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{{b}^{3}x}{32\,{c}^{3}{d}^{3}}}-{\frac{{a}^{3}}{4\,c{d}^{3} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{3\,{b}^{2}{a}^{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}}} \end{align*}

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*a*b*x-1/32/d^3/c^3*b^3*x-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 = 1.16199, size = 198, normalized size = 1.98 \begin{align*} \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}} \end{align*}

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 [B]  time = 1.99976, size = 539, normalized size = 5.39 \begin{align*} \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 )}} \end{align*}

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 + 16*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 = 2.45082, size = 150, normalized size = 1.5 \begin{align*} \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}} \end{align*}

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*log(b + 2*c*x)/(128*c**4*d**3)

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Giac [A]  time = 1.16787, size = 200, normalized size = 2. \begin{align*} \frac{3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \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}} \end{align*}

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)*log(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)