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]

1/256*(-4*a*c+b^2)^3/c^4/d^3/(2*c*x+b)^2-3/256*(-4*a*c+b^2)*(2*c*x+b)^2/c^4/d^3+1/512*(2*c*x+b)^4/c^4/d^3+3/12
8*(-4*a*c+b^2)^2*ln(2*c*x+b)/c^4/d^3

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Rubi [A]  time = 0.10, antiderivative size = 100, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.09, size = 90, normalized size = 0.90 \[ \frac {\frac {\left (b^2-4 a c\right )^3}{c^4 (b+2 c x)^2}+\frac {6 \left (b^2-4 a c\right )^2 \log (b+2 c x)}{c^4}-\frac {8 b x \left (b^2-6 a c\right )}{c^3}+\frac {48 a x^2}{c}+\frac {16 b x^3}{c}+8 x^4}{256 d^3} \]

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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fricas [B]  time = 1.19, size = 252, normalized size = 2.52 \[ \frac {32 \, c^{6} x^{6} + 96 \, b c^{5} x^{5} + b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 24 \, {\left (3 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} - 16 \, {\left (b^{3} c^{3} - 24 \, a b c^{4}\right )} x^{3} - 16 \, {\left (2 \, b^{4} c^{2} - 15 \, a b^{2} c^{3}\right )} x^{2} - 8 \, {\left (b^{5} c - 6 \, a b^{3} c^{2}\right )} x + 6 \, {\left (b^{6} - 8 \, a b^{4} c + 16 \, a^{2} b^{2} c^{2} + 4 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 4 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x\right )} \log \left (2 \, c x + b\right )}{256 \, {\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} \]

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 + 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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giac [A]  time = 0.17, size = 148, normalized size = 1.48 \[ \frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d^{3}} + \frac {b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{256 \, {\left (2 \, c x + b\right )}^{2} c^{4} d^{3}} + \frac {c^{12} d^{9} x^{4} + 2 \, b c^{11} d^{9} x^{3} + 6 \, a c^{11} d^{9} x^{2} - b^{3} c^{9} d^{9} x + 6 \, a b c^{10} d^{9} x}{32 \, c^{12} d^{12}} \]

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)

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maple [B]  time = 0.05, size = 192, normalized size = 1.92 \[ \frac {x^{4}}{32 d^{3}}+\frac {b \,x^{3}}{16 c \,d^{3}}-\frac {a^{3}}{4 \left (2 c x +b \right )^{2} c \,d^{3}}+\frac {3 a^{2} b^{2}}{16 \left (2 c x +b \right )^{2} c^{2} d^{3}}-\frac {3 a \,b^{4}}{64 \left (2 c x +b \right )^{2} c^{3} d^{3}}+\frac {3 a \,x^{2}}{16 c \,d^{3}}+\frac {b^{6}}{256 \left (2 c x +b \right )^{2} c^{4} d^{3}}+\frac {3 a^{2} \ln \left (2 c x +b \right )}{8 c^{2} d^{3}}-\frac {3 a \,b^{2} \ln \left (2 c x +b \right )}{16 c^{3} d^{3}}+\frac {3 a b x}{16 c^{2} d^{3}}+\frac {3 b^{4} \ln \left (2 c x +b \right )}{128 c^{4} d^{3}}-\frac {b^{3} x}{32 c^{3} d^{3}} \]

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*x^2*a+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 = 1.45, size = 147, normalized size = 1.47 \[ \frac {b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}}{256 \, {\left (4 \, c^{6} d^{3} x^{2} + 4 \, b c^{5} d^{3} x + b^{2} c^{4} d^{3}\right )}} + \frac {c^{3} x^{4} + 2 \, b c^{2} x^{3} + 6 \, a c^{2} x^{2} - {\left (b^{3} - 6 \, a b c\right )} x}{32 \, c^{3} d^{3}} + \frac {3 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d^{3}} \]

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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mupad [B]  time = 0.48, size = 223, normalized size = 2.23 \[ x^2\,\left (\frac {3\,\left (b^2+a\,c\right )}{16\,c^2\,d^3}-\frac {3\,b^2}{16\,c^2\,d^3}\right )-x\,\left (\frac {5\,b^3}{32\,c^3\,d^3}-\frac {b^3+6\,a\,c\,b}{8\,c^3\,d^3}+\frac {3\,b\,\left (\frac {3\,\left (b^2+a\,c\right )}{8\,c^2\,d^3}-\frac {3\,b^2}{8\,c^2\,d^3}\right )}{2\,c}\right )+\frac {x^4}{32\,d^3}+\frac {-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}{8\,c\,\left (32\,b^2\,c^3\,d^3+128\,b\,c^4\,d^3\,x+128\,c^5\,d^3\,x^2\right )}+\frac {b\,x^3}{16\,c\,d^3}+\frac {\ln \left (b+2\,c\,x\right )\,\left (48\,a^2\,c^2-24\,a\,b^2\,c+3\,b^4\right )}{128\,c^4\,d^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^2*((3*(a*c + b^2))/(16*c^2*d^3) - (3*b^2)/(16*c^2*d^3)) - x*((5*b^3)/(32*c^3*d^3) - (b^3 + 6*a*b*c)/(8*c^3*d
^3) + (3*b*((3*(a*c + b^2))/(8*c^2*d^3) - (3*b^2)/(8*c^2*d^3)))/(2*c)) + x^4/(32*d^3) + (b^6 - 64*a^3*c^3 + 48
*a^2*b^2*c^2 - 12*a*b^4*c)/(8*c*(32*b^2*c^3*d^3 + 128*c^5*d^3*x^2 + 128*b*c^4*d^3*x)) + (b*x^3)/(16*c*d^3) + (
log(b + 2*c*x)*(3*b^4 + 48*a^2*c^2 - 24*a*b^2*c))/(128*c^4*d^3)

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sympy [A]  time = 1.27, size = 156, normalized size = 1.56 \[ \frac {3 a x^{2}}{16 c d^{3}} + \frac {b x^{3}}{16 c d^{3}} + x \left (\frac {3 a b}{16 c^{2} d^{3}} - \frac {b^{3}}{32 c^{3} d^{3}}\right ) + \frac {- 64 a^{3} c^{3} + 48 a^{2} b^{2} c^{2} - 12 a b^{4} c + b^{6}}{256 b^{2} c^{4} d^{3} + 1024 b c^{5} d^{3} x + 1024 c^{6} d^{3} x^{2}} + \frac {x^{4}}{32 d^{3}} + \frac {3 \left (4 a c - b^{2}\right )^{2} \log {\left (b + 2 c x \right )}}{128 c^{4} d^{3}} \]

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) + x*(3*a*b/(16*c**2*d**3) - b**3/(32*c**3*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) + 3*(4*a*c - b**2)**2*log(b + 2*c*x)/(128*c**4*d**3)

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