3.1146 \(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^5} \, dx\)

Optimal. Leaf size=107 \[ \frac {\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac {3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5}+\frac {b x}{64 c^3 d^5}+\frac {x^2}{64 c^2 d^5} \]

[Out]

1/64*b*x/c^3/d^5+1/64*x^2/c^2/d^5+1/512*(-4*a*c+b^2)^3/c^4/d^5/(2*c*x+b)^4-3/256*(-4*a*c+b^2)^2/c^4/d^5/(2*c*x
+b)^2-3/128*(-4*a*c+b^2)*ln(2*c*x+b)/c^4/d^5

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Rubi [A]  time = 0.10, antiderivative size = 107, 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} \[ \frac {\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac {3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5}+\frac {b x}{64 c^3 d^5}+\frac {x^2}{64 c^2 d^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^5,x]

[Out]

(b*x)/(64*c^3*d^5) + x^2/(64*c^2*d^5) + (b^2 - 4*a*c)^3/(512*c^4*d^5*(b + 2*c*x)^4) - (3*(b^2 - 4*a*c)^2)/(256
*c^4*d^5*(b + 2*c*x)^2) - (3*(b^2 - 4*a*c)*Log[b + 2*c*x])/(128*c^4*d^5)

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)^5} \, dx &=\int \left (\frac {b}{64 c^3 d^5}+\frac {x}{32 c^2 d^5}+\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^5 (b+2 c x)^5}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^5 (b+2 c x)^3}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^5 (b+2 c x)}\right ) \, dx\\ &=\frac {b x}{64 c^3 d^5}+\frac {x^2}{64 c^2 d^5}+\frac {\left (b^2-4 a c\right )^3}{512 c^4 d^5 (b+2 c x)^4}-\frac {3 \left (b^2-4 a c\right )^2}{256 c^4 d^5 (b+2 c x)^2}-\frac {3 \left (b^2-4 a c\right ) \log (b+2 c x)}{128 c^4 d^5}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 80, normalized size = 0.75 \[ \frac {\frac {\left (b^2-4 a c\right )^3}{(b+2 c x)^4}-\frac {6 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}-12 \left (b^2-4 a c\right ) \log (b+2 c x)+8 b c x+8 c^2 x^2}{512 c^4 d^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^5,x]

[Out]

(8*b*c*x + 8*c^2*x^2 + (b^2 - 4*a*c)^3/(b + 2*c*x)^4 - (6*(b^2 - 4*a*c)^2)/(b + 2*c*x)^2 - 12*(b^2 - 4*a*c)*Lo
g[b + 2*c*x])/(512*c^4*d^5)

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fricas [B]  time = 0.91, size = 291, normalized size = 2.72 \[ \frac {128 \, c^{6} x^{6} + 384 \, b c^{5} x^{5} + 448 \, b^{2} c^{4} x^{4} + 256 \, b^{3} c^{3} x^{3} - 5 \, b^{6} + 36 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3} + 48 \, {\left (b^{4} c^{2} + 4 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} x^{2} - 16 \, {\left (b^{5} c - 12 \, a b^{3} c^{2} + 24 \, a^{2} b c^{3}\right )} x - 12 \, {\left (b^{6} - 4 \, a b^{4} c + 16 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 32 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 24 \, {\left (b^{4} c^{2} - 4 \, a b^{2} c^{3}\right )} x^{2} + 8 \, {\left (b^{5} c - 4 \, a b^{3} c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{512 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} \]

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)^5,x, algorithm="fricas")

[Out]

1/512*(128*c^6*x^6 + 384*b*c^5*x^5 + 448*b^2*c^4*x^4 + 256*b^3*c^3*x^3 - 5*b^6 + 36*a*b^4*c - 48*a^2*b^2*c^2 -
64*a^3*c^3 + 48*(b^4*c^2 + 4*a*b^2*c^3 - 8*a^2*c^4)*x^2 - 16*(b^5*c - 12*a*b^3*c^2 + 24*a^2*b*c^3)*x - 12*(b^
6 - 4*a*b^4*c + 16*(b^2*c^4 - 4*a*c^5)*x^4 + 32*(b^3*c^3 - 4*a*b*c^4)*x^3 + 24*(b^4*c^2 - 4*a*b^2*c^3)*x^2 + 8
*(b^5*c - 4*a*b^3*c^2)*x)*log(2*c*x + b))/(16*c^8*d^5*x^4 + 32*b*c^7*d^5*x^3 + 24*b^2*c^6*d^5*x^2 + 8*b^3*c^5*
d^5*x + b^4*c^4*d^5)

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giac [B]  time = 0.22, size = 262, normalized size = 2.45 \[ \frac {3 \, {\left (b^{2} - 4 \, a c\right )} \log \left (\frac {1}{4 \, {\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}\right )}{256 \, c^{4} d^{5}} - \frac {{\left (2 \, c d x + b d\right )}^{2} {\left (\frac {3 \, b^{2} d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {12 \, a c d^{2}}{{\left (2 \, c d x + b d\right )}^{2}} - 1\right )}}{256 \, c^{4} d^{7}} + \frac {\frac {b^{6} c^{8} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {12 \, a b^{4} c^{9} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac {48 \, a^{2} b^{2} c^{10} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {64 \, a^{3} c^{11} d^{17}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac {6 \, b^{4} c^{8} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac {48 \, a b^{2} c^{9} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}} - \frac {96 \, a^{2} c^{10} d^{15}}{{\left (2 \, c d x + b d\right )}^{2}}}{512 \, c^{12} d^{18}} \]

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)^5,x, algorithm="giac")

[Out]

3/256*(b^2 - 4*a*c)*log(1/4/((2*c*d*x + b*d)^2*c^2*d^2))/(c^4*d^5) - 1/256*(2*c*d*x + b*d)^2*(3*b^2*d^2/(2*c*d
*x + b*d)^2 - 12*a*c*d^2/(2*c*d*x + b*d)^2 - 1)/(c^4*d^7) + 1/512*(b^6*c^8*d^17/(2*c*d*x + b*d)^4 - 12*a*b^4*c
^9*d^17/(2*c*d*x + b*d)^4 + 48*a^2*b^2*c^10*d^17/(2*c*d*x + b*d)^4 - 64*a^3*c^11*d^17/(2*c*d*x + b*d)^4 - 6*b^
4*c^8*d^15/(2*c*d*x + b*d)^2 + 48*a*b^2*c^9*d^15/(2*c*d*x + b*d)^2 - 96*a^2*c^10*d^15/(2*c*d*x + b*d)^2)/(c^12
*d^18)

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maple [A]  time = 0.06, size = 195, normalized size = 1.82 \[ -\frac {a^{3}}{8 \left (2 c x +b \right )^{4} c \,d^{5}}+\frac {3 a^{2} b^{2}}{32 \left (2 c x +b \right )^{4} c^{2} d^{5}}-\frac {3 a \,b^{4}}{128 \left (2 c x +b \right )^{4} c^{3} d^{5}}+\frac {b^{6}}{512 \left (2 c x +b \right )^{4} c^{4} d^{5}}-\frac {3 a^{2}}{16 \left (2 c x +b \right )^{2} c^{2} d^{5}}+\frac {3 a \,b^{2}}{32 \left (2 c x +b \right )^{2} c^{3} d^{5}}-\frac {3 b^{4}}{256 \left (2 c x +b \right )^{2} c^{4} d^{5}}+\frac {x^{2}}{64 c^{2} d^{5}}+\frac {3 a \ln \left (2 c x +b \right )}{32 c^{3} d^{5}}-\frac {3 b^{2} \ln \left (2 c x +b \right )}{128 c^{4} d^{5}}+\frac {b x}{64 c^{3} d^{5}} \]

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)^5,x)

[Out]

1/64*x^2/c^2/d^5+1/64*b*x/c^3/d^5-1/8/d^5/c/(2*c*x+b)^4*a^3+3/32/d^5/c^2/(2*c*x+b)^4*a^2*b^2-3/128/d^5/c^3/(2*
c*x+b)^4*a*b^4+1/512/d^5/c^4/(2*c*x+b)^4*b^6-3/16/d^5/c^2/(2*c*x+b)^2*a^2+3/32/d^5/c^3/(2*c*x+b)^2*a*b^2-3/256
/d^5/c^4/(2*c*x+b)^2*b^4+3/32/d^5/c^3*ln(2*c*x+b)*a-3/128/d^5/c^4*ln(2*c*x+b)*b^2

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maxima [A]  time = 1.58, size = 194, normalized size = 1.81 \[ -\frac {5 \, b^{6} - 36 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 64 \, a^{3} c^{3} + 24 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{2} + 24 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x}{512 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} + \frac {c x^{2} + b x}{64 \, c^{3} d^{5}} - \frac {3 \, {\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{128 \, c^{4} d^{5}} \]

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)^5,x, algorithm="maxima")

[Out]

-1/512*(5*b^6 - 36*a*b^4*c + 48*a^2*b^2*c^2 + 64*a^3*c^3 + 24*(b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^2 + 24*(b
^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x)/(16*c^8*d^5*x^4 + 32*b*c^7*d^5*x^3 + 24*b^2*c^6*d^5*x^2 + 8*b^3*c^5*d^5*
x + b^4*c^4*d^5) + 1/64*(c*x^2 + b*x)/(c^3*d^5) - 3/128*(b^2 - 4*a*c)*log(2*c*x + b)/(c^4*d^5)

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mupad [B]  time = 0.50, size = 202, normalized size = 1.89 \[ \frac {x^2}{64\,c^2\,d^5}-\frac {\frac {64\,a^3\,c^3+48\,a^2\,b^2\,c^2-36\,a\,b^4\,c+5\,b^6}{8\,c}+x^2\,\left (48\,a^2\,c^3-24\,a\,b^2\,c^2+3\,b^4\,c\right )+x\,\left (48\,a^2\,b\,c^2-24\,a\,b^3\,c+3\,b^5\right )}{64\,b^4\,c^3\,d^5+512\,b^3\,c^4\,d^5\,x+1536\,b^2\,c^5\,d^5\,x^2+2048\,b\,c^6\,d^5\,x^3+1024\,c^7\,d^5\,x^4}+\frac {b\,x}{64\,c^3\,d^5}+\frac {\ln \left (b+2\,c\,x\right )\,\left (12\,a\,c-3\,b^2\right )}{128\,c^4\,d^5} \]

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)^5,x)

[Out]

x^2/(64*c^2*d^5) - ((5*b^6 + 64*a^3*c^3 + 48*a^2*b^2*c^2 - 36*a*b^4*c)/(8*c) + x^2*(3*b^4*c + 48*a^2*c^3 - 24*
a*b^2*c^2) + x*(3*b^5 + 48*a^2*b*c^2 - 24*a*b^3*c))/(64*b^4*c^3*d^5 + 1024*c^7*d^5*x^4 + 512*b^3*c^4*d^5*x + 2
048*b*c^6*d^5*x^3 + 1536*b^2*c^5*d^5*x^2) + (b*x)/(64*c^3*d^5) + (log(b + 2*c*x)*(12*a*c - 3*b^2))/(128*c^4*d^
5)

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sympy [A]  time = 3.03, size = 209, normalized size = 1.95 \[ \frac {b x}{64 c^{3} d^{5}} + \frac {- 64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 36 a b^{4} c - 5 b^{6} + x^{2} \left (- 384 a^{2} c^{4} + 192 a b^{2} c^{3} - 24 b^{4} c^{2}\right ) + x \left (- 384 a^{2} b c^{3} + 192 a b^{3} c^{2} - 24 b^{5} c\right )}{512 b^{4} c^{4} d^{5} + 4096 b^{3} c^{5} d^{5} x + 12288 b^{2} c^{6} d^{5} x^{2} + 16384 b c^{7} d^{5} x^{3} + 8192 c^{8} d^{5} x^{4}} + \frac {x^{2}}{64 c^{2} d^{5}} + \frac {3 \left (4 a c - b^{2}\right ) \log {\left (b + 2 c x \right )}}{128 c^{4} d^{5}} \]

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)**5,x)

[Out]

b*x/(64*c**3*d**5) + (-64*a**3*c**3 - 48*a**2*b**2*c**2 + 36*a*b**4*c - 5*b**6 + x**2*(-384*a**2*c**4 + 192*a*
b**2*c**3 - 24*b**4*c**2) + x*(-384*a**2*b*c**3 + 192*a*b**3*c**2 - 24*b**5*c))/(512*b**4*c**4*d**5 + 4096*b**
3*c**5*d**5*x + 12288*b**2*c**6*d**5*x**2 + 16384*b*c**7*d**5*x**3 + 8192*c**8*d**5*x**4) + x**2/(64*c**2*d**5
) + 3*(4*a*c - b**2)*log(b + 2*c*x)/(128*c**4*d**5)

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