3.10.65 \(\int \frac {(a+b x+c x^2)^3}{(b d+2 c d x)^{12}} \, dx\)

Optimal. Leaf size=101 \[ \frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5} \]

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Rubi [A]  time = 0.08, antiderivative size = 101, 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} \begin {gather*} \frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^12,x]

[Out]

(b^2 - 4*a*c)^3/(1408*c^4*d^12*(b + 2*c*x)^11) - (b^2 - 4*a*c)^2/(384*c^4*d^12*(b + 2*c*x)^9) + (3*(b^2 - 4*a*
c))/(896*c^4*d^12*(b + 2*c*x)^7) - 1/(640*c^4*d^12*(b + 2*c*x)^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)^{12}} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^{12} (b+2 c x)^{12}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{12} (b+2 c x)^{10}}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^{12} (b+2 c x)^8}+\frac {1}{64 c^3 d^{12} (b+2 c x)^6}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac {\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac {3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac {1}{640 c^4 d^{12} (b+2 c x)^5}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.78 \begin {gather*} \frac {495 \left (b^2-4 a c\right ) (b+2 c x)^4-385 \left (b^2-4 a c\right )^2 (b+2 c x)^2+105 \left (b^2-4 a c\right )^3-231 (b+2 c x)^6}{147840 c^4 d^{12} (b+2 c x)^{11}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^12,x]

[Out]

(105*(b^2 - 4*a*c)^3 - 385*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 495*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 231*(b + 2*c*x)^6
)/(147840*c^4*d^12*(b + 2*c*x)^11)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^12,x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^12, x]

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fricas [B]  time = 0.39, size = 306, normalized size = 3.03 \begin {gather*} -\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \, {\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \, {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \, {\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \end {gather*}

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)^12,x, algorithm="fricas")

[Out]

-1/9240*(924*c^6*x^6 + 2772*b*c^5*x^5 + b^6 + 10*a*b^4*c + 70*a^2*b^2*c^2 + 420*a^3*c^3 + 990*(3*b^2*c^4 + 2*a
*c^5)*x^4 + 1320*(b^3*c^3 + 3*a*b*c^4)*x^3 + 220*(b^4*c^2 + 10*a*b^2*c^3 + 7*a^2*c^4)*x^2 + 22*(b^5*c + 10*a*b
^3*c^2 + 70*a^2*b*c^3)*x)/(2048*c^15*d^12*x^11 + 11264*b*c^14*d^12*x^10 + 28160*b^2*c^13*d^12*x^9 + 42240*b^3*
c^12*d^12*x^8 + 42240*b^4*c^11*d^12*x^7 + 29568*b^5*c^10*d^12*x^6 + 14784*b^6*c^9*d^12*x^5 + 5280*b^7*c^8*d^12
*x^4 + 1320*b^8*c^7*d^12*x^3 + 220*b^9*c^6*d^12*x^2 + 22*b^10*c^5*d^12*x + b^11*c^4*d^12)

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giac [A]  time = 0.16, size = 165, normalized size = 1.63 \begin {gather*} -\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + 2970 \, b^{2} c^{4} x^{4} + 1980 \, a c^{5} x^{4} + 1320 \, b^{3} c^{3} x^{3} + 3960 \, a b c^{4} x^{3} + 220 \, b^{4} c^{2} x^{2} + 2200 \, a b^{2} c^{3} x^{2} + 1540 \, a^{2} c^{4} x^{2} + 22 \, b^{5} c x + 220 \, a b^{3} c^{2} x + 1540 \, a^{2} b c^{3} x + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3}}{9240 \, {\left (2 \, c x + b\right )}^{11} c^{4} d^{12}} \end {gather*}

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)^12,x, algorithm="giac")

[Out]

-1/9240*(924*c^6*x^6 + 2772*b*c^5*x^5 + 2970*b^2*c^4*x^4 + 1980*a*c^5*x^4 + 1320*b^3*c^3*x^3 + 3960*a*b*c^4*x^
3 + 220*b^4*c^2*x^2 + 2200*a*b^2*c^3*x^2 + 1540*a^2*c^4*x^2 + 22*b^5*c*x + 220*a*b^3*c^2*x + 1540*a^2*b*c^3*x
+ b^6 + 10*a*b^4*c + 70*a^2*b^2*c^2 + 420*a^3*c^3)/((2*c*x + b)^11*c^4*d^12)

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maple [A]  time = 0.05, size = 121, normalized size = 1.20 \begin {gather*} \frac {-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{1408 \left (2 c x +b \right )^{11} c^{4}}-\frac {12 a c -3 b^{2}}{896 \left (2 c x +b \right )^{7} c^{4}}-\frac {48 a^{2} c^{2}-24 a \,b^{2} c +3 b^{4}}{1152 \left (2 c x +b \right )^{9} c^{4}}-\frac {1}{640 \left (2 c x +b \right )^{5} c^{4}}}{d^{12}} \end {gather*}

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)^12,x)

[Out]

1/d^12*(-1/1408*(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)^11-1/896*(12*a*c-3*b^2)/c^4/(2*c*x+b)
^7-1/1152*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2*c*x+b)^9-1/640/c^4/(2*c*x+b)^5)

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maxima [B]  time = 1.70, size = 306, normalized size = 3.03 \begin {gather*} -\frac {924 \, c^{6} x^{6} + 2772 \, b c^{5} x^{5} + b^{6} + 10 \, a b^{4} c + 70 \, a^{2} b^{2} c^{2} + 420 \, a^{3} c^{3} + 990 \, {\left (3 \, b^{2} c^{4} + 2 \, a c^{5}\right )} x^{4} + 1320 \, {\left (b^{3} c^{3} + 3 \, a b c^{4}\right )} x^{3} + 220 \, {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} + 22 \, {\left (b^{5} c + 10 \, a b^{3} c^{2} + 70 \, a^{2} b c^{3}\right )} x}{9240 \, {\left (2048 \, c^{15} d^{12} x^{11} + 11264 \, b c^{14} d^{12} x^{10} + 28160 \, b^{2} c^{13} d^{12} x^{9} + 42240 \, b^{3} c^{12} d^{12} x^{8} + 42240 \, b^{4} c^{11} d^{12} x^{7} + 29568 \, b^{5} c^{10} d^{12} x^{6} + 14784 \, b^{6} c^{9} d^{12} x^{5} + 5280 \, b^{7} c^{8} d^{12} x^{4} + 1320 \, b^{8} c^{7} d^{12} x^{3} + 220 \, b^{9} c^{6} d^{12} x^{2} + 22 \, b^{10} c^{5} d^{12} x + b^{11} c^{4} d^{12}\right )}} \end {gather*}

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)^12,x, algorithm="maxima")

[Out]

-1/9240*(924*c^6*x^6 + 2772*b*c^5*x^5 + b^6 + 10*a*b^4*c + 70*a^2*b^2*c^2 + 420*a^3*c^3 + 990*(3*b^2*c^4 + 2*a
*c^5)*x^4 + 1320*(b^3*c^3 + 3*a*b*c^4)*x^3 + 220*(b^4*c^2 + 10*a*b^2*c^3 + 7*a^2*c^4)*x^2 + 22*(b^5*c + 10*a*b
^3*c^2 + 70*a^2*b*c^3)*x)/(2048*c^15*d^12*x^11 + 11264*b*c^14*d^12*x^10 + 28160*b^2*c^13*d^12*x^9 + 42240*b^3*
c^12*d^12*x^8 + 42240*b^4*c^11*d^12*x^7 + 29568*b^5*c^10*d^12*x^6 + 14784*b^6*c^9*d^12*x^5 + 5280*b^7*c^8*d^12
*x^4 + 1320*b^8*c^7*d^12*x^3 + 220*b^9*c^6*d^12*x^2 + 22*b^10*c^5*d^12*x + b^11*c^4*d^12)

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mupad [B]  time = 0.78, size = 292, normalized size = 2.89 \begin {gather*} -\frac {\frac {420\,a^3\,c^3+70\,a^2\,b^2\,c^2+10\,a\,b^4\,c+b^6}{9240\,c^4}+x^4\,\left (\frac {9\,b^2}{28}+\frac {3\,a\,c}{14}\right )+\frac {c^2\,x^6}{10}+\frac {x^3\,\left (b^3+3\,a\,c\,b\right )}{7\,c}+\frac {x^2\,\left (7\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{42\,c^2}+\frac {3\,b\,c\,x^5}{10}+\frac {b\,x\,\left (70\,a^2\,c^2+10\,a\,b^2\,c+b^4\right )}{420\,c^3}}{b^{11}\,d^{12}+22\,b^{10}\,c\,d^{12}\,x+220\,b^9\,c^2\,d^{12}\,x^2+1320\,b^8\,c^3\,d^{12}\,x^3+5280\,b^7\,c^4\,d^{12}\,x^4+14784\,b^6\,c^5\,d^{12}\,x^5+29568\,b^5\,c^6\,d^{12}\,x^6+42240\,b^4\,c^7\,d^{12}\,x^7+42240\,b^3\,c^8\,d^{12}\,x^8+28160\,b^2\,c^9\,d^{12}\,x^9+11264\,b\,c^{10}\,d^{12}\,x^{10}+2048\,c^{11}\,d^{12}\,x^{11}} \end {gather*}

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)^12,x)

[Out]

-((b^6 + 420*a^3*c^3 + 70*a^2*b^2*c^2 + 10*a*b^4*c)/(9240*c^4) + x^4*((3*a*c)/14 + (9*b^2)/28) + (c^2*x^6)/10
+ (x^3*(b^3 + 3*a*b*c))/(7*c) + (x^2*(b^4 + 7*a^2*c^2 + 10*a*b^2*c))/(42*c^2) + (3*b*c*x^5)/10 + (b*x*(b^4 + 7
0*a^2*c^2 + 10*a*b^2*c))/(420*c^3))/(b^11*d^12 + 2048*c^11*d^12*x^11 + 11264*b*c^10*d^12*x^10 + 220*b^9*c^2*d^
12*x^2 + 1320*b^8*c^3*d^12*x^3 + 5280*b^7*c^4*d^12*x^4 + 14784*b^6*c^5*d^12*x^5 + 29568*b^5*c^6*d^12*x^6 + 422
40*b^4*c^7*d^12*x^7 + 42240*b^3*c^8*d^12*x^8 + 28160*b^2*c^9*d^12*x^9 + 22*b^10*c*d^12*x)

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sympy [B]  time = 10.00, size = 328, normalized size = 3.25 \begin {gather*} \frac {- 420 a^{3} c^{3} - 70 a^{2} b^{2} c^{2} - 10 a b^{4} c - b^{6} - 2772 b c^{5} x^{5} - 924 c^{6} x^{6} + x^{4} \left (- 1980 a c^{5} - 2970 b^{2} c^{4}\right ) + x^{3} \left (- 3960 a b c^{4} - 1320 b^{3} c^{3}\right ) + x^{2} \left (- 1540 a^{2} c^{4} - 2200 a b^{2} c^{3} - 220 b^{4} c^{2}\right ) + x \left (- 1540 a^{2} b c^{3} - 220 a b^{3} c^{2} - 22 b^{5} c\right )}{9240 b^{11} c^{4} d^{12} + 203280 b^{10} c^{5} d^{12} x + 2032800 b^{9} c^{6} d^{12} x^{2} + 12196800 b^{8} c^{7} d^{12} x^{3} + 48787200 b^{7} c^{8} d^{12} x^{4} + 136604160 b^{6} c^{9} d^{12} x^{5} + 273208320 b^{5} c^{10} d^{12} x^{6} + 390297600 b^{4} c^{11} d^{12} x^{7} + 390297600 b^{3} c^{12} d^{12} x^{8} + 260198400 b^{2} c^{13} d^{12} x^{9} + 104079360 b c^{14} d^{12} x^{10} + 18923520 c^{15} d^{12} x^{11}} \end {gather*}

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)**12,x)

[Out]

(-420*a**3*c**3 - 70*a**2*b**2*c**2 - 10*a*b**4*c - b**6 - 2772*b*c**5*x**5 - 924*c**6*x**6 + x**4*(-1980*a*c*
*5 - 2970*b**2*c**4) + x**3*(-3960*a*b*c**4 - 1320*b**3*c**3) + x**2*(-1540*a**2*c**4 - 2200*a*b**2*c**3 - 220
*b**4*c**2) + x*(-1540*a**2*b*c**3 - 220*a*b**3*c**2 - 22*b**5*c))/(9240*b**11*c**4*d**12 + 203280*b**10*c**5*
d**12*x + 2032800*b**9*c**6*d**12*x**2 + 12196800*b**8*c**7*d**12*x**3 + 48787200*b**7*c**8*d**12*x**4 + 13660
4160*b**6*c**9*d**12*x**5 + 273208320*b**5*c**10*d**12*x**6 + 390297600*b**4*c**11*d**12*x**7 + 390297600*b**3
*c**12*d**12*x**8 + 260198400*b**2*c**13*d**12*x**9 + 104079360*b*c**14*d**12*x**10 + 18923520*c**15*d**12*x**
11)

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