3.1153 $$\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^{12}} \, dx$$

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

[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)

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Rubi [A]  time = 0.0846991, antiderivative size = 101, 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} $\frac{\left (b^2-4 a c\right )^3}{1408 c^4 d^{12} (b+2 c x)^{11}}-\frac{\left (b^2-4 a c\right )^2}{384 c^4 d^{12} (b+2 c x)^9}+\frac{3 \left (b^2-4 a c\right )}{896 c^4 d^{12} (b+2 c x)^7}-\frac{1}{640 c^4 d^{12} (b+2 c x)^5}$

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*}

Mathematica [A]  time = 0.0626427, size = 79, normalized size = 0.78 $\frac{495 \left (b^2-4 a c\right ) (b+2 c x)^4-385 \left (b^2-4 a c\right )^2 (b+2 c x)^2+105 \left (b^2-4 a c\right )^3-231 (b+2 c x)^6}{147840 c^4 d^{12} (b+2 c x)^{11}}$

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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Maple [A]  time = 0.047, size = 121, normalized size = 1.2 \begin{align*}{\frac{1}{{d}^{12}} \left ( -{\frac{1}{640\,{c}^{4} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{12\,ac-3\,{b}^{2}}{896\,{c}^{4} \left ( 2\,cx+b \right ) ^{7}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{1152\,{c}^{4} \left ( 2\,cx+b \right ) ^{9}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{1408\,{c}^{4} \left ( 2\,cx+b \right ) ^{11}}} \right ) } \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)^12,x)

[Out]

1/d^12*(-1/640/c^4/(2*c*x+b)^5-1/896*(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)^9-1/1408*(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)

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Maxima [B]  time = 1.36038, size = 413, normalized size = 4.09 \begin{align*} -\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{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)^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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Fricas [B]  time = 2.07111, size = 717, normalized size = 7.1 \begin{align*} -\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{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)^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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Sympy [B]  time = 20.5477, size = 323, normalized size = 3.2 \begin{align*} - \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{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)**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**1
2*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)

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Giac [A]  time = 1.17052, size = 223, normalized size = 2.21 \begin{align*} -\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{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)^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)