3.1151 \(\int \frac{(a+b x+c x^2)^3}{(b d+2 c d x)^{10}} \, dx\)

Optimal. Leaf size=101 \[ \frac{\left (b^2-4 a c\right )^3}{1152 c^4 d^{10} (b+2 c x)^9}-\frac{3 \left (b^2-4 a c\right )^2}{896 c^4 d^{10} (b+2 c x)^7}+\frac{3 \left (b^2-4 a c\right )}{640 c^4 d^{10} (b+2 c x)^5}-\frac{1}{384 c^4 d^{10} (b+2 c x)^3} \]

[Out]

(b^2 - 4*a*c)^3/(1152*c^4*d^10*(b + 2*c*x)^9) - (3*(b^2 - 4*a*c)^2)/(896*c^4*d^10*(b + 2*c*x)^7) + (3*(b^2 - 4
*a*c))/(640*c^4*d^10*(b + 2*c*x)^5) - 1/(384*c^4*d^10*(b + 2*c*x)^3)

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Rubi [A]  time = 0.0816032, 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} \[ \frac{\left (b^2-4 a c\right )^3}{1152 c^4 d^{10} (b+2 c x)^9}-\frac{3 \left (b^2-4 a c\right )^2}{896 c^4 d^{10} (b+2 c x)^7}+\frac{3 \left (b^2-4 a c\right )}{640 c^4 d^{10} (b+2 c x)^5}-\frac{1}{384 c^4 d^{10} (b+2 c x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^10,x]

[Out]

(b^2 - 4*a*c)^3/(1152*c^4*d^10*(b + 2*c*x)^9) - (3*(b^2 - 4*a*c)^2)/(896*c^4*d^10*(b + 2*c*x)^7) + (3*(b^2 - 4
*a*c))/(640*c^4*d^10*(b + 2*c*x)^5) - 1/(384*c^4*d^10*(b + 2*c*x)^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)^{10}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^3}{64 c^3 d^{10} (b+2 c x)^{10}}+\frac{3 \left (-b^2+4 a c\right )^2}{64 c^3 d^{10} (b+2 c x)^8}+\frac{3 \left (-b^2+4 a c\right )}{64 c^3 d^{10} (b+2 c x)^6}+\frac{1}{64 c^3 d^{10} (b+2 c x)^4}\right ) \, dx\\ &=\frac{\left (b^2-4 a c\right )^3}{1152 c^4 d^{10} (b+2 c x)^9}-\frac{3 \left (b^2-4 a c\right )^2}{896 c^4 d^{10} (b+2 c x)^7}+\frac{3 \left (b^2-4 a c\right )}{640 c^4 d^{10} (b+2 c x)^5}-\frac{1}{384 c^4 d^{10} (b+2 c x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0577125, size = 79, normalized size = 0.78 \[ \frac{189 \left (b^2-4 a c\right ) (b+2 c x)^4-135 \left (b^2-4 a c\right )^2 (b+2 c x)^2+35 \left (b^2-4 a c\right )^3-105 (b+2 c x)^6}{40320 c^4 d^{10} (b+2 c x)^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/(b*d + 2*c*d*x)^10,x]

[Out]

(35*(b^2 - 4*a*c)^3 - 135*(b^2 - 4*a*c)^2*(b + 2*c*x)^2 + 189*(b^2 - 4*a*c)*(b + 2*c*x)^4 - 105*(b + 2*c*x)^6)
/(40320*c^4*d^10*(b + 2*c*x)^9)

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Maple [A]  time = 0.048, size = 121, normalized size = 1.2 \begin{align*}{\frac{1}{{d}^{10}} \left ( -{\frac{1}{384\,{c}^{4} \left ( 2\,cx+b \right ) ^{3}}}-{\frac{12\,ac-3\,{b}^{2}}{640\,{c}^{4} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{48\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+3\,{b}^{4}}{896\,{c}^{4} \left ( 2\,cx+b \right ) ^{7}}}-{\frac{64\,{a}^{3}{c}^{3}-48\,{a}^{2}{b}^{2}{c}^{2}+12\,a{b}^{4}c-{b}^{6}}{1152\,{c}^{4} \left ( 2\,cx+b \right ) ^{9}}} \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)^10,x)

[Out]

1/d^10*(-1/384/c^4/(2*c*x+b)^3-1/640*(12*a*c-3*b^2)/c^4/(2*c*x+b)^5-1/896*(48*a^2*c^2-24*a*b^2*c+3*b^4)/c^4/(2
*c*x+b)^7-1/1152*(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)^9)

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Maxima [B]  time = 1.89643, size = 378, normalized size = 3.74 \begin{align*} -\frac{420 \, c^{6} x^{6} + 1260 \, b c^{5} x^{5} + b^{6} + 6 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} + 140 \, a^{3} c^{3} + 126 \,{\left (11 \, b^{2} c^{4} + 6 \, a c^{5}\right )} x^{4} + 168 \,{\left (4 \, b^{3} c^{3} + 9 \, a b c^{4}\right )} x^{3} + 36 \,{\left (4 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 15 \, a^{2} c^{4}\right )} x^{2} + 18 \,{\left (b^{5} c + 6 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x}{2520 \,{\left (512 \, c^{13} d^{10} x^{9} + 2304 \, b c^{12} d^{10} x^{8} + 4608 \, b^{2} c^{11} d^{10} x^{7} + 5376 \, b^{3} c^{10} d^{10} x^{6} + 4032 \, b^{4} c^{9} d^{10} x^{5} + 2016 \, b^{5} c^{8} d^{10} x^{4} + 672 \, b^{6} c^{7} d^{10} x^{3} + 144 \, b^{7} c^{6} d^{10} x^{2} + 18 \, b^{8} c^{5} d^{10} x + b^{9} c^{4} d^{10}\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)^10,x, algorithm="maxima")

[Out]

-1/2520*(420*c^6*x^6 + 1260*b*c^5*x^5 + b^6 + 6*a*b^4*c + 30*a^2*b^2*c^2 + 140*a^3*c^3 + 126*(11*b^2*c^4 + 6*a
*c^5)*x^4 + 168*(4*b^3*c^3 + 9*a*b*c^4)*x^3 + 36*(4*b^4*c^2 + 24*a*b^2*c^3 + 15*a^2*c^4)*x^2 + 18*(b^5*c + 6*a
*b^3*c^2 + 30*a^2*b*c^3)*x)/(512*c^13*d^10*x^9 + 2304*b*c^12*d^10*x^8 + 4608*b^2*c^11*d^10*x^7 + 5376*b^3*c^10
*d^10*x^6 + 4032*b^4*c^9*d^10*x^5 + 2016*b^5*c^8*d^10*x^4 + 672*b^6*c^7*d^10*x^3 + 144*b^7*c^6*d^10*x^2 + 18*b
^8*c^5*d^10*x + b^9*c^4*d^10)

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Fricas [B]  time = 1.9383, size = 636, normalized size = 6.3 \begin{align*} -\frac{420 \, c^{6} x^{6} + 1260 \, b c^{5} x^{5} + b^{6} + 6 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} + 140 \, a^{3} c^{3} + 126 \,{\left (11 \, b^{2} c^{4} + 6 \, a c^{5}\right )} x^{4} + 168 \,{\left (4 \, b^{3} c^{3} + 9 \, a b c^{4}\right )} x^{3} + 36 \,{\left (4 \, b^{4} c^{2} + 24 \, a b^{2} c^{3} + 15 \, a^{2} c^{4}\right )} x^{2} + 18 \,{\left (b^{5} c + 6 \, a b^{3} c^{2} + 30 \, a^{2} b c^{3}\right )} x}{2520 \,{\left (512 \, c^{13} d^{10} x^{9} + 2304 \, b c^{12} d^{10} x^{8} + 4608 \, b^{2} c^{11} d^{10} x^{7} + 5376 \, b^{3} c^{10} d^{10} x^{6} + 4032 \, b^{4} c^{9} d^{10} x^{5} + 2016 \, b^{5} c^{8} d^{10} x^{4} + 672 \, b^{6} c^{7} d^{10} x^{3} + 144 \, b^{7} c^{6} d^{10} x^{2} + 18 \, b^{8} c^{5} d^{10} x + b^{9} c^{4} d^{10}\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)^10,x, algorithm="fricas")

[Out]

-1/2520*(420*c^6*x^6 + 1260*b*c^5*x^5 + b^6 + 6*a*b^4*c + 30*a^2*b^2*c^2 + 140*a^3*c^3 + 126*(11*b^2*c^4 + 6*a
*c^5)*x^4 + 168*(4*b^3*c^3 + 9*a*b*c^4)*x^3 + 36*(4*b^4*c^2 + 24*a*b^2*c^3 + 15*a^2*c^4)*x^2 + 18*(b^5*c + 6*a
*b^3*c^2 + 30*a^2*b*c^3)*x)/(512*c^13*d^10*x^9 + 2304*b*c^12*d^10*x^8 + 4608*b^2*c^11*d^10*x^7 + 5376*b^3*c^10
*d^10*x^6 + 4032*b^4*c^9*d^10*x^5 + 2016*b^5*c^8*d^10*x^4 + 672*b^6*c^7*d^10*x^3 + 144*b^7*c^6*d^10*x^2 + 18*b
^8*c^5*d^10*x + b^9*c^4*d^10)

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Sympy [B]  time = 12.05, size = 292, normalized size = 2.89 \begin{align*} - \frac{140 a^{3} c^{3} + 30 a^{2} b^{2} c^{2} + 6 a b^{4} c + b^{6} + 1260 b c^{5} x^{5} + 420 c^{6} x^{6} + x^{4} \left (756 a c^{5} + 1386 b^{2} c^{4}\right ) + x^{3} \left (1512 a b c^{4} + 672 b^{3} c^{3}\right ) + x^{2} \left (540 a^{2} c^{4} + 864 a b^{2} c^{3} + 144 b^{4} c^{2}\right ) + x \left (540 a^{2} b c^{3} + 108 a b^{3} c^{2} + 18 b^{5} c\right )}{2520 b^{9} c^{4} d^{10} + 45360 b^{8} c^{5} d^{10} x + 362880 b^{7} c^{6} d^{10} x^{2} + 1693440 b^{6} c^{7} d^{10} x^{3} + 5080320 b^{5} c^{8} d^{10} x^{4} + 10160640 b^{4} c^{9} d^{10} x^{5} + 13547520 b^{3} c^{10} d^{10} x^{6} + 11612160 b^{2} c^{11} d^{10} x^{7} + 5806080 b c^{12} d^{10} x^{8} + 1290240 c^{13} d^{10} x^{9}} \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)**10,x)

[Out]

-(140*a**3*c**3 + 30*a**2*b**2*c**2 + 6*a*b**4*c + b**6 + 1260*b*c**5*x**5 + 420*c**6*x**6 + x**4*(756*a*c**5
+ 1386*b**2*c**4) + x**3*(1512*a*b*c**4 + 672*b**3*c**3) + x**2*(540*a**2*c**4 + 864*a*b**2*c**3 + 144*b**4*c*
*2) + x*(540*a**2*b*c**3 + 108*a*b**3*c**2 + 18*b**5*c))/(2520*b**9*c**4*d**10 + 45360*b**8*c**5*d**10*x + 362
880*b**7*c**6*d**10*x**2 + 1693440*b**6*c**7*d**10*x**3 + 5080320*b**5*c**8*d**10*x**4 + 10160640*b**4*c**9*d*
*10*x**5 + 13547520*b**3*c**10*d**10*x**6 + 11612160*b**2*c**11*d**10*x**7 + 5806080*b*c**12*d**10*x**8 + 1290
240*c**13*d**10*x**9)

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Giac [A]  time = 1.1778, size = 223, normalized size = 2.21 \begin{align*} -\frac{420 \, c^{6} x^{6} + 1260 \, b c^{5} x^{5} + 1386 \, b^{2} c^{4} x^{4} + 756 \, a c^{5} x^{4} + 672 \, b^{3} c^{3} x^{3} + 1512 \, a b c^{4} x^{3} + 144 \, b^{4} c^{2} x^{2} + 864 \, a b^{2} c^{3} x^{2} + 540 \, a^{2} c^{4} x^{2} + 18 \, b^{5} c x + 108 \, a b^{3} c^{2} x + 540 \, a^{2} b c^{3} x + b^{6} + 6 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} + 140 \, a^{3} c^{3}}{2520 \,{\left (2 \, c x + b\right )}^{9} c^{4} d^{10}} \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)^10,x, algorithm="giac")

[Out]

-1/2520*(420*c^6*x^6 + 1260*b*c^5*x^5 + 1386*b^2*c^4*x^4 + 756*a*c^5*x^4 + 672*b^3*c^3*x^3 + 1512*a*b*c^4*x^3
+ 144*b^4*c^2*x^2 + 864*a*b^2*c^3*x^2 + 540*a^2*c^4*x^2 + 18*b^5*c*x + 108*a*b^3*c^2*x + 540*a^2*b*c^3*x + b^6
+ 6*a*b^4*c + 30*a^2*b^2*c^2 + 140*a^3*c^3)/((2*c*x + b)^9*c^4*d^10)