3.1135 \(\int \frac{(a+b x+c x^2)^2}{(b d+2 c d x)^{10}} \, dx\)

Optimal. Leaf size=73 \[ -\frac{\left (b^2-4 a c\right )^2}{288 c^3 d^{10} (b+2 c x)^9}+\frac{b^2-4 a c}{112 c^3 d^{10} (b+2 c x)^7}-\frac{1}{160 c^3 d^{10} (b+2 c x)^5} \]

[Out]

-(b^2 - 4*a*c)^2/(288*c^3*d^10*(b + 2*c*x)^9) + (b^2 - 4*a*c)/(112*c^3*d^10*(b + 2*c*x)^7) - 1/(160*c^3*d^10*(
b + 2*c*x)^5)

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Rubi [A]  time = 0.0575813, antiderivative size = 73, 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} \[ -\frac{\left (b^2-4 a c\right )^2}{288 c^3 d^{10} (b+2 c x)^9}+\frac{b^2-4 a c}{112 c^3 d^{10} (b+2 c x)^7}-\frac{1}{160 c^3 d^{10} (b+2 c x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2 - 4*a*c)^2/(288*c^3*d^10*(b + 2*c*x)^9) + (b^2 - 4*a*c)/(112*c^3*d^10*(b + 2*c*x)^7) - 1/(160*c^3*d^10*(
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 )^2}{(b d+2 c d x)^{10}} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^{10} (b+2 c x)^{10}}+\frac{-b^2+4 a c}{8 c^2 d^{10} (b+2 c x)^8}+\frac{1}{16 c^2 d^{10} (b+2 c x)^6}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{288 c^3 d^{10} (b+2 c x)^9}+\frac{b^2-4 a c}{112 c^3 d^{10} (b+2 c x)^7}-\frac{1}{160 c^3 d^{10} (b+2 c x)^5}\\ \end{align*}

Mathematica [A]  time = 0.0404149, size = 59, normalized size = 0.81 \[ \frac{90 \left (b^2-4 a c\right ) (b+2 c x)^2-35 \left (b^2-4 a c\right )^2-63 (b+2 c x)^4}{10080 c^3 d^{10} (b+2 c x)^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-35*(b^2 - 4*a*c)^2 + 90*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 63*(b + 2*c*x)^4)/(10080*c^3*d^10*(b + 2*c*x)^9)

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Maple [A]  time = 0.046, size = 74, normalized size = 1. \begin{align*}{\frac{1}{{d}^{10}} \left ( -{\frac{4\,ac-{b}^{2}}{112\,{c}^{3} \left ( 2\,cx+b \right ) ^{7}}}-{\frac{1}{160\,{c}^{3} \left ( 2\,cx+b \right ) ^{5}}}-{\frac{16\,{a}^{2}{c}^{2}-8\,ac{b}^{2}+{b}^{4}}{288\,{c}^{3} \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)^2/(2*c*d*x+b*d)^10,x)

[Out]

1/d^10*(-1/112*(4*a*c-b^2)/c^3/(2*c*x+b)^7-1/160/c^3/(2*c*x+b)^5-1/288*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b
)^9)

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Maxima [B]  time = 1.48932, size = 278, normalized size = 3.81 \begin{align*} -\frac{126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2} + 36 \,{\left (4 \, b^{2} c^{2} + 5 \, a c^{3}\right )} x^{2} + 18 \,{\left (b^{3} c + 10 \, a b c^{2}\right )} x}{1260 \,{\left (512 \, c^{12} d^{10} x^{9} + 2304 \, b c^{11} d^{10} x^{8} + 4608 \, b^{2} c^{10} d^{10} x^{7} + 5376 \, b^{3} c^{9} d^{10} x^{6} + 4032 \, b^{4} c^{8} d^{10} x^{5} + 2016 \, b^{5} c^{7} d^{10} x^{4} + 672 \, b^{6} c^{6} d^{10} x^{3} + 144 \, b^{7} c^{5} d^{10} x^{2} + 18 \, b^{8} c^{4} d^{10} x + b^{9} c^{3} d^{10}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="maxima")

[Out]

-1/1260*(126*c^4*x^4 + 252*b*c^3*x^3 + b^4 + 10*a*b^2*c + 70*a^2*c^2 + 36*(4*b^2*c^2 + 5*a*c^3)*x^2 + 18*(b^3*
c + 10*a*b*c^2)*x)/(512*c^12*d^10*x^9 + 2304*b*c^11*d^10*x^8 + 4608*b^2*c^10*d^10*x^7 + 5376*b^3*c^9*d^10*x^6
+ 4032*b^4*c^8*d^10*x^5 + 2016*b^5*c^7*d^10*x^4 + 672*b^6*c^6*d^10*x^3 + 144*b^7*c^5*d^10*x^2 + 18*b^8*c^4*d^1
0*x + b^9*c^3*d^10)

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Fricas [B]  time = 2.05418, size = 474, normalized size = 6.49 \begin{align*} -\frac{126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2} + 36 \,{\left (4 \, b^{2} c^{2} + 5 \, a c^{3}\right )} x^{2} + 18 \,{\left (b^{3} c + 10 \, a b c^{2}\right )} x}{1260 \,{\left (512 \, c^{12} d^{10} x^{9} + 2304 \, b c^{11} d^{10} x^{8} + 4608 \, b^{2} c^{10} d^{10} x^{7} + 5376 \, b^{3} c^{9} d^{10} x^{6} + 4032 \, b^{4} c^{8} d^{10} x^{5} + 2016 \, b^{5} c^{7} d^{10} x^{4} + 672 \, b^{6} c^{6} d^{10} x^{3} + 144 \, b^{7} c^{5} d^{10} x^{2} + 18 \, b^{8} c^{4} d^{10} x + b^{9} c^{3} d^{10}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="fricas")

[Out]

-1/1260*(126*c^4*x^4 + 252*b*c^3*x^3 + b^4 + 10*a*b^2*c + 70*a^2*c^2 + 36*(4*b^2*c^2 + 5*a*c^3)*x^2 + 18*(b^3*
c + 10*a*b*c^2)*x)/(512*c^12*d^10*x^9 + 2304*b*c^11*d^10*x^8 + 4608*b^2*c^10*d^10*x^7 + 5376*b^3*c^9*d^10*x^6
+ 4032*b^4*c^8*d^10*x^5 + 2016*b^5*c^7*d^10*x^4 + 672*b^6*c^6*d^10*x^3 + 144*b^7*c^5*d^10*x^2 + 18*b^8*c^4*d^1
0*x + b^9*c^3*d^10)

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Sympy [B]  time = 4.73929, size = 218, normalized size = 2.99 \begin{align*} - \frac{70 a^{2} c^{2} + 10 a b^{2} c + b^{4} + 252 b c^{3} x^{3} + 126 c^{4} x^{4} + x^{2} \left (180 a c^{3} + 144 b^{2} c^{2}\right ) + x \left (180 a b c^{2} + 18 b^{3} c\right )}{1260 b^{9} c^{3} d^{10} + 22680 b^{8} c^{4} d^{10} x + 181440 b^{7} c^{5} d^{10} x^{2} + 846720 b^{6} c^{6} d^{10} x^{3} + 2540160 b^{5} c^{7} d^{10} x^{4} + 5080320 b^{4} c^{8} d^{10} x^{5} + 6773760 b^{3} c^{9} d^{10} x^{6} + 5806080 b^{2} c^{10} d^{10} x^{7} + 2903040 b c^{11} d^{10} x^{8} + 645120 c^{12} d^{10} x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**10,x)

[Out]

-(70*a**2*c**2 + 10*a*b**2*c + b**4 + 252*b*c**3*x**3 + 126*c**4*x**4 + x**2*(180*a*c**3 + 144*b**2*c**2) + x*
(180*a*b*c**2 + 18*b**3*c))/(1260*b**9*c**3*d**10 + 22680*b**8*c**4*d**10*x + 181440*b**7*c**5*d**10*x**2 + 84
6720*b**6*c**6*d**10*x**3 + 2540160*b**5*c**7*d**10*x**4 + 5080320*b**4*c**8*d**10*x**5 + 6773760*b**3*c**9*d*
*10*x**6 + 5806080*b**2*c**10*d**10*x**7 + 2903040*b*c**11*d**10*x**8 + 645120*c**12*d**10*x**9)

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Giac [A]  time = 1.13713, size = 117, normalized size = 1.6 \begin{align*} -\frac{126 \, c^{4} x^{4} + 252 \, b c^{3} x^{3} + 144 \, b^{2} c^{2} x^{2} + 180 \, a c^{3} x^{2} + 18 \, b^{3} c x + 180 \, a b c^{2} x + b^{4} + 10 \, a b^{2} c + 70 \, a^{2} c^{2}}{1260 \,{\left (2 \, c x + b\right )}^{9} c^{3} d^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^10,x, algorithm="giac")

[Out]

-1/1260*(126*c^4*x^4 + 252*b*c^3*x^3 + 144*b^2*c^2*x^2 + 180*a*c^3*x^2 + 18*b^3*c*x + 180*a*b*c^2*x + b^4 + 10
*a*b^2*c + 70*a^2*c^2)/((2*c*x + b)^9*c^3*d^10)