3.10.43 \(\int \frac {(a+b x+c x^2)^2}{(b d+2 c d x)^6} \, dx\)

Optimal. Leaf size=73 \[ -\frac {\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 73, 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 )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^6,x]

[Out]

-(b^2 - 4*a*c)^2/(160*c^3*d^6*(b + 2*c*x)^5) + (b^2 - 4*a*c)/(48*c^3*d^6*(b + 2*c*x)^3) - 1/(32*c^3*d^6*(b + 2
*c*x))

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)^6} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^6 (b+2 c x)^6}+\frac {-b^2+4 a c}{8 c^2 d^6 (b+2 c x)^4}+\frac {1}{16 c^2 d^6 (b+2 c x)^2}\right ) \, dx\\ &=-\frac {\left (b^2-4 a c\right )^2}{160 c^3 d^6 (b+2 c x)^5}+\frac {b^2-4 a c}{48 c^3 d^6 (b+2 c x)^3}-\frac {1}{32 c^3 d^6 (b+2 c x)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 59, normalized size = 0.81 \begin {gather*} \frac {10 \left (b^2-4 a c\right ) (b+2 c x)^2-3 \left (b^2-4 a c\right )^2-15 (b+2 c x)^4}{480 c^3 d^6 (b+2 c x)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^6,x]

[Out]

(-3*(b^2 - 4*a*c)^2 + 10*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 15*(b + 2*c*x)^4)/(480*c^3*d^6*(b + 2*c*x)^5)

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^6,x]

[Out]

IntegrateAlgebraic[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^6, x]

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 149, normalized size = 2.04 \begin {gather*} -\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \, {\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \, {\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \end {gather*}

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)^6,x, algorithm="fricas")

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + b^4 + 2*a*b^2*c + 6*a^2*c^2 + 20*(2*b^2*c^2 + a*c^3)*x^2 + 10*(b^3*c + 2*a*
b*c^2)*x)/(32*c^8*d^6*x^5 + 80*b*c^7*d^6*x^4 + 80*b^2*c^6*d^6*x^3 + 40*b^3*c^5*d^6*x^2 + 10*b^4*c^4*d^6*x + b^
5*c^3*d^6)

________________________________________________________________________________________

giac [A]  time = 0.18, size = 87, normalized size = 1.19 \begin {gather*} -\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + 40 \, b^{2} c^{2} x^{2} + 20 \, a c^{3} x^{2} + 10 \, b^{3} c x + 20 \, a b c^{2} x + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2}}{60 \, {\left (2 \, c x + b\right )}^{5} c^{3} d^{6}} \end {gather*}

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)^6,x, algorithm="giac")

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + 40*b^2*c^2*x^2 + 20*a*c^3*x^2 + 10*b^3*c*x + 20*a*b*c^2*x + b^4 + 2*a*b^2*c
+ 6*a^2*c^2)/((2*c*x + b)^5*c^3*d^6)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 74, normalized size = 1.01 \begin {gather*} \frac {-\frac {4 a c -b^{2}}{48 \left (2 c x +b \right )^{3} c^{3}}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{160 \left (2 c x +b \right )^{5} c^{3}}-\frac {1}{32 \left (2 c x +b \right ) c^{3}}}{d^{6}} \end {gather*}

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)^6,x)

[Out]

1/d^6*(-1/48*(4*a*c-b^2)/c^3/(2*c*x+b)^3-1/160*(16*a^2*c^2-8*a*b^2*c+b^4)/c^3/(2*c*x+b)^5-1/32/c^3/(2*c*x+b))

________________________________________________________________________________________

maxima [B]  time = 1.43, size = 149, normalized size = 2.04 \begin {gather*} -\frac {30 \, c^{4} x^{4} + 60 \, b c^{3} x^{3} + b^{4} + 2 \, a b^{2} c + 6 \, a^{2} c^{2} + 20 \, {\left (2 \, b^{2} c^{2} + a c^{3}\right )} x^{2} + 10 \, {\left (b^{3} c + 2 \, a b c^{2}\right )} x}{60 \, {\left (32 \, c^{8} d^{6} x^{5} + 80 \, b c^{7} d^{6} x^{4} + 80 \, b^{2} c^{6} d^{6} x^{3} + 40 \, b^{3} c^{5} d^{6} x^{2} + 10 \, b^{4} c^{4} d^{6} x + b^{5} c^{3} d^{6}\right )}} \end {gather*}

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)^6,x, algorithm="maxima")

[Out]

-1/60*(30*c^4*x^4 + 60*b*c^3*x^3 + b^4 + 2*a*b^2*c + 6*a^2*c^2 + 20*(2*b^2*c^2 + a*c^3)*x^2 + 10*(b^3*c + 2*a*
b*c^2)*x)/(32*c^8*d^6*x^5 + 80*b*c^7*d^6*x^4 + 80*b^2*c^6*d^6*x^3 + 40*b^3*c^5*d^6*x^2 + 10*b^4*c^4*d^6*x + b^
5*c^3*d^6)

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 141, normalized size = 1.93 \begin {gather*} -\frac {\frac {6\,a^2\,c^2+2\,a\,b^2\,c+b^4}{60\,c^3}+b\,x^3+\frac {c\,x^4}{2}+\frac {x^2\,\left (2\,b^2+a\,c\right )}{3\,c}+\frac {b\,x\,\left (b^2+2\,a\,c\right )}{6\,c^2}}{b^5\,d^6+10\,b^4\,c\,d^6\,x+40\,b^3\,c^2\,d^6\,x^2+80\,b^2\,c^3\,d^6\,x^3+80\,b\,c^4\,d^6\,x^4+32\,c^5\,d^6\,x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^6,x)

[Out]

-((b^4 + 6*a^2*c^2 + 2*a*b^2*c)/(60*c^3) + b*x^3 + (c*x^4)/2 + (x^2*(a*c + 2*b^2))/(3*c) + (b*x*(2*a*c + b^2))
/(6*c^2))/(b^5*d^6 + 32*c^5*d^6*x^5 + 80*b*c^4*d^6*x^4 + 40*b^3*c^2*d^6*x^2 + 80*b^2*c^3*d^6*x^3 + 10*b^4*c*d^
6*x)

________________________________________________________________________________________

sympy [B]  time = 1.36, size = 158, normalized size = 2.16 \begin {gather*} \frac {- 6 a^{2} c^{2} - 2 a b^{2} c - b^{4} - 60 b c^{3} x^{3} - 30 c^{4} x^{4} + x^{2} \left (- 20 a c^{3} - 40 b^{2} c^{2}\right ) + x \left (- 20 a b c^{2} - 10 b^{3} c\right )}{60 b^{5} c^{3} d^{6} + 600 b^{4} c^{4} d^{6} x + 2400 b^{3} c^{5} d^{6} x^{2} + 4800 b^{2} c^{6} d^{6} x^{3} + 4800 b c^{7} d^{6} x^{4} + 1920 c^{8} d^{6} x^{5}} \end {gather*}

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)**6,x)

[Out]

(-6*a**2*c**2 - 2*a*b**2*c - b**4 - 60*b*c**3*x**3 - 30*c**4*x**4 + x**2*(-20*a*c**3 - 40*b**2*c**2) + x*(-20*
a*b*c**2 - 10*b**3*c))/(60*b**5*c**3*d**6 + 600*b**4*c**4*d**6*x + 2400*b**3*c**5*d**6*x**2 + 4800*b**2*c**6*d
**6*x**3 + 4800*b*c**7*d**6*x**4 + 1920*c**8*d**6*x**5)

________________________________________________________________________________________