3.1128 \(\int \frac {(a+b x+c x^2)^2}{(b d+2 c d x)^3} \, dx\)

Optimal. Leaf size=79 \[ -\frac {\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3}+\frac {b x}{16 c^2 d^3}+\frac {x^2}{16 c d^3} \]

[Out]

1/16*b*x/c^2/d^3+1/16*x^2/c/d^3-1/64*(-4*a*c+b^2)^2/c^3/d^3/(2*c*x+b)^2-1/16*(-4*a*c+b^2)*ln(2*c*x+b)/c^3/d^3

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 79, 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} \[ -\frac {\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3}+\frac {b x}{16 c^2 d^3}+\frac {x^2}{16 c d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x)/(16*c^2*d^3) + x^2/(16*c*d^3) - (b^2 - 4*a*c)^2/(64*c^3*d^3*(b + 2*c*x)^2) - ((b^2 - 4*a*c)*Log[b + 2*c*
x])/(16*c^3*d^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 )^2}{(b d+2 c d x)^3} \, dx &=\int \left (\frac {b}{16 c^2 d^3}+\frac {x}{8 c d^3}+\frac {\left (-b^2+4 a c\right )^2}{16 c^2 d^3 (b+2 c x)^3}+\frac {-b^2+4 a c}{8 c^2 d^3 (b+2 c x)}\right ) \, dx\\ &=\frac {b x}{16 c^2 d^3}+\frac {x^2}{16 c d^3}-\frac {\left (b^2-4 a c\right )^2}{64 c^3 d^3 (b+2 c x)^2}-\frac {\left (b^2-4 a c\right ) \log (b+2 c x)}{16 c^3 d^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 61, normalized size = 0.77 \[ \frac {-\frac {\left (b^2-4 a c\right )^2}{(b+2 c x)^2}-4 \left (b^2-4 a c\right ) \log (b+2 c x)+4 b c x+4 c^2 x^2}{64 c^3 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b*c*x + 4*c^2*x^2 - (b^2 - 4*a*c)^2/(b + 2*c*x)^2 - 4*(b^2 - 4*a*c)*Log[b + 2*c*x])/(64*c^3*d^3)

________________________________________________________________________________________

fricas [B]  time = 1.15, size = 147, normalized size = 1.86 \[ \frac {16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 20 \, b^{2} c^{2} x^{2} + 4 \, b^{3} c x - b^{4} + 8 \, a b^{2} c - 16 \, a^{2} c^{2} - 4 \, {\left (b^{4} - 4 \, a b^{2} c + 4 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 4 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \log \left (2 \, c x + b\right )}{64 \, {\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/64*(16*c^4*x^4 + 32*b*c^3*x^3 + 20*b^2*c^2*x^2 + 4*b^3*c*x - b^4 + 8*a*b^2*c - 16*a^2*c^2 - 4*(b^4 - 4*a*b^2
*c + 4*(b^2*c^2 - 4*a*c^3)*x^2 + 4*(b^3*c - 4*a*b*c^2)*x)*log(2*c*x + b))/(4*c^5*d^3*x^2 + 4*b*c^4*d^3*x + b^2
*c^3*d^3)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 88, normalized size = 1.11 \[ -\frac {{\left (b^{2} - 4 \, a c\right )} \log \left ({\left | 2 \, c x + b \right |}\right )}{16 \, c^{3} d^{3}} - \frac {b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{64 \, {\left (2 \, c x + b\right )}^{2} c^{3} d^{3}} + \frac {c^{5} d^{3} x^{2} + b c^{4} d^{3} x}{16 \, c^{6} d^{6}} \]

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)^3,x, algorithm="giac")

[Out]

-1/16*(b^2 - 4*a*c)*log(abs(2*c*x + b))/(c^3*d^3) - 1/64*(b^4 - 8*a*b^2*c + 16*a^2*c^2)/((2*c*x + b)^2*c^3*d^3
) + 1/16*(c^5*d^3*x^2 + b*c^4*d^3*x)/(c^6*d^6)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 115, normalized size = 1.46 \[ -\frac {a^{2}}{4 \left (2 c x +b \right )^{2} c \,d^{3}}+\frac {a \,b^{2}}{8 \left (2 c x +b \right )^{2} c^{2} d^{3}}-\frac {b^{4}}{64 \left (2 c x +b \right )^{2} c^{3} d^{3}}+\frac {x^{2}}{16 c \,d^{3}}+\frac {a \ln \left (2 c x +b \right )}{4 c^{2} d^{3}}-\frac {b^{2} \ln \left (2 c x +b \right )}{16 c^{3} d^{3}}+\frac {b x}{16 c^{2} d^{3}} \]

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)^3,x)

[Out]

1/16*x^2/c/d^3+1/16*b*x/c^2/d^3-1/4/d^3/c/(2*c*x+b)^2*a^2+1/8/d^3/c^2/(2*c*x+b)^2*a*b^2-1/64/d^3/c^3/(2*c*x+b)
^2*b^4+1/4/d^3/c^2*ln(2*c*x+b)*a-1/16/d^3/c^3*ln(2*c*x+b)*b^2

________________________________________________________________________________________

maxima [A]  time = 1.38, size = 96, normalized size = 1.22 \[ -\frac {b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{64 \, {\left (4 \, c^{5} d^{3} x^{2} + 4 \, b c^{4} d^{3} x + b^{2} c^{3} d^{3}\right )}} + \frac {c x^{2} + b x}{16 \, c^{2} d^{3}} - \frac {{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{16 \, c^{3} d^{3}} \]

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)^3,x, algorithm="maxima")

[Out]

-1/64*(b^4 - 8*a*b^2*c + 16*a^2*c^2)/(4*c^5*d^3*x^2 + 4*b*c^4*d^3*x + b^2*c^3*d^3) + 1/16*(c*x^2 + b*x)/(c^2*d
^3) - 1/16*(b^2 - 4*a*c)*log(2*c*x + b)/(c^3*d^3)

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 106, normalized size = 1.34 \[ \frac {x^2}{16\,c\,d^3}-\frac {16\,a^2\,c^2-8\,a\,b^2\,c+b^4}{4\,c\,\left (16\,b^2\,c^2\,d^3+64\,b\,c^3\,d^3\,x+64\,c^4\,d^3\,x^2\right )}+\frac {b\,x}{16\,c^2\,d^3}+\frac {\ln \left (b+2\,c\,x\right )\,\left (4\,a\,c-b^2\right )}{16\,c^3\,d^3} \]

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)^3,x)

[Out]

x^2/(16*c*d^3) - (b^4 + 16*a^2*c^2 - 8*a*b^2*c)/(4*c*(16*b^2*c^2*d^3 + 64*c^4*d^3*x^2 + 64*b*c^3*d^3*x)) + (b*
x)/(16*c^2*d^3) + (log(b + 2*c*x)*(4*a*c - b^2))/(16*c^3*d^3)

________________________________________________________________________________________

sympy [A]  time = 0.79, size = 102, normalized size = 1.29 \[ \frac {b x}{16 c^{2} d^{3}} + \frac {- 16 a^{2} c^{2} + 8 a b^{2} c - b^{4}}{64 b^{2} c^{3} d^{3} + 256 b c^{4} d^{3} x + 256 c^{5} d^{3} x^{2}} + \frac {x^{2}}{16 c d^{3}} + \frac {\left (4 a c - b^{2}\right ) \log {\left (b + 2 c x \right )}}{16 c^{3} d^{3}} \]

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)**3,x)

[Out]

b*x/(16*c**2*d**3) + (-16*a**2*c**2 + 8*a*b**2*c - b**4)/(64*b**2*c**3*d**3 + 256*b*c**4*d**3*x + 256*c**5*d**
3*x**2) + x**2/(16*c*d**3) + (4*a*c - b**2)*log(b + 2*c*x)/(16*c**3*d**3)

________________________________________________________________________________________