### 3.1130 $$\int \frac{(a+b x+c x^2)^2}{(b d+2 c d x)^5} \, dx$$

Optimal. Leaf size=72 $-\frac{\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac{b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac{\log (b+2 c x)}{32 c^3 d^5}$

[Out]

-(b^2 - 4*a*c)^2/(128*c^3*d^5*(b + 2*c*x)^4) + (b^2 - 4*a*c)/(32*c^3*d^5*(b + 2*c*x)^2) + Log[b + 2*c*x]/(32*c
^3*d^5)

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Rubi [A]  time = 0.0576435, antiderivative size = 72, 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} $-\frac{\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac{b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac{\log (b+2 c x)}{32 c^3 d^5}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^5,x]

[Out]

-(b^2 - 4*a*c)^2/(128*c^3*d^5*(b + 2*c*x)^4) + (b^2 - 4*a*c)/(32*c^3*d^5*(b + 2*c*x)^2) + Log[b + 2*c*x]/(32*c
^3*d^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)^5} \, dx &=\int \left (\frac{\left (-b^2+4 a c\right )^2}{16 c^2 d^5 (b+2 c x)^5}+\frac{-b^2+4 a c}{8 c^2 d^5 (b+2 c x)^3}+\frac{1}{16 c^2 d^5 (b+2 c x)}\right ) \, dx\\ &=-\frac{\left (b^2-4 a c\right )^2}{128 c^3 d^5 (b+2 c x)^4}+\frac{b^2-4 a c}{32 c^3 d^5 (b+2 c x)^2}+\frac{\log (b+2 c x)}{32 c^3 d^5}\\ \end{align*}

Mathematica [A]  time = 0.0348702, size = 59, normalized size = 0.82 $\frac{\frac{\left (b^2-4 a c\right ) \left (4 c \left (a+4 c x^2\right )+3 b^2+16 b c x\right )}{(b+2 c x)^4}+4 \log (b+2 c x)}{128 c^3 d^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^5,x]

[Out]

(((b^2 - 4*a*c)*(3*b^2 + 16*b*c*x + 4*c*(a + 4*c*x^2)))/(b + 2*c*x)^4 + 4*Log[b + 2*c*x])/(128*c^3*d^5)

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Maple [A]  time = 0.045, size = 111, normalized size = 1.5 \begin{align*} -{\frac{a}{8\,{c}^{2}{d}^{5} \left ( 2\,cx+b \right ) ^{2}}}+{\frac{{b}^{2}}{32\,{c}^{3}{d}^{5} \left ( 2\,cx+b \right ) ^{2}}}-{\frac{{a}^{2}}{8\,{d}^{5}c \left ( 2\,cx+b \right ) ^{4}}}+{\frac{{b}^{2}a}{16\,{c}^{2}{d}^{5} \left ( 2\,cx+b \right ) ^{4}}}-{\frac{{b}^{4}}{128\,{c}^{3}{d}^{5} \left ( 2\,cx+b \right ) ^{4}}}+{\frac{\ln \left ( 2\,cx+b \right ) }{32\,{c}^{3}{d}^{5}}} \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)^5,x)

[Out]

-1/8/d^5/c^2/(2*c*x+b)^2*a+1/32/d^5/c^3/(2*c*x+b)^2*b^2-1/8/d^5/c/(2*c*x+b)^4*a^2+1/16/d^5/c^2/(2*c*x+b)^4*a*b
^2-1/128/d^5/c^3/(2*c*x+b)^4*b^4+1/32*ln(2*c*x+b)/c^3/d^5

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Maxima [B]  time = 1.20123, size = 184, normalized size = 2.56 \begin{align*} \frac{3 \, b^{4} - 8 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x}{128 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )}} + \frac{\log \left (2 \, c x + b\right )}{32 \, c^{3} d^{5}} \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)^5,x, algorithm="maxima")

[Out]

1/128*(3*b^4 - 8*a*b^2*c - 16*a^2*c^2 + 16*(b^2*c^2 - 4*a*c^3)*x^2 + 16*(b^3*c - 4*a*b*c^2)*x)/(16*c^7*d^5*x^4
+ 32*b*c^6*d^5*x^3 + 24*b^2*c^5*d^5*x^2 + 8*b^3*c^4*d^5*x + b^4*c^3*d^5) + 1/32*log(2*c*x + b)/(c^3*d^5)

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Fricas [B]  time = 2.05401, size = 360, normalized size = 5. \begin{align*} \frac{3 \, b^{4} - 8 \, a b^{2} c - 16 \, a^{2} c^{2} + 16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} + 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} x + 4 \,{\left (16 \, c^{4} x^{4} + 32 \, b c^{3} x^{3} + 24 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x + b^{4}\right )} \log \left (2 \, c x + b\right )}{128 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\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)^5,x, algorithm="fricas")

[Out]

1/128*(3*b^4 - 8*a*b^2*c - 16*a^2*c^2 + 16*(b^2*c^2 - 4*a*c^3)*x^2 + 16*(b^3*c - 4*a*b*c^2)*x + 4*(16*c^4*x^4
+ 32*b*c^3*x^3 + 24*b^2*c^2*x^2 + 8*b^3*c*x + b^4)*log(2*c*x + b))/(16*c^7*d^5*x^4 + 32*b*c^6*d^5*x^3 + 24*b^2
*c^5*d^5*x^2 + 8*b^3*c^4*d^5*x + b^4*c^3*d^5)

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Sympy [B]  time = 2.07894, size = 139, normalized size = 1.93 \begin{align*} - \frac{16 a^{2} c^{2} + 8 a b^{2} c - 3 b^{4} + x^{2} \left (64 a c^{3} - 16 b^{2} c^{2}\right ) + x \left (64 a b c^{2} - 16 b^{3} c\right )}{128 b^{4} c^{3} d^{5} + 1024 b^{3} c^{4} d^{5} x + 3072 b^{2} c^{5} d^{5} x^{2} + 4096 b c^{6} d^{5} x^{3} + 2048 c^{7} d^{5} x^{4}} + \frac{\log{\left (b + 2 c x \right )}}{32 c^{3} d^{5}} \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)**5,x)

[Out]

-(16*a**2*c**2 + 8*a*b**2*c - 3*b**4 + x**2*(64*a*c**3 - 16*b**2*c**2) + x*(64*a*b*c**2 - 16*b**3*c))/(128*b**
4*c**3*d**5 + 1024*b**3*c**4*d**5*x + 3072*b**2*c**5*d**5*x**2 + 4096*b*c**6*d**5*x**3 + 2048*c**7*d**5*x**4)
+ log(b + 2*c*x)/(32*c**3*d**5)

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Giac [B]  time = 1.17544, size = 197, normalized size = 2.74 \begin{align*} -\frac{\log \left (\frac{1}{4 \,{\left (2 \, c d x + b d\right )}^{2} c^{2} d^{2}}\right )}{64 \, c^{3} d^{5}} - \frac{\frac{b^{4} c^{3} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{8 \, a b^{2} c^{4} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} + \frac{16 \, a^{2} c^{5} d^{9}}{{\left (2 \, c d x + b d\right )}^{4}} - \frac{4 \, b^{2} c^{3} d^{7}}{{\left (2 \, c d x + b d\right )}^{2}} + \frac{16 \, a c^{4} d^{7}}{{\left (2 \, c d x + b d\right )}^{2}}}{128 \, c^{6} 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)^5,x, algorithm="giac")

[Out]

-1/64*log(1/4/((2*c*d*x + b*d)^2*c^2*d^2))/(c^3*d^5) - 1/128*(b^4*c^3*d^9/(2*c*d*x + b*d)^4 - 8*a*b^2*c^4*d^9/
(2*c*d*x + b*d)^4 + 16*a^2*c^5*d^9/(2*c*d*x + b*d)^4 - 4*b^2*c^3*d^7/(2*c*d*x + b*d)^2 + 16*a*c^4*d^7/(2*c*d*x
+ b*d)^2)/(c^6*d^10)