### 3.2128 $$\int \frac{(a+b x+c x^2)^2}{(d+e x)^6} \, dx$$

Optimal. Leaf size=151 $-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)}$

[Out]

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

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Rubi [A]  time = 0.111422, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.05, Rules used = {698} $-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{3 e^5 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{2 e^5 (d+e x)^4}-\frac{\left (a e^2-b d e+c d^2\right )^2}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

Rule 698

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] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^6} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^6}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^5}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^4}-\frac{2 c (2 c d-b e)}{e^4 (d+e x)^3}+\frac{c^2}{e^4 (d+e x)^2}\right ) \, dx\\ &=-\frac{\left (c d^2-b d e+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{2 e^5 (d+e x)^4}-\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{3 e^5 (d+e x)^3}+\frac{c (2 c d-b e)}{e^5 (d+e x)^2}-\frac{c^2}{e^5 (d+e x)}\\ \end{align*}

Mathematica [A]  time = 0.0722244, size = 160, normalized size = 1.06 $-\frac{e^2 \left (6 a^2 e^2+3 a b e (d+5 e x)+b^2 \left (d^2+5 d e x+10 e^2 x^2\right )\right )+c e \left (2 a e \left (d^2+5 d e x+10 e^2 x^2\right )+3 b \left (5 d^2 e x+d^3+10 d e^2 x^2+10 e^3 x^3\right )\right )+6 c^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )}{30 e^5 (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(6*c^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + e^2*(6*a^2*e^2 + 3*a*b*e*(d + 5*e*x) +
b^2*(d^2 + 5*d*e*x + 10*e^2*x^2)) + c*e*(2*a*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*b*(d^3 + 5*d^2*e*x + 10*d*e^2
*x^2 + 10*e^3*x^3)))/(30*e^5*(d + e*x)^5)

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Maple [A]  time = 0.044, size = 195, normalized size = 1.3 \begin{align*} -{\frac{2\,ab{e}^{3}-4\,ad{e}^{2}c-2\,{b}^{2}d{e}^{2}+6\,{d}^{2}ebc-4\,{c}^{2}{d}^{3}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{c \left ( be-2\,cd \right ) }{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{a}^{2}{e}^{4}-2\,d{e}^{3}ab+2\,ac{d}^{2}{e}^{2}+{b}^{2}{d}^{2}{e}^{2}-2\,{d}^{3}ebc+{c}^{2}{d}^{4}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.987873, size = 296, normalized size = 1.96 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/30*(30*c^2*e^4*x^4 + 6*c^2*d^4 + 3*b*c*d^3*e + 3*a*b*d*e^3 + 6*a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 30*(2*c^2*
d*e^3 + b*c*e^4)*x^3 + 10*(6*c^2*d^2*e^2 + 3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 5*(6*c^2*d^3*e + 3*b*c*d^2*e
^2 + 3*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6
*x + d^5*e^5)

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Fricas [A]  time = 2.03934, size = 466, normalized size = 3.09 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 6 \, c^{2} d^{4} + 3 \, b c d^{3} e + 3 \, a b d e^{3} + 6 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 30 \,{\left (2 \, c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 10 \,{\left (6 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 5 \,{\left (6 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 3 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{30 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/30*(30*c^2*e^4*x^4 + 6*c^2*d^4 + 3*b*c*d^3*e + 3*a*b*d*e^3 + 6*a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 30*(2*c^2*
d*e^3 + b*c*e^4)*x^3 + 10*(6*c^2*d^2*e^2 + 3*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 5*(6*c^2*d^3*e + 3*b*c*d^2*e
^2 + 3*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6
*x + d^5*e^5)

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Sympy [A]  time = 38.7147, size = 250, normalized size = 1.66 \begin{align*} - \frac{6 a^{2} e^{4} + 3 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 3 b c d^{3} e + 6 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (30 b c e^{4} + 60 c^{2} d e^{3}\right ) + x^{2} \left (20 a c e^{4} + 10 b^{2} e^{4} + 30 b c d e^{3} + 60 c^{2} d^{2} e^{2}\right ) + x \left (15 a b e^{4} + 10 a c d e^{3} + 5 b^{2} d e^{3} + 15 b c d^{2} e^{2} + 30 c^{2} d^{3} e\right )}{30 d^{5} e^{5} + 150 d^{4} e^{6} x + 300 d^{3} e^{7} x^{2} + 300 d^{2} e^{8} x^{3} + 150 d e^{9} x^{4} + 30 e^{10} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**6,x)

[Out]

-(6*a**2*e**4 + 3*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 + 3*b*c*d**3*e + 6*c**2*d**4 + 30*c**2*e**4*x*
*4 + x**3*(30*b*c*e**4 + 60*c**2*d*e**3) + x**2*(20*a*c*e**4 + 10*b**2*e**4 + 30*b*c*d*e**3 + 60*c**2*d**2*e**
2) + x*(15*a*b*e**4 + 10*a*c*d*e**3 + 5*b**2*d*e**3 + 15*b*c*d**2*e**2 + 30*c**2*d**3*e))/(30*d**5*e**5 + 150*
d**4*e**6*x + 300*d**3*e**7*x**2 + 300*d**2*e**8*x**3 + 150*d*e**9*x**4 + 30*e**10*x**5)

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Giac [A]  time = 1.11779, size = 242, normalized size = 1.6 \begin{align*} -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 60 \, c^{2} d x^{3} e^{3} + 60 \, c^{2} d^{2} x^{2} e^{2} + 30 \, c^{2} d^{3} x e + 6 \, c^{2} d^{4} + 30 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 15 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 10 \, b^{2} x^{2} e^{4} + 20 \, a c x^{2} e^{4} + 5 \, b^{2} d x e^{3} + 10 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 15 \, a b x e^{4} + 3 \, a b d e^{3} + 6 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{30 \,{\left (x e + d\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/30*(30*c^2*x^4*e^4 + 60*c^2*d*x^3*e^3 + 60*c^2*d^2*x^2*e^2 + 30*c^2*d^3*x*e + 6*c^2*d^4 + 30*b*c*x^3*e^4 +
30*b*c*d*x^2*e^3 + 15*b*c*d^2*x*e^2 + 3*b*c*d^3*e + 10*b^2*x^2*e^4 + 20*a*c*x^2*e^4 + 5*b^2*d*x*e^3 + 10*a*c*d
*x*e^3 + b^2*d^2*e^2 + 2*a*c*d^2*e^2 + 15*a*b*x*e^4 + 3*a*b*d*e^3 + 6*a^2*e^4)*e^(-5)/(x*e + d)^5