### 3.2130 $$\int \frac{(a+b x+c x^2)^2}{(d+e x)^8} \, dx$$

Optimal. Leaf size=156 $-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{5 e^5 (d+e x)^5}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{3 e^5 (d+e x)^3}$

[Out]

-(c*d^2 - b*d*e + a*e^2)^2/(7*e^5*(d + e*x)^7) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(3*e^5*(d + e*x)^6) -
(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))/(5*e^5*(d + e*x)^5) + (c*(2*c*d - b*e))/(2*e^5*(d + e*x)^4) - c^2
/(3*e^5*(d + e*x)^3)

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Rubi [A]  time = 0.1081, antiderivative size = 156, 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} $-\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{5 e^5 (d+e x)^5}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{3 e^5 (d+e x)^6}-\frac{\left (a e^2-b d e+c d^2\right )^2}{7 e^5 (d+e x)^7}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{3 e^5 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^2/(d + e*x)^8,x]

[Out]

-(c*d^2 - b*d*e + a*e^2)^2/(7*e^5*(d + e*x)^7) + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(3*e^5*(d + e*x)^6) -
(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))/(5*e^5*(d + e*x)^5) + (c*(2*c*d - b*e))/(2*e^5*(d + e*x)^4) - c^2
/(3*e^5*(d + e*x)^3)

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)^8} \, dx &=\int \left (\frac{\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^8}+\frac{2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)^7}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^4 (d+e x)^6}-\frac{2 c (2 c d-b e)}{e^4 (d+e x)^5}+\frac{c^2}{e^4 (d+e x)^4}\right ) \, dx\\ &=-\frac{\left (c d^2-b d e+a e^2\right )^2}{7 e^5 (d+e x)^7}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{3 e^5 (d+e x)^6}-\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{5 e^5 (d+e x)^5}+\frac{c (2 c d-b e)}{2 e^5 (d+e x)^4}-\frac{c^2}{3 e^5 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.067985, size = 161, normalized size = 1.03 $-\frac{2 e^2 \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )+c e \left (4 a e \left (d^2+7 d e x+21 e^2 x^2\right )+3 b \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )}{210 e^5 (d+e x)^7}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + b*x + c*x^2)^2/(d + e*x)^8,x]

[Out]

-(2*c^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 2*e^2*(15*a^2*e^2 + 5*a*b*e*(d + 7*e*
x) + b^2*(d^2 + 7*d*e*x + 21*e^2*x^2)) + c*e*(4*a*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*(d^3 + 7*d^2*e*x + 21*d
*e^2*x^2 + 35*e^3*x^3)))/(210*e^5*(d + e*x)^7)

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Maple [A]  time = 0.046, size = 195, normalized size = 1.3 \begin{align*} -{\frac{c \left ( be-2\,cd \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\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}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\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}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{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)^8,x)

[Out]

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

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Maxima [A]  time = 1.06428, size = 331, normalized size = 2.12 \begin{align*} -\frac{70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 10 \, a b d e^{3} + 30 \, a^{2} e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 35 \,{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \,{\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 7 \,{\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 10 \, a b e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{210 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} 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)^8,x, algorithm="maxima")

[Out]

-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 10*a*b*d*e^3 + 30*a^2*e^4 + 2*(b^2 + 2*a*c)*d^2*e^2 + 35*(2
*c^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^2*d^2*e^2 + 3*b*c*d*e^3 + 2*(b^2 + 2*a*c)*e^4)*x^2 + 7*(2*c^2*d^3*e + 3*
b*c*d^2*e^2 + 10*a*b*e^4 + 2*(b^2 + 2*a*c)*d*e^3)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x
^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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Fricas [A]  time = 2.013, size = 531, normalized size = 3.4 \begin{align*} -\frac{70 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 3 \, b c d^{3} e + 10 \, a b d e^{3} + 30 \, a^{2} e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 35 \,{\left (2 \, c^{2} d e^{3} + 3 \, b c e^{4}\right )} x^{3} + 21 \,{\left (2 \, c^{2} d^{2} e^{2} + 3 \, b c d e^{3} + 2 \,{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 7 \,{\left (2 \, c^{2} d^{3} e + 3 \, b c d^{2} e^{2} + 10 \, a b e^{4} + 2 \,{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{210 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} 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)^8,x, algorithm="fricas")

[Out]

-1/210*(70*c^2*e^4*x^4 + 2*c^2*d^4 + 3*b*c*d^3*e + 10*a*b*d*e^3 + 30*a^2*e^4 + 2*(b^2 + 2*a*c)*d^2*e^2 + 35*(2
*c^2*d*e^3 + 3*b*c*e^4)*x^3 + 21*(2*c^2*d^2*e^2 + 3*b*c*d*e^3 + 2*(b^2 + 2*a*c)*e^4)*x^2 + 7*(2*c^2*d^3*e + 3*
b*c*d^2*e^2 + 10*a*b*e^4 + 2*(b^2 + 2*a*c)*d*e^3)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x
^4 + 35*d^4*e^8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.09075, size = 243, normalized size = 1.56 \begin{align*} -\frac{{\left (70 \, c^{2} x^{4} e^{4} + 70 \, c^{2} d x^{3} e^{3} + 42 \, c^{2} d^{2} x^{2} e^{2} + 14 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 105 \, b c x^{3} e^{4} + 63 \, b c d x^{2} e^{3} + 21 \, b c d^{2} x e^{2} + 3 \, b c d^{3} e + 42 \, b^{2} x^{2} e^{4} + 84 \, a c x^{2} e^{4} + 14 \, b^{2} d x e^{3} + 28 \, a c d x e^{3} + 2 \, b^{2} d^{2} e^{2} + 4 \, a c d^{2} e^{2} + 70 \, a b x e^{4} + 10 \, a b d e^{3} + 30 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{210 \,{\left (x e + d\right )}^{7}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/210*(70*c^2*x^4*e^4 + 70*c^2*d*x^3*e^3 + 42*c^2*d^2*x^2*e^2 + 14*c^2*d^3*x*e + 2*c^2*d^4 + 105*b*c*x^3*e^4
+ 63*b*c*d*x^2*e^3 + 21*b*c*d^2*x*e^2 + 3*b*c*d^3*e + 42*b^2*x^2*e^4 + 84*a*c*x^2*e^4 + 14*b^2*d*x*e^3 + 28*a*
c*d*x*e^3 + 2*b^2*d^2*e^2 + 4*a*c*d^2*e^2 + 70*a*b*x*e^4 + 10*a*b*d*e^3 + 30*a^2*e^4)*e^(-5)/(x*e + d)^7