3.2129 \(\int \frac{(a+b x+c x^2)^2}{(d+e x)^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.109025, 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} \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{4 e^5 (d+e x)^4}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac{c^2}{2 e^5 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^2/(d + e*x)^7,x]

[Out]

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

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

Mathematica [A]  time = 0.0636362, size = 159, normalized size = 1.02 \[ -\frac{e^2 \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+6 d e x+15 e^2 x^2\right )+b \left (6 d^2 e x+d^3+15 d e^2 x^2+20 e^3 x^3\right )\right )+2 c^2 \left (15 d^2 e^2 x^2+6 d^3 e x+d^4+20 d e^3 x^3+15 e^4 x^4\right )}{60 e^5 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^2/(d + e*x)^7,x]

[Out]

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

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

[Out]

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

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Maxima [A]  time = 1.00818, size = 309, normalized size = 1.98 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + 4 \, a b d e^{3} + 10 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + 4 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} 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)^7,x, algorithm="maxima")

[Out]

-1/60*(30*c^2*e^4*x^4 + 2*c^2*d^4 + 2*b*c*d^3*e + 4*a*b*d*e^3 + 10*a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 40*(c^2*d
*e^3 + b*c*e^4)*x^3 + 15*(2*c^2*d^2*e^2 + 2*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 6*(2*c^2*d^3*e + 2*b*c*d^2*e^
2 + 4*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^
7*x^2 + 6*d^5*e^6*x + d^6*e^5)

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Fricas [A]  time = 2.02112, size = 489, normalized size = 3.13 \begin{align*} -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + 4 \, a b d e^{3} + 10 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + 4 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} 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)^7,x, algorithm="fricas")

[Out]

-1/60*(30*c^2*e^4*x^4 + 2*c^2*d^4 + 2*b*c*d^3*e + 4*a*b*d*e^3 + 10*a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2 + 40*(c^2*d
*e^3 + b*c*e^4)*x^3 + 15*(2*c^2*d^2*e^2 + 2*b*c*d*e^3 + (b^2 + 2*a*c)*e^4)*x^2 + 6*(2*c^2*d^3*e + 2*b*c*d^2*e^
2 + 4*a*b*e^4 + (b^2 + 2*a*c)*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^
7*x^2 + 6*d^5*e^6*x + d^6*e^5)

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Sympy [A]  time = 85.4175, size = 262, normalized size = 1.68 \begin{align*} - \frac{10 a^{2} e^{4} + 4 a b d e^{3} + 2 a c d^{2} e^{2} + b^{2} d^{2} e^{2} + 2 b c d^{3} e + 2 c^{2} d^{4} + 30 c^{2} e^{4} x^{4} + x^{3} \left (40 b c e^{4} + 40 c^{2} d e^{3}\right ) + x^{2} \left (30 a c e^{4} + 15 b^{2} e^{4} + 30 b c d e^{3} + 30 c^{2} d^{2} e^{2}\right ) + x \left (24 a b e^{4} + 12 a c d e^{3} + 6 b^{2} d e^{3} + 12 b c d^{2} e^{2} + 12 c^{2} d^{3} e\right )}{60 d^{6} e^{5} + 360 d^{5} e^{6} x + 900 d^{4} e^{7} x^{2} + 1200 d^{3} e^{8} x^{3} + 900 d^{2} e^{9} x^{4} + 360 d e^{10} x^{5} + 60 e^{11} x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**2/(e*x+d)**7,x)

[Out]

-(10*a**2*e**4 + 4*a*b*d*e**3 + 2*a*c*d**2*e**2 + b**2*d**2*e**2 + 2*b*c*d**3*e + 2*c**2*d**4 + 30*c**2*e**4*x
**4 + x**3*(40*b*c*e**4 + 40*c**2*d*e**3) + x**2*(30*a*c*e**4 + 15*b**2*e**4 + 30*b*c*d*e**3 + 30*c**2*d**2*e*
*2) + x*(24*a*b*e**4 + 12*a*c*d*e**3 + 6*b**2*d*e**3 + 12*b*c*d**2*e**2 + 12*c**2*d**3*e))/(60*d**6*e**5 + 360
*d**5*e**6*x + 900*d**4*e**7*x**2 + 1200*d**3*e**8*x**3 + 900*d**2*e**9*x**4 + 360*d*e**10*x**5 + 60*e**11*x**
6)

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Giac [A]  time = 1.09241, size = 242, normalized size = 1.55 \begin{align*} -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 40 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 12 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 40 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 12 \, b c d^{2} x e^{2} + 2 \, b c d^{3} e + 15 \, b^{2} x^{2} e^{4} + 30 \, a c x^{2} e^{4} + 6 \, b^{2} d x e^{3} + 12 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 24 \, a b x e^{4} + 4 \, a b d e^{3} + 10 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^2/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/60*(30*c^2*x^4*e^4 + 40*c^2*d*x^3*e^3 + 30*c^2*d^2*x^2*e^2 + 12*c^2*d^3*x*e + 2*c^2*d^4 + 40*b*c*x^3*e^4 +
30*b*c*d*x^2*e^3 + 12*b*c*d^2*x*e^2 + 2*b*c*d^3*e + 15*b^2*x^2*e^4 + 30*a*c*x^2*e^4 + 6*b^2*d*x*e^3 + 12*a*c*d
*x*e^3 + b^2*d^2*e^2 + 2*a*c*d^2*e^2 + 24*a*b*x*e^4 + 4*a*b*d*e^3 + 10*a^2*e^4)*e^(-5)/(x*e + d)^6