3.2118 \(\int \frac{\left (a+b x+c x^2\right )^2}{(d+e x)^7} \, dx\)

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 = 0.321087, 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 \[ -\frac{-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2}{4 e^5 (d+e x)^4}+\frac{2 (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{5 e^5 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right )^2}{6 e^5 (d+e x)^6}+\frac{2 c (2 c d-b e)}{3 e^5 (d+e x)^3}-\frac{c^2}{2 e^5 (d+e x)^2} \]

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)

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Rubi in Sympy [A]  time = 48.1743, size = 150, normalized size = 0.96 \[ - \frac{c^{2}}{2 e^{5} \left (d + e x\right )^{2}} - \frac{2 c \left (b e - 2 c d\right )}{3 e^{5} \left (d + e x\right )^{3}} - \frac{2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}}{4 e^{5} \left (d + e x\right )^{4}} - \frac{2 \left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )}{5 e^{5} \left (d + e x\right )^{5}} - \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2}}{6 e^{5} \left (d + e x\right )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.127451, size = 159, normalized size = 1.02 \[ -\frac{e^2 \left (10 a^2 e^2+4 a b e (d+6 e x)+b^2 \left (d^2+6 d e x+15 e^2 x^2\right )\right )+2 c e \left (a e \left (d^2+6 d e x+15 e^2 x^2\right )+b \left (d^3+6 d^2 e x+15 d e^2 x^2+20 e^3 x^3\right )\right )+2 c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )}{60 e^5 (d+e x)^6} \]

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 = 0.009, size = 195, normalized size = 1.3 \[ -{\frac{{c}^{2}}{2\,{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{2\,c \left ( be-2\,cd \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{2\,ab{e}^{3}-4\,ad{e}^{2}c-2\,{b}^{2}d{e}^{2}+6\,bc{d}^{2}e-4\,{c}^{2}{d}^{3}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{a}^{2}{e}^{4}-2\,d{e}^{3}ab+2\,ac{d}^{2}{e}^{2}+{b}^{2}{d}^{2}{e}^{2}-2\,{d}^{3}ebc+{c}^{2}{d}^{4}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{2\,ac{e}^{2}+{b}^{2}{e}^{2}-6\,bcde+6\,{c}^{2}{d}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}} \]

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]

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

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Maxima [A]  time = 0.818063, size = 309, normalized size = 1.98 \[ -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + 4 \, a b d e^{3} + 10 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + 4 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

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 = 0.204338, size = 309, normalized size = 1.98 \[ -\frac{30 \, c^{2} e^{4} x^{4} + 2 \, c^{2} d^{4} + 2 \, b c d^{3} e + 4 \, a b d e^{3} + 10 \, a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} + 40 \,{\left (c^{2} d e^{3} + b c e^{4}\right )} x^{3} + 15 \,{\left (2 \, c^{2} d^{2} e^{2} + 2 \, b c d e^{3} +{\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} + 6 \,{\left (2 \, c^{2} d^{3} e + 2 \, b c d^{2} e^{2} + 4 \, a b e^{4} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x}{60 \,{\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

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

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]

Timed out

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GIAC/XCAS [A]  time = 0.205996, size = 242, normalized size = 1.55 \[ -\frac{{\left (30 \, c^{2} x^{4} e^{4} + 40 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 12 \, c^{2} d^{3} x e + 2 \, c^{2} d^{4} + 40 \, b c x^{3} e^{4} + 30 \, b c d x^{2} e^{3} + 12 \, b c d^{2} x e^{2} + 2 \, b c d^{3} e + 15 \, b^{2} x^{2} e^{4} + 30 \, a c x^{2} e^{4} + 6 \, b^{2} d x e^{3} + 12 \, a c d x e^{3} + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} + 24 \, a b x e^{4} + 4 \, a b d e^{3} + 10 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{60 \,{\left (x e + d\right )}^{6}} \]

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