3.1466 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=89 \[ \frac{b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac{b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac{(a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]

[Out]

(a + b*x)^5/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a + b*x)^5)/(21*(b*d - a*e)^2*(d +
e*x)^6) + (b^2*(a + b*x)^5)/(105*(b*d - a*e)^3*(d + e*x)^5)

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Rubi [A]  time = 0.0814133, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{b^2 (a+b x)^5}{105 (d+e x)^5 (b d-a e)^3}+\frac{b (a+b x)^5}{21 (d+e x)^6 (b d-a e)^2}+\frac{(a+b x)^5}{7 (d+e x)^7 (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^5/(7*(b*d - a*e)*(d + e*x)^7) + (b*(a + b*x)^5)/(21*(b*d - a*e)^2*(d +
e*x)^6) + (b^2*(a + b*x)^5)/(105*(b*d - a*e)^3*(d + e*x)^5)

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Rubi in Sympy [A]  time = 30.5838, size = 73, normalized size = 0.82 \[ - \frac{b^{2} \left (a + b x\right )^{5}}{105 \left (d + e x\right )^{5} \left (a e - b d\right )^{3}} + \frac{b \left (a + b x\right )^{5}}{21 \left (d + e x\right )^{6} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{5}}{7 \left (d + e x\right )^{7} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**8,x)

[Out]

-b**2*(a + b*x)**5/(105*(d + e*x)**5*(a*e - b*d)**3) + b*(a + b*x)**5/(21*(d + e
*x)**6*(a*e - b*d)**2) - (a + b*x)**5/(7*(d + e*x)**7*(a*e - b*d))

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Mathematica [A]  time = 0.10328, size = 144, normalized size = 1.62 \[ -\frac{15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+6 a^2 b^2 e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+3 a b^3 e \left (d^3+7 d^2 e x+21 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (d^4+7 d^3 e x+21 d^2 e^2 x^2+35 d e^3 x^3+35 e^4 x^4\right )}{105 e^5 (d+e x)^7} \]

Antiderivative was successfully verified.

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

[Out]

-(15*a^4*e^4 + 10*a^3*b*e^3*(d + 7*e*x) + 6*a^2*b^2*e^2*(d^2 + 7*d*e*x + 21*e^2*
x^2) + 3*a*b^3*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + b^4*(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))/(105*e^5*(d + e*x)^7)

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Maple [B]  time = 0.009, size = 186, normalized size = 2.1 \[ -{\frac{{e}^{4}{a}^{4}-4\,d{e}^{3}{a}^{3}b+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,{d}^{3}ea{b}^{3}+{b}^{4}{d}^{4}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{6\,{b}^{2} \left ({a}^{2}{e}^{2}-2\,deab+{b}^{2}{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{2\,b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{3} \left ( ae-bd \right ) }{{e}^{5} \left ( ex+d \right ) ^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.699926, size = 333, normalized size = 3.74 \[ -\frac{35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \,{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \,{\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^8,x, algorithm="maxima")

[Out]

-1/105*(35*b^4*e^4*x^4 + b^4*d^4 + 3*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 10*a^3*b*
d*e^3 + 15*a^4*e^4 + 35*(b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 21*(b^4*d^2*e^2 + 3*a*b^
3*d*e^3 + 6*a^2*b^2*e^4)*x^2 + 7*(b^4*d^3*e + 3*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3
+ 10*a^3*b*e^4)*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 = 0.198663, size = 333, normalized size = 3.74 \[ -\frac{35 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 3 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4} + 35 \,{\left (b^{4} d e^{3} + 3 \, a b^{3} e^{4}\right )} x^{3} + 21 \,{\left (b^{4} d^{2} e^{2} + 3 \, a b^{3} d e^{3} + 6 \, a^{2} b^{2} e^{4}\right )} x^{2} + 7 \,{\left (b^{4} d^{3} e + 3 \, a b^{3} d^{2} e^{2} + 6 \, a^{2} b^{2} d e^{3} + 10 \, a^{3} b e^{4}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^2/(e*x + d)^8,x, algorithm="fricas")

[Out]

-1/105*(35*b^4*e^4*x^4 + b^4*d^4 + 3*a*b^3*d^3*e + 6*a^2*b^2*d^2*e^2 + 10*a^3*b*
d*e^3 + 15*a^4*e^4 + 35*(b^4*d*e^3 + 3*a*b^3*e^4)*x^3 + 21*(b^4*d^2*e^2 + 3*a*b^
3*d*e^3 + 6*a^2*b^2*e^4)*x^2 + 7*(b^4*d^3*e + 3*a*b^3*d^2*e^2 + 6*a^2*b^2*d*e^3
+ 10*a^3*b*e^4)*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 [A]  time = 35.7083, size = 264, normalized size = 2.97 \[ - \frac{15 a^{4} e^{4} + 10 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} + 3 a b^{3} d^{3} e + b^{4} d^{4} + 35 b^{4} e^{4} x^{4} + x^{3} \left (105 a b^{3} e^{4} + 35 b^{4} d e^{3}\right ) + x^{2} \left (126 a^{2} b^{2} e^{4} + 63 a b^{3} d e^{3} + 21 b^{4} d^{2} e^{2}\right ) + x \left (70 a^{3} b e^{4} + 42 a^{2} b^{2} d e^{3} + 21 a b^{3} d^{2} e^{2} + 7 b^{4} d^{3} e\right )}{105 d^{7} e^{5} + 735 d^{6} e^{6} x + 2205 d^{5} e^{7} x^{2} + 3675 d^{4} e^{8} x^{3} + 3675 d^{3} e^{9} x^{4} + 2205 d^{2} e^{10} x^{5} + 735 d e^{11} x^{6} + 105 e^{12} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**8,x)

[Out]

-(15*a**4*e**4 + 10*a**3*b*d*e**3 + 6*a**2*b**2*d**2*e**2 + 3*a*b**3*d**3*e + b*
*4*d**4 + 35*b**4*e**4*x**4 + x**3*(105*a*b**3*e**4 + 35*b**4*d*e**3) + x**2*(12
6*a**2*b**2*e**4 + 63*a*b**3*d*e**3 + 21*b**4*d**2*e**2) + x*(70*a**3*b*e**4 + 4
2*a**2*b**2*d*e**3 + 21*a*b**3*d**2*e**2 + 7*b**4*d**3*e))/(105*d**7*e**5 + 735*
d**6*e**6*x + 2205*d**5*e**7*x**2 + 3675*d**4*e**8*x**3 + 3675*d**3*e**9*x**4 +
2205*d**2*e**10*x**5 + 735*d*e**11*x**6 + 105*e**12*x**7)

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GIAC/XCAS [A]  time = 0.211148, size = 235, normalized size = 2.64 \[ -\frac{{\left (35 \, b^{4} x^{4} e^{4} + 35 \, b^{4} d x^{3} e^{3} + 21 \, b^{4} d^{2} x^{2} e^{2} + 7 \, b^{4} d^{3} x e + b^{4} d^{4} + 105 \, a b^{3} x^{3} e^{4} + 63 \, a b^{3} d x^{2} e^{3} + 21 \, a b^{3} d^{2} x e^{2} + 3 \, a b^{3} d^{3} e + 126 \, a^{2} b^{2} x^{2} e^{4} + 42 \, a^{2} b^{2} d x e^{3} + 6 \, a^{2} b^{2} d^{2} e^{2} + 70 \, a^{3} b x e^{4} + 10 \, a^{3} b d e^{3} + 15 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{105 \,{\left (x e + d\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/105*(35*b^4*x^4*e^4 + 35*b^4*d*x^3*e^3 + 21*b^4*d^2*x^2*e^2 + 7*b^4*d^3*x*e +
b^4*d^4 + 105*a*b^3*x^3*e^4 + 63*a*b^3*d*x^2*e^3 + 21*a*b^3*d^2*x*e^2 + 3*a*b^3
*d^3*e + 126*a^2*b^2*x^2*e^4 + 42*a^2*b^2*d*x*e^3 + 6*a^2*b^2*d^2*e^2 + 70*a^3*b
*x*e^4 + 10*a^3*b*d*e^3 + 15*a^4*e^4)*e^(-5)/(x*e + d)^7