3.1488 \(\int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{10}} \, dx\)

Optimal. Leaf size=89 \[ \frac{b^2 (a+b x)^7}{252 (d+e x)^7 (b d-a e)^3}+\frac{b (a+b x)^7}{36 (d+e x)^8 (b d-a e)^2}+\frac{(a+b x)^7}{9 (d+e x)^9 (b d-a e)} \]

[Out]

(a + b*x)^7/(9*(b*d - a*e)*(d + e*x)^9) + (b*(a + b*x)^7)/(36*(b*d - a*e)^2*(d +
e*x)^8) + (b^2*(a + b*x)^7)/(252*(b*d - a*e)^3*(d + e*x)^7)

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Rubi [A]  time = 0.0789017, 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 \[ \frac{b^2 (a+b x)^7}{252 (d+e x)^7 (b d-a e)^3}+\frac{b (a+b x)^7}{36 (d+e x)^8 (b d-a e)^2}+\frac{(a+b x)^7}{9 (d+e x)^9 (b d-a e)} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^10,x]

[Out]

(a + b*x)^7/(9*(b*d - a*e)*(d + e*x)^9) + (b*(a + b*x)^7)/(36*(b*d - a*e)^2*(d +
e*x)^8) + (b^2*(a + b*x)^7)/(252*(b*d - a*e)^3*(d + e*x)^7)

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Rubi in Sympy [A]  time = 30.6394, size = 73, normalized size = 0.82 \[ - \frac{b^{2} \left (a + b x\right )^{7}}{252 \left (d + e x\right )^{7} \left (a e - b d\right )^{3}} + \frac{b \left (a + b x\right )^{7}}{36 \left (d + e x\right )^{8} \left (a e - b d\right )^{2}} - \frac{\left (a + b x\right )^{7}}{9 \left (d + e x\right )^{9} \left (a e - b d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**10,x)

[Out]

-b**2*(a + b*x)**7/(252*(d + e*x)**7*(a*e - b*d)**3) + b*(a + b*x)**7/(36*(d + e
*x)**8*(a*e - b*d)**2) - (a + b*x)**7/(9*(d + e*x)**9*(a*e - b*d))

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Mathematica [B]  time = 0.225806, size = 277, normalized size = 3.11 \[ -\frac{28 a^6 e^6+21 a^5 b e^5 (d+9 e x)+15 a^4 b^2 e^4 \left (d^2+9 d e x+36 e^2 x^2\right )+10 a^3 b^3 e^3 \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+6 a^2 b^4 e^2 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )+3 a b^5 e \left (d^5+9 d^4 e x+36 d^3 e^2 x^2+84 d^2 e^3 x^3+126 d e^4 x^4+126 e^5 x^5\right )+b^6 \left (d^6+9 d^5 e x+36 d^4 e^2 x^2+84 d^3 e^3 x^3+126 d^2 e^4 x^4+126 d e^5 x^5+84 e^6 x^6\right )}{252 e^7 (d+e x)^9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^10,x]

[Out]

-(28*a^6*e^6 + 21*a^5*b*e^5*(d + 9*e*x) + 15*a^4*b^2*e^4*(d^2 + 9*d*e*x + 36*e^2
*x^2) + 10*a^3*b^3*e^3*(d^3 + 9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + 6*a^2*b^4
*e^2*(d^4 + 9*d^3*e*x + 36*d^2*e^2*x^2 + 84*d*e^3*x^3 + 126*e^4*x^4) + 3*a*b^5*e
*(d^5 + 9*d^4*e*x + 36*d^3*e^2*x^2 + 84*d^2*e^3*x^3 + 126*d*e^4*x^4 + 126*e^5*x^
5) + b^6*(d^6 + 9*d^5*e*x + 36*d^4*e^2*x^2 + 84*d^3*e^3*x^3 + 126*d^2*e^4*x^4 +
126*d*e^5*x^5 + 84*e^6*x^6))/(252*e^7*(d + e*x)^9)

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Maple [B]  time = 0.011, size = 357, normalized size = 4. \[ -{\frac{3\,b \left ({a}^{5}{e}^{5}-5\,{a}^{4}bd{e}^{4}+10\,{a}^{3}{b}^{2}{d}^{2}{e}^{3}-10\,{a}^{2}{b}^{3}{d}^{3}{e}^{2}+5\,a{b}^{4}{d}^{4}e-{b}^{5}{d}^{5} \right ) }{4\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{15\,{b}^{2} \left ({e}^{4}{a}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{a}^{2}{b}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-3\,{\frac{{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{{e}^{7} \left ( ex+d \right ) ^{5}}}-{\frac{10\,{b}^{3} \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{3\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{6}}{3\,{e}^{7} \left ( ex+d \right ) ^{3}}}-{\frac{3\,{b}^{5} \left ( ae-bd \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{e}^{6}{a}^{6}-6\,d{e}^{5}{a}^{5}b+15\,{d}^{2}{e}^{4}{b}^{2}{a}^{4}-20\,{d}^{3}{e}^{3}{a}^{3}{b}^{3}+15\,{d}^{4}{e}^{2}{a}^{2}{b}^{4}-6\,{d}^{5}ea{b}^{5}+{d}^{6}{b}^{6}}{9\,{e}^{7} \left ( ex+d \right ) ^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^3/(e*x+d)^10,x)

[Out]

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

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Maxima [A]  time = 0.704093, size = 595, normalized size = 6.69 \[ -\frac{84 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6} + 126 \,{\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 126 \,{\left (b^{6} d^{2} e^{4} + 3 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} + 84 \,{\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 36 \,{\left (b^{6} d^{4} e^{2} + 3 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 10 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (b^{6} d^{5} e + 3 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 21 \, a^{5} b e^{6}\right )} x}{252 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^10,x, algorithm="maxima")

[Out]

-1/252*(84*b^6*e^6*x^6 + b^6*d^6 + 3*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 + 10*a^3*b^
3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 21*a^5*b*d*e^5 + 28*a^6*e^6 + 126*(b^6*d*e^5 +
3*a*b^5*e^6)*x^5 + 126*(b^6*d^2*e^4 + 3*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 84*(b
^6*d^3*e^3 + 3*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 10*a^3*b^3*e^6)*x^3 + 36*(b^6*d
^4*e^2 + 3*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 10*a^3*b^3*d*e^5 + 15*a^4*b^2*e^6
)*x^2 + 9*(b^6*d^5*e + 3*a*b^5*d^4*e^2 + 6*a^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*e^4
+ 15*a^4*b^2*d*e^5 + 21*a^5*b*e^6)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7
+ 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*
d^7*e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)

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Fricas [A]  time = 0.203235, size = 595, normalized size = 6.69 \[ -\frac{84 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 3 \, a b^{5} d^{5} e + 6 \, a^{2} b^{4} d^{4} e^{2} + 10 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6} + 126 \,{\left (b^{6} d e^{5} + 3 \, a b^{5} e^{6}\right )} x^{5} + 126 \,{\left (b^{6} d^{2} e^{4} + 3 \, a b^{5} d e^{5} + 6 \, a^{2} b^{4} e^{6}\right )} x^{4} + 84 \,{\left (b^{6} d^{3} e^{3} + 3 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 10 \, a^{3} b^{3} e^{6}\right )} x^{3} + 36 \,{\left (b^{6} d^{4} e^{2} + 3 \, a b^{5} d^{3} e^{3} + 6 \, a^{2} b^{4} d^{2} e^{4} + 10 \, a^{3} b^{3} d e^{5} + 15 \, a^{4} b^{2} e^{6}\right )} x^{2} + 9 \,{\left (b^{6} d^{5} e + 3 \, a b^{5} d^{4} e^{2} + 6 \, a^{2} b^{4} d^{3} e^{3} + 10 \, a^{3} b^{3} d^{2} e^{4} + 15 \, a^{4} b^{2} d e^{5} + 21 \, a^{5} b e^{6}\right )} x}{252 \,{\left (e^{16} x^{9} + 9 \, d e^{15} x^{8} + 36 \, d^{2} e^{14} x^{7} + 84 \, d^{3} e^{13} x^{6} + 126 \, d^{4} e^{12} x^{5} + 126 \, d^{5} e^{11} x^{4} + 84 \, d^{6} e^{10} x^{3} + 36 \, d^{7} e^{9} x^{2} + 9 \, d^{8} e^{8} x + d^{9} e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^10,x, algorithm="fricas")

[Out]

-1/252*(84*b^6*e^6*x^6 + b^6*d^6 + 3*a*b^5*d^5*e + 6*a^2*b^4*d^4*e^2 + 10*a^3*b^
3*d^3*e^3 + 15*a^4*b^2*d^2*e^4 + 21*a^5*b*d*e^5 + 28*a^6*e^6 + 126*(b^6*d*e^5 +
3*a*b^5*e^6)*x^5 + 126*(b^6*d^2*e^4 + 3*a*b^5*d*e^5 + 6*a^2*b^4*e^6)*x^4 + 84*(b
^6*d^3*e^3 + 3*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 10*a^3*b^3*e^6)*x^3 + 36*(b^6*d
^4*e^2 + 3*a*b^5*d^3*e^3 + 6*a^2*b^4*d^2*e^4 + 10*a^3*b^3*d*e^5 + 15*a^4*b^2*e^6
)*x^2 + 9*(b^6*d^5*e + 3*a*b^5*d^4*e^2 + 6*a^2*b^4*d^3*e^3 + 10*a^3*b^3*d^2*e^4
+ 15*a^4*b^2*d*e^5 + 21*a^5*b*e^6)*x)/(e^16*x^9 + 9*d*e^15*x^8 + 36*d^2*e^14*x^7
+ 84*d^3*e^13*x^6 + 126*d^4*e^12*x^5 + 126*d^5*e^11*x^4 + 84*d^6*e^10*x^3 + 36*
d^7*e^9*x^2 + 9*d^8*e^8*x + d^9*e^7)

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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((b**2*x**2+2*a*b*x+a**2)**3/(e*x+d)**10,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213026, size = 475, normalized size = 5.34 \[ -\frac{{\left (84 \, b^{6} x^{6} e^{6} + 126 \, b^{6} d x^{5} e^{5} + 126 \, b^{6} d^{2} x^{4} e^{4} + 84 \, b^{6} d^{3} x^{3} e^{3} + 36 \, b^{6} d^{4} x^{2} e^{2} + 9 \, b^{6} d^{5} x e + b^{6} d^{6} + 378 \, a b^{5} x^{5} e^{6} + 378 \, a b^{5} d x^{4} e^{5} + 252 \, a b^{5} d^{2} x^{3} e^{4} + 108 \, a b^{5} d^{3} x^{2} e^{3} + 27 \, a b^{5} d^{4} x e^{2} + 3 \, a b^{5} d^{5} e + 756 \, a^{2} b^{4} x^{4} e^{6} + 504 \, a^{2} b^{4} d x^{3} e^{5} + 216 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 54 \, a^{2} b^{4} d^{3} x e^{3} + 6 \, a^{2} b^{4} d^{4} e^{2} + 840 \, a^{3} b^{3} x^{3} e^{6} + 360 \, a^{3} b^{3} d x^{2} e^{5} + 90 \, a^{3} b^{3} d^{2} x e^{4} + 10 \, a^{3} b^{3} d^{3} e^{3} + 540 \, a^{4} b^{2} x^{2} e^{6} + 135 \, a^{4} b^{2} d x e^{5} + 15 \, a^{4} b^{2} d^{2} e^{4} + 189 \, a^{5} b x e^{6} + 21 \, a^{5} b d e^{5} + 28 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{252 \,{\left (x e + d\right )}^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^2 + 2*a*b*x + a^2)^3/(e*x + d)^10,x, algorithm="giac")

[Out]

-1/252*(84*b^6*x^6*e^6 + 126*b^6*d*x^5*e^5 + 126*b^6*d^2*x^4*e^4 + 84*b^6*d^3*x^
3*e^3 + 36*b^6*d^4*x^2*e^2 + 9*b^6*d^5*x*e + b^6*d^6 + 378*a*b^5*x^5*e^6 + 378*a
*b^5*d*x^4*e^5 + 252*a*b^5*d^2*x^3*e^4 + 108*a*b^5*d^3*x^2*e^3 + 27*a*b^5*d^4*x*
e^2 + 3*a*b^5*d^5*e + 756*a^2*b^4*x^4*e^6 + 504*a^2*b^4*d*x^3*e^5 + 216*a^2*b^4*
d^2*x^2*e^4 + 54*a^2*b^4*d^3*x*e^3 + 6*a^2*b^4*d^4*e^2 + 840*a^3*b^3*x^3*e^6 + 3
60*a^3*b^3*d*x^2*e^5 + 90*a^3*b^3*d^2*x*e^4 + 10*a^3*b^3*d^3*e^3 + 540*a^4*b^2*x
^2*e^6 + 135*a^4*b^2*d*x*e^5 + 15*a^4*b^2*d^2*e^4 + 189*a^5*b*x*e^6 + 21*a^5*b*d
*e^5 + 28*a^6*e^6)*e^(-7)/(x*e + d)^9