3.1480 \(\int \frac{(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=119 \[ \frac{4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac{3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac{4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac{(b d-a e)^4}{10 e^5 (d+e x)^{10}}-\frac{b^4}{6 e^5 (d+e x)^6} \]

[Out]

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

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Rubi [A]  time = 0.0682365, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {27, 43} \[ \frac{4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac{3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac{4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac{(b d-a e)^4}{10 e^5 (d+e x)^{10}}-\frac{b^4}{6 e^5 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^11,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx &=\int \frac{(a+b x)^4}{(d+e x)^{11}} \, dx\\ &=\int \left (\frac{(-b d+a e)^4}{e^4 (d+e x)^{11}}-\frac{4 b (b d-a e)^3}{e^4 (d+e x)^{10}}+\frac{6 b^2 (b d-a e)^2}{e^4 (d+e x)^9}-\frac{4 b^3 (b d-a e)}{e^4 (d+e x)^8}+\frac{b^4}{e^4 (d+e x)^7}\right ) \, dx\\ &=-\frac{(b d-a e)^4}{10 e^5 (d+e x)^{10}}+\frac{4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac{3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac{4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac{b^4}{6 e^5 (d+e x)^6}\\ \end{align*}

Mathematica [A]  time = 0.0459065, size = 144, normalized size = 1.21 \[ -\frac{21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+56 a^3 b e^3 (d+10 e x)+126 a^4 e^4+6 a b^3 e \left (10 d^2 e x+d^3+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )}{1260 e^5 (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^2/(d + e*x)^11,x]

[Out]

-(126*a^4*e^4 + 56*a^3*b*e^3*(d + 10*e*x) + 21*a^2*b^2*e^2*(d^2 + 10*d*e*x + 45*e^2*x^2) + 6*a*b^3*e*(d^3 + 10
*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + b^4*(d^4 + 10*d^3*e*x + 45*d^2*e^2*x^2 + 120*d*e^3*x^3 + 210*e^4*x^4)
)/(1260*e^5*(d + e*x)^10)

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Maple [A]  time = 0.045, size = 186, normalized size = 1.6 \begin{align*} -{\frac{4\,b \left ({a}^{3}{e}^{3}-3\,{a}^{2}bd{e}^{2}+3\,a{b}^{2}{d}^{2}e-{b}^{3}{d}^{3} \right ) }{9\,{e}^{5} \left ( ex+d \right ) ^{9}}}-{\frac{{b}^{4}}{6\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{b}^{2} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{4\,{e}^{5} \left ( ex+d \right ) ^{8}}}-{\frac{4\,{b}^{3} \left ( ae-bd \right ) }{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{{a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,{d}^{3}ea{b}^{3}+{b}^{4}{d}^{4}}{10\,{e}^{5} \left ( ex+d \right ) ^{10}}} \end{align*}

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)^11,x)

[Out]

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

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Maxima [B]  time = 1.06797, size = 378, normalized size = 3.18 \begin{align*} -\frac{210 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \,{\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \,{\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \,{\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \end{align*}

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)^11,x, algorithm="maxima")

[Out]

-1/1260*(210*b^4*e^4*x^4 + b^4*d^4 + 6*a*b^3*d^3*e + 21*a^2*b^2*d^2*e^2 + 56*a^3*b*d*e^3 + 126*a^4*e^4 + 120*(
b^4*d*e^3 + 6*a*b^3*e^4)*x^3 + 45*(b^4*d^2*e^2 + 6*a*b^3*d*e^3 + 21*a^2*b^2*e^4)*x^2 + 10*(b^4*d^3*e + 6*a*b^3
*d^2*e^2 + 21*a^2*b^2*d*e^3 + 56*a^3*b*e^4)*x)/(e^15*x^10 + 10*d*e^14*x^9 + 45*d^2*e^13*x^8 + 120*d^3*e^12*x^7
+ 210*d^4*e^11*x^6 + 252*d^5*e^10*x^5 + 210*d^6*e^9*x^4 + 120*d^7*e^8*x^3 + 45*d^8*e^7*x^2 + 10*d^9*e^6*x + d
^10*e^5)

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Fricas [B]  time = 1.70903, size = 608, normalized size = 5.11 \begin{align*} -\frac{210 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \,{\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \,{\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \,{\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \,{\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \end{align*}

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)^11,x, algorithm="fricas")

[Out]

-1/1260*(210*b^4*e^4*x^4 + b^4*d^4 + 6*a*b^3*d^3*e + 21*a^2*b^2*d^2*e^2 + 56*a^3*b*d*e^3 + 126*a^4*e^4 + 120*(
b^4*d*e^3 + 6*a*b^3*e^4)*x^3 + 45*(b^4*d^2*e^2 + 6*a*b^3*d*e^3 + 21*a^2*b^2*e^4)*x^2 + 10*(b^4*d^3*e + 6*a*b^3
*d^2*e^2 + 21*a^2*b^2*d*e^3 + 56*a^3*b*e^4)*x)/(e^15*x^10 + 10*d*e^14*x^9 + 45*d^2*e^13*x^8 + 120*d^3*e^12*x^7
+ 210*d^4*e^11*x^6 + 252*d^5*e^10*x^5 + 210*d^6*e^9*x^4 + 120*d^7*e^8*x^3 + 45*d^8*e^7*x^2 + 10*d^9*e^6*x + d
^10*e^5)

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Sympy [B]  time = 69.0312, size = 299, normalized size = 2.51 \begin{align*} - \frac{126 a^{4} e^{4} + 56 a^{3} b d e^{3} + 21 a^{2} b^{2} d^{2} e^{2} + 6 a b^{3} d^{3} e + b^{4} d^{4} + 210 b^{4} e^{4} x^{4} + x^{3} \left (720 a b^{3} e^{4} + 120 b^{4} d e^{3}\right ) + x^{2} \left (945 a^{2} b^{2} e^{4} + 270 a b^{3} d e^{3} + 45 b^{4} d^{2} e^{2}\right ) + x \left (560 a^{3} b e^{4} + 210 a^{2} b^{2} d e^{3} + 60 a b^{3} d^{2} e^{2} + 10 b^{4} d^{3} e\right )}{1260 d^{10} e^{5} + 12600 d^{9} e^{6} x + 56700 d^{8} e^{7} x^{2} + 151200 d^{7} e^{8} x^{3} + 264600 d^{6} e^{9} x^{4} + 317520 d^{5} e^{10} x^{5} + 264600 d^{4} e^{11} x^{6} + 151200 d^{3} e^{12} x^{7} + 56700 d^{2} e^{13} x^{8} + 12600 d e^{14} x^{9} + 1260 e^{15} x^{10}} \end{align*}

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)**11,x)

[Out]

-(126*a**4*e**4 + 56*a**3*b*d*e**3 + 21*a**2*b**2*d**2*e**2 + 6*a*b**3*d**3*e + b**4*d**4 + 210*b**4*e**4*x**4
+ x**3*(720*a*b**3*e**4 + 120*b**4*d*e**3) + x**2*(945*a**2*b**2*e**4 + 270*a*b**3*d*e**3 + 45*b**4*d**2*e**2
) + x*(560*a**3*b*e**4 + 210*a**2*b**2*d*e**3 + 60*a*b**3*d**2*e**2 + 10*b**4*d**3*e))/(1260*d**10*e**5 + 1260
0*d**9*e**6*x + 56700*d**8*e**7*x**2 + 151200*d**7*e**8*x**3 + 264600*d**6*e**9*x**4 + 317520*d**5*e**10*x**5
+ 264600*d**4*e**11*x**6 + 151200*d**3*e**12*x**7 + 56700*d**2*e**13*x**8 + 12600*d*e**14*x**9 + 1260*e**15*x*
*10)

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Giac [A]  time = 1.12953, size = 235, normalized size = 1.97 \begin{align*} -\frac{{\left (210 \, b^{4} x^{4} e^{4} + 120 \, b^{4} d x^{3} e^{3} + 45 \, b^{4} d^{2} x^{2} e^{2} + 10 \, b^{4} d^{3} x e + b^{4} d^{4} + 720 \, a b^{3} x^{3} e^{4} + 270 \, a b^{3} d x^{2} e^{3} + 60 \, a b^{3} d^{2} x e^{2} + 6 \, a b^{3} d^{3} e + 945 \, a^{2} b^{2} x^{2} e^{4} + 210 \, a^{2} b^{2} d x e^{3} + 21 \, a^{2} b^{2} d^{2} e^{2} + 560 \, a^{3} b x e^{4} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4}\right )} e^{\left (-5\right )}}{1260 \,{\left (x e + d\right )}^{10}} \end{align*}

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)^11,x, algorithm="giac")

[Out]

-1/1260*(210*b^4*x^4*e^4 + 120*b^4*d*x^3*e^3 + 45*b^4*d^2*x^2*e^2 + 10*b^4*d^3*x*e + b^4*d^4 + 720*a*b^3*x^3*e
^4 + 270*a*b^3*d*x^2*e^3 + 60*a*b^3*d^2*x*e^2 + 6*a*b^3*d^3*e + 945*a^2*b^2*x^2*e^4 + 210*a^2*b^2*d*x*e^3 + 21
*a^2*b^2*d^2*e^2 + 560*a^3*b*x*e^4 + 56*a^3*b*d*e^3 + 126*a^4*e^4)*e^(-5)/(x*e + d)^10