3.1500 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{11}} \, dx\)

Optimal. Leaf size=120 \[ \frac{b^3 (a+b x)^7}{840 (d+e x)^7 (b d-a e)^4}+\frac{b^2 (a+b x)^7}{120 (d+e x)^8 (b d-a e)^3}+\frac{b (a+b x)^7}{30 (d+e x)^9 (b d-a e)^2}+\frac{(a+b x)^7}{10 (d+e x)^{10} (b d-a e)} \]

[Out]

(a + b*x)^7/(10*(b*d - a*e)*(d + e*x)^10) + (b*(a + b*x)^7)/(30*(b*d - a*e)^2*(d + e*x)^9) + (b^2*(a + b*x)^7)
/(120*(b*d - a*e)^3*(d + e*x)^8) + (b^3*(a + b*x)^7)/(840*(b*d - a*e)^4*(d + e*x)^7)

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Rubi [A]  time = 0.0350822, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {27, 45, 37} \[ \frac{b^3 (a+b x)^7}{840 (d+e x)^7 (b d-a e)^4}+\frac{b^2 (a+b x)^7}{120 (d+e x)^8 (b d-a e)^3}+\frac{b (a+b x)^7}{30 (d+e x)^9 (b d-a e)^2}+\frac{(a+b x)^7}{10 (d+e x)^{10} (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^11,x]

[Out]

(a + b*x)^7/(10*(b*d - a*e)*(d + e*x)^10) + (b*(a + b*x)^7)/(30*(b*d - a*e)^2*(d + e*x)^9) + (b^2*(a + b*x)^7)
/(120*(b*d - a*e)^3*(d + e*x)^8) + (b^3*(a + b*x)^7)/(840*(b*d - a*e)^4*(d + e*x)^7)

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 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^3}{(d+e x)^{11}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{11}} \, dx\\ &=\frac{(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac{(3 b) \int \frac{(a+b x)^6}{(d+e x)^{10}} \, dx}{10 (b d-a e)}\\ &=\frac{(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac{b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac{b^2 \int \frac{(a+b x)^6}{(d+e x)^9} \, dx}{15 (b d-a e)^2}\\ &=\frac{(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac{b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac{b^2 (a+b x)^7}{120 (b d-a e)^3 (d+e x)^8}+\frac{b^3 \int \frac{(a+b x)^6}{(d+e x)^8} \, dx}{120 (b d-a e)^3}\\ &=\frac{(a+b x)^7}{10 (b d-a e) (d+e x)^{10}}+\frac{b (a+b x)^7}{30 (b d-a e)^2 (d+e x)^9}+\frac{b^2 (a+b x)^7}{120 (b d-a e)^3 (d+e x)^8}+\frac{b^3 (a+b x)^7}{840 (b d-a e)^4 (d+e x)^7}\\ \end{align*}

Mathematica [B]  time = 0.0952054, size = 277, normalized size = 2.31 \[ -\frac{10 a^2 b^4 e^2 \left (45 d^2 e^2 x^2+10 d^3 e x+d^4+120 d e^3 x^3+210 e^4 x^4\right )+20 a^3 b^3 e^3 \left (10 d^2 e x+d^3+45 d e^2 x^2+120 e^3 x^3\right )+35 a^4 b^2 e^4 \left (d^2+10 d e x+45 e^2 x^2\right )+56 a^5 b e^5 (d+10 e x)+84 a^6 e^6+4 a b^5 e \left (45 d^3 e^2 x^2+120 d^2 e^3 x^3+10 d^4 e x+d^5+210 d e^4 x^4+252 e^5 x^5\right )+b^6 \left (45 d^4 e^2 x^2+120 d^3 e^3 x^3+210 d^2 e^4 x^4+10 d^5 e x+d^6+252 d e^5 x^5+210 e^6 x^6\right )}{840 e^7 (d+e x)^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^11,x]

[Out]

-(84*a^6*e^6 + 56*a^5*b*e^5*(d + 10*e*x) + 35*a^4*b^2*e^4*(d^2 + 10*d*e*x + 45*e^2*x^2) + 20*a^3*b^3*e^3*(d^3
+ 10*d^2*e*x + 45*d*e^2*x^2 + 120*e^3*x^3) + 10*a^2*b^4*e^2*(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) + 4*a*b^5*e*(d^5 + 10*d^4*e*x + 45*d^3*e^2*x^2 + 120*d^2*e^3*x^3 + 210*d*e^4*x^4 + 252*e^5*x^5
) + b^6*(d^6 + 10*d^5*e*x + 45*d^4*e^2*x^2 + 120*d^3*e^3*x^3 + 210*d^2*e^4*x^4 + 252*d*e^5*x^5 + 210*e^6*x^6))
/(840*e^7*(d + e*x)^10)

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Maple [B]  time = 0.048, size = 357, normalized size = 3. \begin{align*} -{\frac{5\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{2\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{{b}^{6}}{4\,{e}^{7} \left ( ex+d \right ) ^{4}}}-{\frac{{e}^{6}{a}^{6}-6\,{a}^{5}bd{e}^{5}+15\,{d}^{2}{e}^{4}{a}^{4}{b}^{2}-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+{d}^{6}{b}^{6}}{10\,{e}^{7} \left ( ex+d \right ) ^{10}}}-{\frac{20\,{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 ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\frac{2\,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 ) }{3\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{15\,{b}^{2} \left ({a}^{4}{e}^{4}-4\,{a}^{3}bd{e}^{3}+6\,{d}^{2}{e}^{2}{b}^{2}{a}^{2}-4\,a{b}^{3}{d}^{3}e+{b}^{4}{d}^{4} \right ) }{8\,{e}^{7} \left ( ex+d \right ) ^{8}}}-{\frac{6\,{b}^{5} \left ( ae-bd \right ) }{5\,{e}^{7} \left ( ex+d \right ) ^{5}}} \end{align*}

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)^11,x)

[Out]

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

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Maxima [B]  time = 1.30075, size = 610, normalized size = 5.08 \begin{align*} -\frac{210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \,{\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \,{\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \,{\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \,{\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \,{\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\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)^3/(e*x+d)^11,x, algorithm="maxima")

[Out]

-1/840*(210*b^6*e^6*x^6 + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e
^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6 + 252*(b^6*d*e^5 + 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a
^2*b^4*e^6)*x^4 + 120*(b^6*d^3*e^3 + 4*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 + 20*a^3*b^3*e^6)*x^3 + 45*(b^6*d^4*e^
2 + 4*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 + 20*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 10*(b^6*d^5*e + 4*a*b^5*d^
4*e^2 + 10*a^2*b^4*d^3*e^3 + 20*a^3*b^3*d^2*e^4 + 35*a^4*b^2*d*e^5 + 56*a^5*b*e^6)*x)/(e^17*x^10 + 10*d*e^16*x
^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7*e^1
0*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*e^7)

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Fricas [B]  time = 1.74229, size = 960, normalized size = 8. \begin{align*} -\frac{210 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 4 \, a b^{5} d^{5} e + 10 \, a^{2} b^{4} d^{4} e^{2} + 20 \, a^{3} b^{3} d^{3} e^{3} + 35 \, a^{4} b^{2} d^{2} e^{4} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6} + 252 \,{\left (b^{6} d e^{5} + 4 \, a b^{5} e^{6}\right )} x^{5} + 210 \,{\left (b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 10 \, a^{2} b^{4} e^{6}\right )} x^{4} + 120 \,{\left (b^{6} d^{3} e^{3} + 4 \, a b^{5} d^{2} e^{4} + 10 \, a^{2} b^{4} d e^{5} + 20 \, a^{3} b^{3} e^{6}\right )} x^{3} + 45 \,{\left (b^{6} d^{4} e^{2} + 4 \, a b^{5} d^{3} e^{3} + 10 \, a^{2} b^{4} d^{2} e^{4} + 20 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 10 \,{\left (b^{6} d^{5} e + 4 \, a b^{5} d^{4} e^{2} + 10 \, a^{2} b^{4} d^{3} e^{3} + 20 \, a^{3} b^{3} d^{2} e^{4} + 35 \, a^{4} b^{2} d e^{5} + 56 \, a^{5} b e^{6}\right )} x}{840 \,{\left (e^{17} x^{10} + 10 \, d e^{16} x^{9} + 45 \, d^{2} e^{15} x^{8} + 120 \, d^{3} e^{14} x^{7} + 210 \, d^{4} e^{13} x^{6} + 252 \, d^{5} e^{12} x^{5} + 210 \, d^{6} e^{11} x^{4} + 120 \, d^{7} e^{10} x^{3} + 45 \, d^{8} e^{9} x^{2} + 10 \, d^{9} e^{8} x + d^{10} e^{7}\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)^3/(e*x+d)^11,x, algorithm="fricas")

[Out]

-1/840*(210*b^6*e^6*x^6 + b^6*d^6 + 4*a*b^5*d^5*e + 10*a^2*b^4*d^4*e^2 + 20*a^3*b^3*d^3*e^3 + 35*a^4*b^2*d^2*e
^4 + 56*a^5*b*d*e^5 + 84*a^6*e^6 + 252*(b^6*d*e^5 + 4*a*b^5*e^6)*x^5 + 210*(b^6*d^2*e^4 + 4*a*b^5*d*e^5 + 10*a
^2*b^4*e^6)*x^4 + 120*(b^6*d^3*e^3 + 4*a*b^5*d^2*e^4 + 10*a^2*b^4*d*e^5 + 20*a^3*b^3*e^6)*x^3 + 45*(b^6*d^4*e^
2 + 4*a*b^5*d^3*e^3 + 10*a^2*b^4*d^2*e^4 + 20*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 10*(b^6*d^5*e + 4*a*b^5*d^
4*e^2 + 10*a^2*b^4*d^3*e^3 + 20*a^3*b^3*d^2*e^4 + 35*a^4*b^2*d*e^5 + 56*a^5*b*e^6)*x)/(e^17*x^10 + 10*d*e^16*x
^9 + 45*d^2*e^15*x^8 + 120*d^3*e^14*x^7 + 210*d^4*e^13*x^6 + 252*d^5*e^12*x^5 + 210*d^6*e^11*x^4 + 120*d^7*e^1
0*x^3 + 45*d^8*e^9*x^2 + 10*d^9*e^8*x + d^10*e^7)

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

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)**11,x)

[Out]

Timed out

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Giac [B]  time = 1.14104, size = 475, normalized size = 3.96 \begin{align*} -\frac{{\left (210 \, b^{6} x^{6} e^{6} + 252 \, b^{6} d x^{5} e^{5} + 210 \, b^{6} d^{2} x^{4} e^{4} + 120 \, b^{6} d^{3} x^{3} e^{3} + 45 \, b^{6} d^{4} x^{2} e^{2} + 10 \, b^{6} d^{5} x e + b^{6} d^{6} + 1008 \, a b^{5} x^{5} e^{6} + 840 \, a b^{5} d x^{4} e^{5} + 480 \, a b^{5} d^{2} x^{3} e^{4} + 180 \, a b^{5} d^{3} x^{2} e^{3} + 40 \, a b^{5} d^{4} x e^{2} + 4 \, a b^{5} d^{5} e + 2100 \, a^{2} b^{4} x^{4} e^{6} + 1200 \, a^{2} b^{4} d x^{3} e^{5} + 450 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 100 \, a^{2} b^{4} d^{3} x e^{3} + 10 \, a^{2} b^{4} d^{4} e^{2} + 2400 \, a^{3} b^{3} x^{3} e^{6} + 900 \, a^{3} b^{3} d x^{2} e^{5} + 200 \, a^{3} b^{3} d^{2} x e^{4} + 20 \, a^{3} b^{3} d^{3} e^{3} + 1575 \, a^{4} b^{2} x^{2} e^{6} + 350 \, a^{4} b^{2} d x e^{5} + 35 \, a^{4} b^{2} d^{2} e^{4} + 560 \, a^{5} b x e^{6} + 56 \, a^{5} b d e^{5} + 84 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{840 \,{\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)^3/(e*x+d)^11,x, algorithm="giac")

[Out]

-1/840*(210*b^6*x^6*e^6 + 252*b^6*d*x^5*e^5 + 210*b^6*d^2*x^4*e^4 + 120*b^6*d^3*x^3*e^3 + 45*b^6*d^4*x^2*e^2 +
10*b^6*d^5*x*e + b^6*d^6 + 1008*a*b^5*x^5*e^6 + 840*a*b^5*d*x^4*e^5 + 480*a*b^5*d^2*x^3*e^4 + 180*a*b^5*d^3*x
^2*e^3 + 40*a*b^5*d^4*x*e^2 + 4*a*b^5*d^5*e + 2100*a^2*b^4*x^4*e^6 + 1200*a^2*b^4*d*x^3*e^5 + 450*a^2*b^4*d^2*
x^2*e^4 + 100*a^2*b^4*d^3*x*e^3 + 10*a^2*b^4*d^4*e^2 + 2400*a^3*b^3*x^3*e^6 + 900*a^3*b^3*d*x^2*e^5 + 200*a^3*
b^3*d^2*x*e^4 + 20*a^3*b^3*d^3*e^3 + 1575*a^4*b^2*x^2*e^6 + 350*a^4*b^2*d*x*e^5 + 35*a^4*b^2*d^2*e^4 + 560*a^5
*b*x*e^6 + 56*a^5*b*d*e^5 + 84*a^6*e^6)*e^(-7)/(x*e + d)^10