3.1502 \(\int \frac{(a^2+2 a b x+b^2 x^2)^3}{(d+e x)^{13}} \, dx\)

Optimal. Leaf size=173 \[ \frac{6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac{15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac{20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac{3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac{6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac{(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac{b^6}{6 e^7 (d+e x)^6} \]

[Out]

-(b*d - a*e)^6/(12*e^7*(d + e*x)^12) + (6*b*(b*d - a*e)^5)/(11*e^7*(d + e*x)^11) - (3*b^2*(b*d - a*e)^4)/(2*e^
7*(d + e*x)^10) + (20*b^3*(b*d - a*e)^3)/(9*e^7*(d + e*x)^9) - (15*b^4*(b*d - a*e)^2)/(8*e^7*(d + e*x)^8) + (6
*b^5*(b*d - a*e))/(7*e^7*(d + e*x)^7) - b^6/(6*e^7*(d + e*x)^6)

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Rubi [A]  time = 0.118325, antiderivative size = 173, 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} \[ \frac{6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac{15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac{20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac{3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac{6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac{(b d-a e)^6}{12 e^7 (d+e x)^{12}}-\frac{b^6}{6 e^7 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^13,x]

[Out]

-(b*d - a*e)^6/(12*e^7*(d + e*x)^12) + (6*b*(b*d - a*e)^5)/(11*e^7*(d + e*x)^11) - (3*b^2*(b*d - a*e)^4)/(2*e^
7*(d + e*x)^10) + (20*b^3*(b*d - a*e)^3)/(9*e^7*(d + e*x)^9) - (15*b^4*(b*d - a*e)^2)/(8*e^7*(d + e*x)^8) + (6
*b^5*(b*d - a*e))/(7*e^7*(d + e*x)^7) - b^6/(6*e^7*(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 )^3}{(d+e x)^{13}} \, dx &=\int \frac{(a+b x)^6}{(d+e x)^{13}} \, dx\\ &=\int \left (\frac{(-b d+a e)^6}{e^6 (d+e x)^{13}}-\frac{6 b (b d-a e)^5}{e^6 (d+e x)^{12}}+\frac{15 b^2 (b d-a e)^4}{e^6 (d+e x)^{11}}-\frac{20 b^3 (b d-a e)^3}{e^6 (d+e x)^{10}}+\frac{15 b^4 (b d-a e)^2}{e^6 (d+e x)^9}-\frac{6 b^5 (b d-a e)}{e^6 (d+e x)^8}+\frac{b^6}{e^6 (d+e x)^7}\right ) \, dx\\ &=-\frac{(b d-a e)^6}{12 e^7 (d+e x)^{12}}+\frac{6 b (b d-a e)^5}{11 e^7 (d+e x)^{11}}-\frac{3 b^2 (b d-a e)^4}{2 e^7 (d+e x)^{10}}+\frac{20 b^3 (b d-a e)^3}{9 e^7 (d+e x)^9}-\frac{15 b^4 (b d-a e)^2}{8 e^7 (d+e x)^8}+\frac{6 b^5 (b d-a e)}{7 e^7 (d+e x)^7}-\frac{b^6}{6 e^7 (d+e x)^6}\\ \end{align*}

Mathematica [A]  time = 0.0915019, size = 277, normalized size = 1.6 \[ -\frac{21 a^2 b^4 e^2 \left (66 d^2 e^2 x^2+12 d^3 e x+d^4+220 d e^3 x^3+495 e^4 x^4\right )+56 a^3 b^3 e^3 \left (12 d^2 e x+d^3+66 d e^2 x^2+220 e^3 x^3\right )+126 a^4 b^2 e^4 \left (d^2+12 d e x+66 e^2 x^2\right )+252 a^5 b e^5 (d+12 e x)+462 a^6 e^6+6 a b^5 e \left (66 d^3 e^2 x^2+220 d^2 e^3 x^3+12 d^4 e x+d^5+495 d e^4 x^4+792 e^5 x^5\right )+b^6 \left (66 d^4 e^2 x^2+220 d^3 e^3 x^3+495 d^2 e^4 x^4+12 d^5 e x+d^6+792 d e^5 x^5+924 e^6 x^6\right )}{5544 e^7 (d+e x)^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x + b^2*x^2)^3/(d + e*x)^13,x]

[Out]

-(462*a^6*e^6 + 252*a^5*b*e^5*(d + 12*e*x) + 126*a^4*b^2*e^4*(d^2 + 12*d*e*x + 66*e^2*x^2) + 56*a^3*b^3*e^3*(d
^3 + 12*d^2*e*x + 66*d*e^2*x^2 + 220*e^3*x^3) + 21*a^2*b^4*e^2*(d^4 + 12*d^3*e*x + 66*d^2*e^2*x^2 + 220*d*e^3*
x^3 + 495*e^4*x^4) + 6*a*b^5*e*(d^5 + 12*d^4*e*x + 66*d^3*e^2*x^2 + 220*d^2*e^3*x^3 + 495*d*e^4*x^4 + 792*e^5*
x^5) + b^6*(d^6 + 12*d^5*e*x + 66*d^4*e^2*x^2 + 220*d^3*e^3*x^3 + 495*d^2*e^4*x^4 + 792*d*e^5*x^5 + 924*e^6*x^
6))/(5544*e^7*(d + e*x)^12)

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Maple [B]  time = 0.047, size = 357, normalized size = 2.1 \begin{align*} -{\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 ) }{9\,{e}^{7} \left ( ex+d \right ) ^{9}}}-{\frac{{b}^{6}}{6\,{e}^{7} \left ( ex+d \right ) ^{6}}}-{\frac{3\,{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 ) }{2\,{e}^{7} \left ( ex+d \right ) ^{10}}}-{\frac{6\,{b}^{5} \left ( ae-bd \right ) }{7\,{e}^{7} \left ( ex+d \right ) ^{7}}}-{\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}}{12\,{e}^{7} \left ( ex+d \right ) ^{12}}}-{\frac{6\,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 ) }{11\,{e}^{7} \left ( ex+d \right ) ^{11}}}-{\frac{15\,{b}^{4} \left ({a}^{2}{e}^{2}-2\,abde+{b}^{2}{d}^{2} \right ) }{8\,{e}^{7} \left ( ex+d \right ) ^{8}}} \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)^13,x)

[Out]

-20/9*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)^9-1/6*b^6/e^7/(e*x+d)^6-3/2*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)^10-6/7*b^5*(a*e-b*d)/e^7/(e*x+d)^7-1/12*(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)
^12-6/11*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)^11-
15/8*b^4*(a^2*e^2-2*a*b*d*e+b^2*d^2)/e^7/(e*x+d)^8

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Maxima [B]  time = 1.27283, size = 640, normalized size = 3.7 \begin{align*} -\frac{924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \,{\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \,{\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \,{\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \,{\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \,{\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \,{\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} 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)^13,x, algorithm="maxima")

[Out]

-1/5544*(924*b^6*e^6*x^6 + b^6*d^6 + 6*a*b^5*d^5*e + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 126*a^4*b^2*d^2
*e^4 + 252*a^5*b*d*e^5 + 462*a^6*e^6 + 792*(b^6*d*e^5 + 6*a*b^5*e^6)*x^5 + 495*(b^6*d^2*e^4 + 6*a*b^5*d*e^5 +
21*a^2*b^4*e^6)*x^4 + 220*(b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 + 21*a^2*b^4*d*e^5 + 56*a^3*b^3*e^6)*x^3 + 66*(b^6*d^
4*e^2 + 6*a*b^5*d^3*e^3 + 21*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 126*a^4*b^2*e^6)*x^2 + 12*(b^6*d^5*e + 6*a*b
^5*d^4*e^2 + 21*a^2*b^4*d^3*e^3 + 56*a^3*b^3*d^2*e^4 + 126*a^4*b^2*d*e^5 + 252*a^5*b*e^6)*x)/(e^19*x^12 + 12*d
*e^18*x^11 + 66*d^2*e^17*x^10 + 220*d^3*e^16*x^9 + 495*d^4*e^15*x^8 + 792*d^5*e^14*x^7 + 924*d^6*e^13*x^6 + 79
2*d^7*e^12*x^5 + 495*d^8*e^11*x^4 + 220*d^9*e^10*x^3 + 66*d^10*e^9*x^2 + 12*d^11*e^8*x + d^12*e^7)

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Fricas [B]  time = 1.76447, size = 1026, normalized size = 5.93 \begin{align*} -\frac{924 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 6 \, a b^{5} d^{5} e + 21 \, a^{2} b^{4} d^{4} e^{2} + 56 \, a^{3} b^{3} d^{3} e^{3} + 126 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6} + 792 \,{\left (b^{6} d e^{5} + 6 \, a b^{5} e^{6}\right )} x^{5} + 495 \,{\left (b^{6} d^{2} e^{4} + 6 \, a b^{5} d e^{5} + 21 \, a^{2} b^{4} e^{6}\right )} x^{4} + 220 \,{\left (b^{6} d^{3} e^{3} + 6 \, a b^{5} d^{2} e^{4} + 21 \, a^{2} b^{4} d e^{5} + 56 \, a^{3} b^{3} e^{6}\right )} x^{3} + 66 \,{\left (b^{6} d^{4} e^{2} + 6 \, a b^{5} d^{3} e^{3} + 21 \, a^{2} b^{4} d^{2} e^{4} + 56 \, a^{3} b^{3} d e^{5} + 126 \, a^{4} b^{2} e^{6}\right )} x^{2} + 12 \,{\left (b^{6} d^{5} e + 6 \, a b^{5} d^{4} e^{2} + 21 \, a^{2} b^{4} d^{3} e^{3} + 56 \, a^{3} b^{3} d^{2} e^{4} + 126 \, a^{4} b^{2} d e^{5} + 252 \, a^{5} b e^{6}\right )} x}{5544 \,{\left (e^{19} x^{12} + 12 \, d e^{18} x^{11} + 66 \, d^{2} e^{17} x^{10} + 220 \, d^{3} e^{16} x^{9} + 495 \, d^{4} e^{15} x^{8} + 792 \, d^{5} e^{14} x^{7} + 924 \, d^{6} e^{13} x^{6} + 792 \, d^{7} e^{12} x^{5} + 495 \, d^{8} e^{11} x^{4} + 220 \, d^{9} e^{10} x^{3} + 66 \, d^{10} e^{9} x^{2} + 12 \, d^{11} e^{8} x + d^{12} 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)^13,x, algorithm="fricas")

[Out]

-1/5544*(924*b^6*e^6*x^6 + b^6*d^6 + 6*a*b^5*d^5*e + 21*a^2*b^4*d^4*e^2 + 56*a^3*b^3*d^3*e^3 + 126*a^4*b^2*d^2
*e^4 + 252*a^5*b*d*e^5 + 462*a^6*e^6 + 792*(b^6*d*e^5 + 6*a*b^5*e^6)*x^5 + 495*(b^6*d^2*e^4 + 6*a*b^5*d*e^5 +
21*a^2*b^4*e^6)*x^4 + 220*(b^6*d^3*e^3 + 6*a*b^5*d^2*e^4 + 21*a^2*b^4*d*e^5 + 56*a^3*b^3*e^6)*x^3 + 66*(b^6*d^
4*e^2 + 6*a*b^5*d^3*e^3 + 21*a^2*b^4*d^2*e^4 + 56*a^3*b^3*d*e^5 + 126*a^4*b^2*e^6)*x^2 + 12*(b^6*d^5*e + 6*a*b
^5*d^4*e^2 + 21*a^2*b^4*d^3*e^3 + 56*a^3*b^3*d^2*e^4 + 126*a^4*b^2*d*e^5 + 252*a^5*b*e^6)*x)/(e^19*x^12 + 12*d
*e^18*x^11 + 66*d^2*e^17*x^10 + 220*d^3*e^16*x^9 + 495*d^4*e^15*x^8 + 792*d^5*e^14*x^7 + 924*d^6*e^13*x^6 + 79
2*d^7*e^12*x^5 + 495*d^8*e^11*x^4 + 220*d^9*e^10*x^3 + 66*d^10*e^9*x^2 + 12*d^11*e^8*x + d^12*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)**13,x)

[Out]

Timed out

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Giac [B]  time = 1.17238, size = 475, normalized size = 2.75 \begin{align*} -\frac{{\left (924 \, b^{6} x^{6} e^{6} + 792 \, b^{6} d x^{5} e^{5} + 495 \, b^{6} d^{2} x^{4} e^{4} + 220 \, b^{6} d^{3} x^{3} e^{3} + 66 \, b^{6} d^{4} x^{2} e^{2} + 12 \, b^{6} d^{5} x e + b^{6} d^{6} + 4752 \, a b^{5} x^{5} e^{6} + 2970 \, a b^{5} d x^{4} e^{5} + 1320 \, a b^{5} d^{2} x^{3} e^{4} + 396 \, a b^{5} d^{3} x^{2} e^{3} + 72 \, a b^{5} d^{4} x e^{2} + 6 \, a b^{5} d^{5} e + 10395 \, a^{2} b^{4} x^{4} e^{6} + 4620 \, a^{2} b^{4} d x^{3} e^{5} + 1386 \, a^{2} b^{4} d^{2} x^{2} e^{4} + 252 \, a^{2} b^{4} d^{3} x e^{3} + 21 \, a^{2} b^{4} d^{4} e^{2} + 12320 \, a^{3} b^{3} x^{3} e^{6} + 3696 \, a^{3} b^{3} d x^{2} e^{5} + 672 \, a^{3} b^{3} d^{2} x e^{4} + 56 \, a^{3} b^{3} d^{3} e^{3} + 8316 \, a^{4} b^{2} x^{2} e^{6} + 1512 \, a^{4} b^{2} d x e^{5} + 126 \, a^{4} b^{2} d^{2} e^{4} + 3024 \, a^{5} b x e^{6} + 252 \, a^{5} b d e^{5} + 462 \, a^{6} e^{6}\right )} e^{\left (-7\right )}}{5544 \,{\left (x e + d\right )}^{12}} \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)^13,x, algorithm="giac")

[Out]

-1/5544*(924*b^6*x^6*e^6 + 792*b^6*d*x^5*e^5 + 495*b^6*d^2*x^4*e^4 + 220*b^6*d^3*x^3*e^3 + 66*b^6*d^4*x^2*e^2
+ 12*b^6*d^5*x*e + b^6*d^6 + 4752*a*b^5*x^5*e^6 + 2970*a*b^5*d*x^4*e^5 + 1320*a*b^5*d^2*x^3*e^4 + 396*a*b^5*d^
3*x^2*e^3 + 72*a*b^5*d^4*x*e^2 + 6*a*b^5*d^5*e + 10395*a^2*b^4*x^4*e^6 + 4620*a^2*b^4*d*x^3*e^5 + 1386*a^2*b^4
*d^2*x^2*e^4 + 252*a^2*b^4*d^3*x*e^3 + 21*a^2*b^4*d^4*e^2 + 12320*a^3*b^3*x^3*e^6 + 3696*a^3*b^3*d*x^2*e^5 + 6
72*a^3*b^3*d^2*x*e^4 + 56*a^3*b^3*d^3*e^3 + 8316*a^4*b^2*x^2*e^6 + 1512*a^4*b^2*d*x*e^5 + 126*a^4*b^2*d^2*e^4
+ 3024*a^5*b*x*e^6 + 252*a^5*b*d*e^5 + 462*a^6*e^6)*e^(-7)/(x*e + d)^12