\(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{11}} \, dx\) [1480]

Optimal result

Integrand size = 26, antiderivative size = 119 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {b^4}{6 e^5 (d+e x)^6} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {27, 45} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=\frac {4 b^3 (b d-a e)}{7 e^5 (d+e x)^7}-\frac {3 b^2 (b d-a e)^2}{4 e^5 (d+e x)^8}+\frac {4 b (b d-a e)^3}{9 e^5 (d+e x)^9}-\frac {(b d-a e)^4}{10 e^5 (d+e x)^{10}}-\frac {b^4}{6 e^5 (d+e x)^6} \]

[In]

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

[Out]

-1/10*(b*d - a*e)^4/(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 45

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*} \text {integral}& = \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] (verified)

Time = 0.03 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {126 a^4 e^4+56 a^3 b e^3 (d+10 e x)+21 a^2 b^2 e^2 \left (d^2+10 d e x+45 e^2 x^2\right )+6 a b^3 e \left (d^3+10 d^2 e x+45 d e^2 x^2+120 e^3 x^3\right )+b^4 \left (d^4+10 d^3 e x+45 d^2 e^2 x^2+120 d e^3 x^3+210 e^4 x^4\right )}{1260 e^5 (d+e x)^{10}} \]

[In]

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

[Out]

-1/1260*(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))/(e^5*(d + e*x)^10)

Maple [A] (verified)

Time = 2.48 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.44

method	result	size
risch	\(\frac {-\frac {b^{4} x^{4}}{6 e}-\frac {2 b^{3} \left (6 a e +b d \right ) x^{3}}{21 e^{2}}-\frac {b^{2} \left (21 a^{2} e^{2}+6 a b d e +b^{2} d^{2}\right ) x^{2}}{28 e^{3}}-\frac {b \left (56 a^{3} e^{3}+21 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{126 e^{4}}-\frac {126 e^{4} a^{4}+56 b \,e^{3} d \,a^{3}+21 b^{2} e^{2} d^{2} a^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}}{1260 e^{5}}}{\left (e x +d \right )^{10}}\)	\(171\)
gosper	\(-\frac {210 b^{4} x^{4} e^{4}+720 x^{3} a \,b^{3} e^{4}+120 x^{3} b^{4} d \,e^{3}+945 x^{2} a^{2} b^{2} e^{4}+270 x^{2} a \,b^{3} d \,e^{3}+45 x^{2} b^{4} d^{2} e^{2}+560 x \,a^{3} b \,e^{4}+210 x \,a^{2} b^{2} d \,e^{3}+60 x a \,b^{3} d^{2} e^{2}+10 x \,b^{4} d^{3} e +126 e^{4} a^{4}+56 b \,e^{3} d \,a^{3}+21 b^{2} e^{2} d^{2} a^{2}+6 a \,b^{3} d^{3} e +b^{4} d^{4}}{1260 e^{5} \left (e x +d \right )^{10}}\)	\(185\)
default	\(-\frac {3 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{4 e^{5} \left (e x +d \right )^{8}}-\frac {e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{10 e^{5} \left (e x +d \right )^{10}}-\frac {4 b^{3} \left (a e -b d \right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {b^{4}}{6 e^{5} \left (e x +d \right )^{6}}-\frac {4 b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{9 e^{5} \left (e x +d \right )^{9}}\)	\(186\)
parallelrisch	\(\frac {-210 b^{4} x^{4} e^{9}-720 a \,b^{3} e^{9} x^{3}-120 b^{4} d \,e^{8} x^{3}-945 a^{2} b^{2} e^{9} x^{2}-270 a \,b^{3} d \,e^{8} x^{2}-45 b^{4} d^{2} e^{7} x^{2}-560 a^{3} b \,e^{9} x -210 a^{2} b^{2} d \,e^{8} x -60 a \,b^{3} d^{2} e^{7} x -10 b^{4} d^{3} e^{6} x -126 a^{4} e^{9}-56 a^{3} b d \,e^{8}-21 a^{2} b^{2} d^{2} e^{7}-6 a \,b^{3} d^{3} e^{6}-b^{4} d^{4} e^{5}}{1260 e^{10} \left (e x +d \right )^{10}}\)	\(193\)
norman	\(\frac {-\frac {b^{4} x^{4}}{6 e}-\frac {2 \left (6 a \,b^{3} e^{6}+b^{4} d \,e^{5}\right ) x^{3}}{21 e^{7}}-\frac {\left (21 a^{2} b^{2} e^{7}+6 a \,b^{3} d \,e^{6}+b^{4} d^{2} e^{5}\right ) x^{2}}{28 e^{8}}-\frac {\left (56 a^{3} b \,e^{8}+21 a^{2} b^{2} d \,e^{7}+6 a \,b^{3} d^{2} e^{6}+b^{4} d^{3} e^{5}\right ) x}{126 e^{9}}-\frac {126 a^{4} e^{9}+56 a^{3} b d \,e^{8}+21 a^{2} b^{2} d^{2} e^{7}+6 a \,b^{3} d^{3} e^{6}+b^{4} d^{4} e^{5}}{1260 e^{10}}}{\left (e x +d \right )^{10}}\)	\(197\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^11,x,method=_RETURNVERBOSE)

[Out]

(-1/6/e*b^4*x^4-2/21*b^3/e^2*(6*a*e+b*d)*x^3-1/28*b^2/e^3*(21*a^2*e^2+6*a*b*d*e+b^2*d^2)*x^2-1/126*b/e^4*(56*a
^3*e^3+21*a^2*b*d*e^2+6*a*b^2*d^2*e+b^3*d^3)*x-1/1260/e^5*(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))/(e*x+d)^10

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (109) = 218\).

Time = 0.32 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \, {\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \, {\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \, {\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=\text {Timed out} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2)**2/(e*x+d)**11,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 280 vs. \(2 (109) = 218\).

Time = 0.21 (sec) , antiderivative size = 280, normalized size of antiderivative = 2.35 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4} + 120 \, {\left (b^{4} d e^{3} + 6 \, a b^{3} e^{4}\right )} x^{3} + 45 \, {\left (b^{4} d^{2} e^{2} + 6 \, a b^{3} d e^{3} + 21 \, a^{2} b^{2} e^{4}\right )} x^{2} + 10 \, {\left (b^{4} d^{3} e + 6 \, a b^{3} d^{2} e^{2} + 21 \, a^{2} b^{2} d e^{3} + 56 \, a^{3} b e^{4}\right )} x}{1260 \, {\left (e^{15} x^{10} + 10 \, d e^{14} x^{9} + 45 \, d^{2} e^{13} x^{8} + 120 \, d^{3} e^{12} x^{7} + 210 \, d^{4} e^{11} x^{6} + 252 \, d^{5} e^{10} x^{5} + 210 \, d^{6} e^{9} x^{4} + 120 \, d^{7} e^{8} x^{3} + 45 \, d^{8} e^{7} x^{2} + 10 \, d^{9} e^{6} x + d^{10} e^{5}\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {210 \, b^{4} e^{4} x^{4} + 120 \, b^{4} d e^{3} x^{3} + 720 \, a b^{3} e^{4} x^{3} + 45 \, b^{4} d^{2} e^{2} x^{2} + 270 \, a b^{3} d e^{3} x^{2} + 945 \, a^{2} b^{2} e^{4} x^{2} + 10 \, b^{4} d^{3} e x + 60 \, a b^{3} d^{2} e^{2} x + 210 \, a^{2} b^{2} d e^{3} x + 560 \, a^{3} b e^{4} x + b^{4} d^{4} + 6 \, a b^{3} d^{3} e + 21 \, a^{2} b^{2} d^{2} e^{2} + 56 \, a^{3} b d e^{3} + 126 \, a^{4} e^{4}}{1260 \, {\left (e x + d\right )}^{10} e^{5}} \]

[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*e^4*x^4 + 120*b^4*d*e^3*x^3 + 720*a*b^3*e^4*x^3 + 45*b^4*d^2*e^2*x^2 + 270*a*b^3*d*e^3*x^2 +
945*a^2*b^2*e^4*x^2 + 10*b^4*d^3*e*x + 60*a*b^3*d^2*e^2*x + 210*a^2*b^2*d*e^3*x + 560*a^3*b*e^4*x + 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)/((e*x + d)^10*e^5)

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.27 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{11}} \, dx=-\frac {\frac {126\,a^4\,e^4+56\,a^3\,b\,d\,e^3+21\,a^2\,b^2\,d^2\,e^2+6\,a\,b^3\,d^3\,e+b^4\,d^4}{1260\,e^5}+\frac {b^4\,x^4}{6\,e}+\frac {2\,b^3\,x^3\,\left (6\,a\,e+b\,d\right )}{21\,e^2}+\frac {b\,x\,\left (56\,a^3\,e^3+21\,a^2\,b\,d\,e^2+6\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{126\,e^4}+\frac {b^2\,x^2\,\left (21\,a^2\,e^2+6\,a\,b\,d\,e+b^2\,d^2\right )}{28\,e^3}}{d^{10}+10\,d^9\,e\,x+45\,d^8\,e^2\,x^2+120\,d^7\,e^3\,x^3+210\,d^6\,e^4\,x^4+252\,d^5\,e^5\,x^5+210\,d^4\,e^6\,x^6+120\,d^3\,e^7\,x^7+45\,d^2\,e^8\,x^8+10\,d\,e^9\,x^9+e^{10}\,x^{10}} \]

[In]

int((a^2 + b^2*x^2 + 2*a*b*x)^2/(d + e*x)^11,x)

[Out]

-((126*a^4*e^4 + b^4*d^4 + 21*a^2*b^2*d^2*e^2 + 6*a*b^3*d^3*e + 56*a^3*b*d*e^3)/(1260*e^5) + (b^4*x^4)/(6*e) +
(2*b^3*x^3*(6*a*e + b*d))/(21*e^2) + (b*x*(56*a^3*e^3 + b^3*d^3 + 6*a*b^2*d^2*e + 21*a^2*b*d*e^2))/(126*e^4)
+ (b^2*x^2*(21*a^2*e^2 + b^2*d^2 + 6*a*b*d*e))/(28*e^3))/(d^10 + e^10*x^10 + 10*d*e^9*x^9 + 45*d^8*e^2*x^2 + 1
20*d^7*e^3*x^3 + 210*d^6*e^4*x^4 + 252*d^5*e^5*x^5 + 210*d^4*e^6*x^6 + 120*d^3*e^7*x^7 + 45*d^2*e^8*x^8 + 10*d
^9*e*x)