\(\int \frac {(a^2+2 a b x+b^2 x^2)^2}{(d+e x)^{10}} \, dx\) [1479]

Optimal result

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

[Out]

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

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

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {70 a^4 e^4+35 a^3 b e^3 (d+9 e x)+15 a^2 b^2 e^2 \left (d^2+9 d e x+36 e^2 x^2\right )+5 a b^3 e \left (d^3+9 d^2 e x+36 d e^2 x^2+84 e^3 x^3\right )+b^4 \left (d^4+9 d^3 e x+36 d^2 e^2 x^2+84 d e^3 x^3+126 e^4 x^4\right )}{630 e^5 (d+e x)^9} \]

[In]

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

[Out]

-1/630*(70*a^4*e^4 + 35*a^3*b*e^3*(d + 9*e*x) + 15*a^2*b^2*e^2*(d^2 + 9*d*e*x + 36*e^2*x^2) + 5*a*b^3*e*(d^3 +
9*d^2*e*x + 36*d*e^2*x^2 + 84*e^3*x^3) + b^4*(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))
/(e^5*(d + e*x)^9)

Maple [A] (verified)

Time = 2.56 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.44

method	result	size
risch	\(\frac {-\frac {b^{4} x^{4}}{5 e}-\frac {2 b^{3} \left (5 a e +b d \right ) x^{3}}{15 e^{2}}-\frac {2 b^{2} \left (15 a^{2} e^{2}+5 a b d e +b^{2} d^{2}\right ) x^{2}}{35 e^{3}}-\frac {b \left (35 a^{3} e^{3}+15 a^{2} b d \,e^{2}+5 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x}{70 e^{4}}-\frac {70 e^{4} a^{4}+35 b \,e^{3} d \,a^{3}+15 b^{2} e^{2} d^{2} a^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}}{630 e^{5}}}{\left (e x +d \right )^{9}}\)	\(171\)
gosper	\(-\frac {126 b^{4} x^{4} e^{4}+420 x^{3} a \,b^{3} e^{4}+84 x^{3} b^{4} d \,e^{3}+540 x^{2} a^{2} b^{2} e^{4}+180 x^{2} a \,b^{3} d \,e^{3}+36 x^{2} b^{4} d^{2} e^{2}+315 x \,a^{3} b \,e^{4}+135 x \,a^{2} b^{2} d \,e^{3}+45 x a \,b^{3} d^{2} e^{2}+9 x \,b^{4} d^{3} e +70 e^{4} a^{4}+35 b \,e^{3} d \,a^{3}+15 b^{2} e^{2} d^{2} a^{2}+5 a \,b^{3} d^{3} e +b^{4} d^{4}}{630 e^{5} \left (e x +d \right )^{9}}\)	\(185\)
default	\(-\frac {b^{4}}{5 e^{5} \left (e x +d \right )^{5}}-\frac {b \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{2 e^{5} \left (e x +d \right )^{8}}-\frac {6 b^{2} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}{7 e^{5} \left (e x +d \right )^{7}}-\frac {2 b^{3} \left (a e -b d \right )}{3 e^{5} \left (e x +d \right )^{6}}-\frac {e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}}{9 e^{5} \left (e x +d \right )^{9}}\)	\(186\)
parallelrisch	\(\frac {-126 b^{4} x^{4} e^{8}-420 a \,b^{3} e^{8} x^{3}-84 b^{4} d \,e^{7} x^{3}-540 a^{2} b^{2} e^{8} x^{2}-180 a \,b^{3} d \,e^{7} x^{2}-36 b^{4} d^{2} e^{6} x^{2}-315 a^{3} b \,e^{8} x -135 a^{2} b^{2} d \,e^{7} x -45 a \,b^{3} d^{2} e^{6} x -9 b^{4} d^{3} e^{5} x -70 a^{4} e^{8}-35 a^{3} b d \,e^{7}-15 a^{2} b^{2} d^{2} e^{6}-5 a \,b^{3} d^{3} e^{5}-b^{4} d^{4} e^{4}}{630 e^{9} \left (e x +d \right )^{9}}\)	\(193\)
norman	\(\frac {-\frac {b^{4} x^{4}}{5 e}-\frac {2 \left (5 a \,b^{3} e^{5}+b^{4} d \,e^{4}\right ) x^{3}}{15 e^{6}}-\frac {2 \left (15 a^{2} b^{2} e^{6}+5 a \,b^{3} d \,e^{5}+b^{4} d^{2} e^{4}\right ) x^{2}}{35 e^{7}}-\frac {\left (35 a^{3} b \,e^{7}+15 a^{2} b^{2} d \,e^{6}+5 a \,b^{3} d^{2} e^{5}+b^{4} d^{3} e^{4}\right ) x}{70 e^{8}}-\frac {70 a^{4} e^{8}+35 a^{3} b d \,e^{7}+15 a^{2} b^{2} d^{2} e^{6}+5 a \,b^{3} d^{3} e^{5}+b^{4} d^{4} e^{4}}{630 e^{9}}}{\left (e x +d \right )^{9}}\)	\(197\)

[In]

int((b^2*x^2+2*a*b*x+a^2)^2/(e*x+d)^10,x,method=_RETURNVERBOSE)

[Out]

(-1/5/e*b^4*x^4-2/15*b^3/e^2*(5*a*e+b*d)*x^3-2/35*b^2/e^3*(15*a^2*e^2+5*a*b*d*e+b^2*d^2)*x^2-1/70*b/e^4*(35*a^
3*e^3+15*a^2*b*d*e^2+5*a*b^2*d^2*e+b^3*d^3)*x-1/630/e^5*(70*a^4*e^4+35*a^3*b*d*e^3+15*a^2*b^2*d^2*e^2+5*a*b^3*
d^3*e+b^4*d^4))/(e*x+d)^9

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (109) = 218\).

Time = 0.30 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \, {\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \, {\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \, {\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \]

[In]

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

[Out]

-1/630*(126*b^4*e^4*x^4 + b^4*d^4 + 5*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 35*a^3*b*d*e^3 + 70*a^4*e^4 + 84*(b^4
*d*e^3 + 5*a*b^3*e^4)*x^3 + 36*(b^4*d^2*e^2 + 5*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 9*(b^4*d^3*e + 5*a*b^3*d^2
*e^2 + 15*a^2*b^2*d*e^3 + 35*a^3*b*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e^12*x^7 + 84*d^3*e^11*x^6 + 126*
d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x + d^9*e^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (109) = 218\).

Time = 0.21 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.26 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {126 \, b^{4} e^{4} x^{4} + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4} + 84 \, {\left (b^{4} d e^{3} + 5 \, a b^{3} e^{4}\right )} x^{3} + 36 \, {\left (b^{4} d^{2} e^{2} + 5 \, a b^{3} d e^{3} + 15 \, a^{2} b^{2} e^{4}\right )} x^{2} + 9 \, {\left (b^{4} d^{3} e + 5 \, a b^{3} d^{2} e^{2} + 15 \, a^{2} b^{2} d e^{3} + 35 \, a^{3} b e^{4}\right )} x}{630 \, {\left (e^{14} x^{9} + 9 \, d e^{13} x^{8} + 36 \, d^{2} e^{12} x^{7} + 84 \, d^{3} e^{11} x^{6} + 126 \, d^{4} e^{10} x^{5} + 126 \, d^{5} e^{9} x^{4} + 84 \, d^{6} e^{8} x^{3} + 36 \, d^{7} e^{7} x^{2} + 9 \, d^{8} e^{6} x + d^{9} e^{5}\right )}} \]

[In]

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

[Out]

-1/630*(126*b^4*e^4*x^4 + b^4*d^4 + 5*a*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 35*a^3*b*d*e^3 + 70*a^4*e^4 + 84*(b^4
*d*e^3 + 5*a*b^3*e^4)*x^3 + 36*(b^4*d^2*e^2 + 5*a*b^3*d*e^3 + 15*a^2*b^2*e^4)*x^2 + 9*(b^4*d^3*e + 5*a*b^3*d^2
*e^2 + 15*a^2*b^2*d*e^3 + 35*a^3*b*e^4)*x)/(e^14*x^9 + 9*d*e^13*x^8 + 36*d^2*e^12*x^7 + 84*d^3*e^11*x^6 + 126*
d^4*e^10*x^5 + 126*d^5*e^9*x^4 + 84*d^6*e^8*x^3 + 36*d^7*e^7*x^2 + 9*d^8*e^6*x + d^9*e^5)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.55 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {126 \, b^{4} e^{4} x^{4} + 84 \, b^{4} d e^{3} x^{3} + 420 \, a b^{3} e^{4} x^{3} + 36 \, b^{4} d^{2} e^{2} x^{2} + 180 \, a b^{3} d e^{3} x^{2} + 540 \, a^{2} b^{2} e^{4} x^{2} + 9 \, b^{4} d^{3} e x + 45 \, a b^{3} d^{2} e^{2} x + 135 \, a^{2} b^{2} d e^{3} x + 315 \, a^{3} b e^{4} x + b^{4} d^{4} + 5 \, a b^{3} d^{3} e + 15 \, a^{2} b^{2} d^{2} e^{2} + 35 \, a^{3} b d e^{3} + 70 \, a^{4} e^{4}}{630 \, {\left (e x + d\right )}^{9} e^{5}} \]

[In]

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

[Out]

-1/630*(126*b^4*e^4*x^4 + 84*b^4*d*e^3*x^3 + 420*a*b^3*e^4*x^3 + 36*b^4*d^2*e^2*x^2 + 180*a*b^3*d*e^3*x^2 + 54
0*a^2*b^2*e^4*x^2 + 9*b^4*d^3*e*x + 45*a*b^3*d^2*e^2*x + 135*a^2*b^2*d*e^3*x + 315*a^3*b*e^4*x + b^4*d^4 + 5*a
*b^3*d^3*e + 15*a^2*b^2*d^2*e^2 + 35*a^3*b*d*e^3 + 70*a^4*e^4)/((e*x + d)^9*e^5)

Mupad [B] (verification not implemented)

Time = 9.72 (sec) , antiderivative size = 259, normalized size of antiderivative = 2.18 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{10}} \, dx=-\frac {\frac {70\,a^4\,e^4+35\,a^3\,b\,d\,e^3+15\,a^2\,b^2\,d^2\,e^2+5\,a\,b^3\,d^3\,e+b^4\,d^4}{630\,e^5}+\frac {b^4\,x^4}{5\,e}+\frac {2\,b^3\,x^3\,\left (5\,a\,e+b\,d\right )}{15\,e^2}+\frac {b\,x\,\left (35\,a^3\,e^3+15\,a^2\,b\,d\,e^2+5\,a\,b^2\,d^2\,e+b^3\,d^3\right )}{70\,e^4}+\frac {2\,b^2\,x^2\,\left (15\,a^2\,e^2+5\,a\,b\,d\,e+b^2\,d^2\right )}{35\,e^3}}{d^9+9\,d^8\,e\,x+36\,d^7\,e^2\,x^2+84\,d^6\,e^3\,x^3+126\,d^5\,e^4\,x^4+126\,d^4\,e^5\,x^5+84\,d^3\,e^6\,x^6+36\,d^2\,e^7\,x^7+9\,d\,e^8\,x^8+e^9\,x^9} \]

[In]

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

[Out]

-((70*a^4*e^4 + b^4*d^4 + 15*a^2*b^2*d^2*e^2 + 5*a*b^3*d^3*e + 35*a^3*b*d*e^3)/(630*e^5) + (b^4*x^4)/(5*e) + (
2*b^3*x^3*(5*a*e + b*d))/(15*e^2) + (b*x*(35*a^3*e^3 + b^3*d^3 + 5*a*b^2*d^2*e + 15*a^2*b*d*e^2))/(70*e^4) + (
2*b^2*x^2*(15*a^2*e^2 + b^2*d^2 + 5*a*b*d*e))/(35*e^3))/(d^9 + e^9*x^9 + 9*d*e^8*x^8 + 36*d^7*e^2*x^2 + 84*d^6
*e^3*x^3 + 126*d^5*e^4*x^4 + 126*d^4*e^5*x^5 + 84*d^3*e^6*x^6 + 36*d^2*e^7*x^7 + 9*d^8*e*x)