Integrand size = 19, antiderivative size = 52 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx=-\frac {d (1+x)^{11}}{13 x^{13}}+\frac {(2 d-13 e) (1+x)^{11}}{156 x^{12}}-\frac {(2 d-13 e) (1+x)^{11}}{1716 x^{11}} \] Output:
-1/13*d*(1+x)^11/x^13+1/156*(2*d-13*e)*(1+x)^11/x^12-1/1716*(2*d-13*e)*(1+ x)^11/x^11
Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(52)=104\).
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.21 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx=-\frac {13 e x \left (11+120 x+594 x^2+1760 x^3+3465 x^4+4752 x^5+4620 x^6+3168 x^7+1485 x^8+440 x^9+66 x^{10}\right )+2 d \left (66+715 x+3510 x^2+10296 x^3+20020 x^4+27027 x^5+25740 x^6+17160 x^7+7722 x^8+2145 x^9+286 x^{10}\right )}{1716 x^{13}} \] Input:
Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^14,x]
Output:
-1/1716*(13*e*x*(11 + 120*x + 594*x^2 + 1760*x^3 + 3465*x^4 + 4752*x^5 + 4 620*x^6 + 3168*x^7 + 1485*x^8 + 440*x^9 + 66*x^10) + 2*d*(66 + 715*x + 351 0*x^2 + 10296*x^3 + 20020*x^4 + 27027*x^5 + 25740*x^6 + 17160*x^7 + 7722*x ^8 + 2145*x^9 + 286*x^10))/x^13
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1184, 87, 55, 48}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (x^2+2 x+1\right )^5 (d+e x)}{x^{14}} \, dx\) |
\(\Big \downarrow \) 1184 |
\(\displaystyle \int \frac {(x+1)^{10} (d+e x)}{x^{14}}dx\) |
\(\Big \downarrow \) 87 |
\(\displaystyle -\frac {1}{13} (2 d-13 e) \int \frac {(x+1)^{10}}{x^{13}}dx-\frac {d (x+1)^{11}}{13 x^{13}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {1}{13} (2 d-13 e) \left (-\frac {1}{12} \int \frac {(x+1)^{10}}{x^{12}}dx-\frac {(x+1)^{11}}{12 x^{12}}\right )-\frac {d (x+1)^{11}}{13 x^{13}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {1}{13} \left (\frac {(x+1)^{11}}{132 x^{11}}-\frac {(x+1)^{11}}{12 x^{12}}\right ) (2 d-13 e)-\frac {d (x+1)^{11}}{13 x^{13}}\) |
Input:
Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^14,x]
Output:
-1/13*(d*(1 + x)^11)/x^13 - ((2*d - 13*e)*(-1/12*(1 + x)^11/x^12 + (1 + x) ^11/(132*x^11)))/13
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] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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] - Simp[d*(S implify[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] && 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] || !SumSimp lerQ[n, 1])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/c^p Int[(d + e*x)^m*(f + g*x )^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(122\) vs. \(2(46)=92\).
Time = 0.86 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.37
method | result | size |
norman | \(\frac {-\frac {e \,x^{11}}{2}+\left (-\frac {d}{3}-\frac {10 e}{3}\right ) x^{10}+\left (-\frac {5 d}{2}-\frac {45 e}{4}\right ) x^{9}+\left (-9 d -24 e \right ) x^{8}+\left (-20 d -35 e \right ) x^{7}+\left (-30 d -36 e \right ) x^{6}+\left (-\frac {63 d}{2}-\frac {105 e}{4}\right ) x^{5}+\left (-\frac {70 d}{3}-\frac {40 e}{3}\right ) x^{4}+\left (-12 d -\frac {9 e}{2}\right ) x^{3}+\left (-\frac {45 d}{11}-\frac {10 e}{11}\right ) x^{2}+\left (-\frac {5 d}{6}-\frac {e}{12}\right ) x -\frac {d}{13}}{x^{13}}\) | \(123\) |
risch | \(\frac {-\frac {e \,x^{11}}{2}+\left (-\frac {d}{3}-\frac {10 e}{3}\right ) x^{10}+\left (-\frac {5 d}{2}-\frac {45 e}{4}\right ) x^{9}+\left (-9 d -24 e \right ) x^{8}+\left (-20 d -35 e \right ) x^{7}+\left (-30 d -36 e \right ) x^{6}+\left (-\frac {63 d}{2}-\frac {105 e}{4}\right ) x^{5}+\left (-\frac {70 d}{3}-\frac {40 e}{3}\right ) x^{4}+\left (-12 d -\frac {9 e}{2}\right ) x^{3}+\left (-\frac {45 d}{11}-\frac {10 e}{11}\right ) x^{2}+\left (-\frac {5 d}{6}-\frac {e}{12}\right ) x -\frac {d}{13}}{x^{13}}\) | \(123\) |
default | \(-\frac {45 d +120 e}{5 x^{5}}-\frac {120 d +210 e}{6 x^{6}}-\frac {d +10 e}{3 x^{3}}-\frac {e}{2 x^{2}}-\frac {d}{13 x^{13}}-\frac {252 d +210 e}{8 x^{8}}-\frac {10 d +45 e}{4 x^{4}}-\frac {210 d +120 e}{9 x^{9}}-\frac {210 d +252 e}{7 x^{7}}-\frac {120 d +45 e}{10 x^{10}}-\frac {45 d +10 e}{11 x^{11}}-\frac {10 d +e}{12 x^{12}}\) | \(130\) |
gosper | \(-\frac {858 e \,x^{11}+572 d \,x^{10}+5720 e \,x^{10}+4290 d \,x^{9}+19305 e \,x^{9}+15444 d \,x^{8}+41184 e \,x^{8}+34320 d \,x^{7}+60060 e \,x^{7}+51480 d \,x^{6}+61776 e \,x^{6}+54054 d \,x^{5}+45045 x^{5} e +40040 d \,x^{4}+22880 x^{4} e +20592 d \,x^{3}+7722 x^{3} e +7020 d \,x^{2}+1560 e \,x^{2}+1430 d x +143 e x +132 d}{1716 x^{13}}\) | \(132\) |
parallelrisch | \(\frac {-858 e \,x^{11}-572 d \,x^{10}-5720 e \,x^{10}-4290 d \,x^{9}-19305 e \,x^{9}-15444 d \,x^{8}-41184 e \,x^{8}-34320 d \,x^{7}-60060 e \,x^{7}-51480 d \,x^{6}-61776 e \,x^{6}-54054 d \,x^{5}-45045 x^{5} e -40040 d \,x^{4}-22880 x^{4} e -20592 d \,x^{3}-7722 x^{3} e -7020 d \,x^{2}-1560 e \,x^{2}-1430 d x -143 e x -132 d}{1716 x^{13}}\) | \(132\) |
orering | \(-\frac {\left (858 e \,x^{11}+572 d \,x^{10}+5720 e \,x^{10}+4290 d \,x^{9}+19305 e \,x^{9}+15444 d \,x^{8}+41184 e \,x^{8}+34320 d \,x^{7}+60060 e \,x^{7}+51480 d \,x^{6}+61776 e \,x^{6}+54054 d \,x^{5}+45045 x^{5} e +40040 d \,x^{4}+22880 x^{4} e +20592 d \,x^{3}+7722 x^{3} e +7020 d \,x^{2}+1560 e \,x^{2}+1430 d x +143 e x +132 d \right ) \left (x^{2}+2 x +1\right )^{5}}{1716 x^{13} \left (x +1\right )^{10}}\) | \(147\) |
Input:
int((e*x+d)*(x^2+2*x+1)^5/x^14,x,method=_RETURNVERBOSE)
Output:
(-1/2*e*x^11+(-1/3*d-10/3*e)*x^10+(-5/2*d-45/4*e)*x^9+(-9*d-24*e)*x^8+(-20 *d-35*e)*x^7+(-30*d-36*e)*x^6+(-63/2*d-105/4*e)*x^5+(-70/3*d-40/3*e)*x^4+( -12*d-9/2*e)*x^3+(-45/11*d-10/11*e)*x^2+(-5/6*d-1/12*e)*x-1/13*d)/x^13
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (46) = 92\).
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx=-\frac {858 \, e x^{11} + 572 \, {\left (d + 10 \, e\right )} x^{10} + 2145 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 5148 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 8580 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 10296 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 9009 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5720 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2574 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 780 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 143 \, {\left (10 \, d + e\right )} x + 132 \, d}{1716 \, x^{13}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^14,x, algorithm="fricas")
Output:
-1/1716*(858*e*x^11 + 572*(d + 10*e)*x^10 + 2145*(2*d + 9*e)*x^9 + 5148*(3 *d + 8*e)*x^8 + 8580*(4*d + 7*e)*x^7 + 10296*(5*d + 6*e)*x^6 + 9009*(6*d + 5*e)*x^5 + 5720*(7*d + 4*e)*x^4 + 2574*(8*d + 3*e)*x^3 + 780*(9*d + 2*e)* x^2 + 143*(10*d + e)*x + 132*d)/x^13
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (41) = 82\).
Time = 7.93 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.52 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx=\frac {- 132 d - 858 e x^{11} + x^{10} \left (- 572 d - 5720 e\right ) + x^{9} \left (- 4290 d - 19305 e\right ) + x^{8} \left (- 15444 d - 41184 e\right ) + x^{7} \left (- 34320 d - 60060 e\right ) + x^{6} \left (- 51480 d - 61776 e\right ) + x^{5} \left (- 54054 d - 45045 e\right ) + x^{4} \left (- 40040 d - 22880 e\right ) + x^{3} \left (- 20592 d - 7722 e\right ) + x^{2} \left (- 7020 d - 1560 e\right ) + x \left (- 1430 d - 143 e\right )}{1716 x^{13}} \] Input:
integrate((e*x+d)*(x**2+2*x+1)**5/x**14,x)
Output:
(-132*d - 858*e*x**11 + x**10*(-572*d - 5720*e) + x**9*(-4290*d - 19305*e) + x**8*(-15444*d - 41184*e) + x**7*(-34320*d - 60060*e) + x**6*(-51480*d - 61776*e) + x**5*(-54054*d - 45045*e) + x**4*(-40040*d - 22880*e) + x**3* (-20592*d - 7722*e) + x**2*(-7020*d - 1560*e) + x*(-1430*d - 143*e))/(1716 *x**13)
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (46) = 92\).
Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 2.48 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx=-\frac {858 \, e x^{11} + 572 \, {\left (d + 10 \, e\right )} x^{10} + 2145 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 5148 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 8580 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 10296 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 9009 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 5720 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 2574 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 780 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 143 \, {\left (10 \, d + e\right )} x + 132 \, d}{1716 \, x^{13}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^14,x, algorithm="maxima")
Output:
-1/1716*(858*e*x^11 + 572*(d + 10*e)*x^10 + 2145*(2*d + 9*e)*x^9 + 5148*(3 *d + 8*e)*x^8 + 8580*(4*d + 7*e)*x^7 + 10296*(5*d + 6*e)*x^6 + 9009*(6*d + 5*e)*x^5 + 5720*(7*d + 4*e)*x^4 + 2574*(8*d + 3*e)*x^3 + 780*(9*d + 2*e)* x^2 + 143*(10*d + e)*x + 132*d)/x^13
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (46) = 92\).
Time = 0.16 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.52 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx=-\frac {858 \, e x^{11} + 572 \, d x^{10} + 5720 \, e x^{10} + 4290 \, d x^{9} + 19305 \, e x^{9} + 15444 \, d x^{8} + 41184 \, e x^{8} + 34320 \, d x^{7} + 60060 \, e x^{7} + 51480 \, d x^{6} + 61776 \, e x^{6} + 54054 \, d x^{5} + 45045 \, e x^{5} + 40040 \, d x^{4} + 22880 \, e x^{4} + 20592 \, d x^{3} + 7722 \, e x^{3} + 7020 \, d x^{2} + 1560 \, e x^{2} + 1430 \, d x + 143 \, e x + 132 \, d}{1716 \, x^{13}} \] Input:
integrate((e*x+d)*(x^2+2*x+1)^5/x^14,x, algorithm="giac")
Output:
-1/1716*(858*e*x^11 + 572*d*x^10 + 5720*e*x^10 + 4290*d*x^9 + 19305*e*x^9 + 15444*d*x^8 + 41184*e*x^8 + 34320*d*x^7 + 60060*e*x^7 + 51480*d*x^6 + 61 776*e*x^6 + 54054*d*x^5 + 45045*e*x^5 + 40040*d*x^4 + 22880*e*x^4 + 20592* d*x^3 + 7722*e*x^3 + 7020*d*x^2 + 1560*e*x^2 + 1430*d*x + 143*e*x + 132*d) /x^13
Time = 10.70 (sec) , antiderivative size = 123, normalized size of antiderivative = 2.37 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx=-\frac {\frac {e\,x^{11}}{2}+\left (\frac {d}{3}+\frac {10\,e}{3}\right )\,x^{10}+\left (\frac {5\,d}{2}+\frac {45\,e}{4}\right )\,x^9+\left (9\,d+24\,e\right )\,x^8+\left (20\,d+35\,e\right )\,x^7+\left (30\,d+36\,e\right )\,x^6+\left (\frac {63\,d}{2}+\frac {105\,e}{4}\right )\,x^5+\left (\frac {70\,d}{3}+\frac {40\,e}{3}\right )\,x^4+\left (12\,d+\frac {9\,e}{2}\right )\,x^3+\left (\frac {45\,d}{11}+\frac {10\,e}{11}\right )\,x^2+\left (\frac {5\,d}{6}+\frac {e}{12}\right )\,x+\frac {d}{13}}{x^{13}} \] Input:
int(((d + e*x)*(2*x + x^2 + 1)^5)/x^14,x)
Output:
-(d/13 + x^3*(12*d + (9*e)/2) + x^10*(d/3 + (10*e)/3) + x^8*(9*d + 24*e) + x^7*(20*d + 35*e) + x^9*((5*d)/2 + (45*e)/4) + x^6*(30*d + 36*e) + x^2*(( 45*d)/11 + (10*e)/11) + x^4*((70*d)/3 + (40*e)/3) + x^5*((63*d)/2 + (105*e )/4) + (e*x^11)/2 + x*((5*d)/6 + e/12))/x^13
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.52 \[ \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx=\frac {-858 e \,x^{11}-572 d \,x^{10}-5720 e \,x^{10}-4290 d \,x^{9}-19305 e \,x^{9}-15444 d \,x^{8}-41184 e \,x^{8}-34320 d \,x^{7}-60060 e \,x^{7}-51480 d \,x^{6}-61776 e \,x^{6}-54054 d \,x^{5}-45045 e \,x^{5}-40040 d \,x^{4}-22880 e \,x^{4}-20592 d \,x^{3}-7722 e \,x^{3}-7020 d \,x^{2}-1560 e \,x^{2}-1430 d x -143 e x -132 d}{1716 x^{13}} \] Input:
int((e*x+d)*(x^2+2*x+1)^5/x^14,x)
Output:
( - 572*d*x**10 - 4290*d*x**9 - 15444*d*x**8 - 34320*d*x**7 - 51480*d*x**6 - 54054*d*x**5 - 40040*d*x**4 - 20592*d*x**3 - 7020*d*x**2 - 1430*d*x - 1 32*d - 858*e*x**11 - 5720*e*x**10 - 19305*e*x**9 - 41184*e*x**8 - 60060*e* x**7 - 61776*e*x**6 - 45045*e*x**5 - 22880*e*x**4 - 7722*e*x**3 - 1560*e*x **2 - 143*e*x)/(1716*x**13)