Integrand size = 18, antiderivative size = 171 \[ \int x^3 (c+d x)^n \left (a+b x^2\right ) \, dx=-\frac {c^3 \left (b c^2+a d^2\right ) (c+d x)^{1+n}}{d^6 (1+n)}+\frac {c^2 \left (5 b c^2+3 a d^2\right ) (c+d x)^{2+n}}{d^6 (2+n)}-\frac {c \left (10 b c^2+3 a d^2\right ) (c+d x)^{3+n}}{d^6 (3+n)}+\frac {\left (10 b c^2+a d^2\right ) (c+d x)^{4+n}}{d^6 (4+n)}-\frac {5 b c (c+d x)^{5+n}}{d^6 (5+n)}+\frac {b (c+d x)^{6+n}}{d^6 (6+n)} \] Output:
-c^3*(a*d^2+b*c^2)*(d*x+c)^(1+n)/d^6/(1+n)+c^2*(3*a*d^2+5*b*c^2)*(d*x+c)^( 2+n)/d^6/(2+n)-c*(3*a*d^2+10*b*c^2)*(d*x+c)^(3+n)/d^6/(3+n)+(a*d^2+10*b*c^ 2)*(d*x+c)^(4+n)/d^6/(4+n)-5*b*c*(d*x+c)^(5+n)/d^6/(5+n)+b*(d*x+c)^(6+n)/d ^6/(6+n)
Time = 0.06 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.85 \[ \int x^3 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {(c+d x)^{1+n} \left (-\frac {b c^5+a c^3 d^2}{1+n}+\frac {c^2 \left (5 b c^2+3 a d^2\right ) (c+d x)}{2+n}-\frac {c \left (10 b c^2+3 a d^2\right ) (c+d x)^2}{3+n}+\frac {\left (10 b c^2+a d^2\right ) (c+d x)^3}{4+n}-\frac {5 b c (c+d x)^4}{5+n}+\frac {b (c+d x)^5}{6+n}\right )}{d^6} \] Input:
Integrate[x^3*(c + d*x)^n*(a + b*x^2),x]
Output:
((c + d*x)^(1 + n)*(-((b*c^5 + a*c^3*d^2)/(1 + n)) + (c^2*(5*b*c^2 + 3*a*d ^2)*(c + d*x))/(2 + n) - (c*(10*b*c^2 + 3*a*d^2)*(c + d*x)^2)/(3 + n) + (( 10*b*c^2 + a*d^2)*(c + d*x)^3)/(4 + n) - (5*b*c*(c + d*x)^4)/(5 + n) + (b* (c + d*x)^5)/(6 + n)))/d^6
Time = 0.33 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {522, 2009}
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.
| \(\displaystyle \int x^3 \left (a+b x^2\right ) (c+d x)^n \, dx\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \int \left (\frac {\left (-3 a c d^2-10 b c^3\right ) (c+d x)^{n+2}}{d^5}+\frac {\left (a d^2+10 b c^2\right ) (c+d x)^{n+3}}{d^5}+\frac {\left (3 a c^2 d^2+5 b c^4\right ) (c+d x)^{n+1}}{d^5}+\frac {c^3 \left (-a d^2-b c^2\right ) (c+d x)^n}{d^5}-\frac {5 b c (c+d x)^{n+4}}{d^5}+\frac {b (c+d x)^{n+5}}{d^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {c^2 \left (3 a d^2+5 b c^2\right ) (c+d x)^{n+2}}{d^6 (n+2)}-\frac {c \left (3 a d^2+10 b c^2\right ) (c+d x)^{n+3}}{d^6 (n+3)}+\frac {\left (a d^2+10 b c^2\right ) (c+d x)^{n+4}}{d^6 (n+4)}-\frac {c^3 \left (a d^2+b c^2\right ) (c+d x)^{n+1}}{d^6 (n+1)}-\frac {5 b c (c+d x)^{n+5}}{d^6 (n+5)}+\frac {b (c+d x)^{n+6}}{d^6 (n+6)}\) |
Input:
Int[x^3*(c + d*x)^n*(a + b*x^2),x]
Output:
-((c^3*(b*c^2 + a*d^2)*(c + d*x)^(1 + n))/(d^6*(1 + n))) + (c^2*(5*b*c^2 + 3*a*d^2)*(c + d*x)^(2 + n))/(d^6*(2 + n)) - (c*(10*b*c^2 + 3*a*d^2)*(c + d*x)^(3 + n))/(d^6*(3 + n)) + ((10*b*c^2 + a*d^2)*(c + d*x)^(4 + n))/(d^6* (4 + n)) - (5*b*c*(c + d*x)^(5 + n))/(d^6*(5 + n)) + (b*(c + d*x)^(6 + n)) /(d^6*(6 + n))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(401\) vs. \(2(171)=342\).
Time = 0.31 (sec) , antiderivative size = 402, normalized size of antiderivative = 2.35
| method | result | size |
| norman | \(\frac {b \,x^{6} {\mathrm e}^{n \ln \left (d x +c \right )}}{6+n}+\frac {\left (a \,d^{2} n^{2}+11 a \,d^{2} n -5 b \,c^{2} n +30 a \,d^{2}\right ) x^{4} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{2} \left (n^{3}+15 n^{2}+74 n +120\right )}+\frac {n b c \,x^{5} {\mathrm e}^{n \ln \left (d x +c \right )}}{d \left (n^{2}+11 n +30\right )}+\frac {n c \left (a \,d^{2} n^{2}+11 a \,d^{2} n +30 a \,d^{2}+20 b \,c^{2}\right ) x^{3} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{3} \left (n^{4}+18 n^{3}+119 n^{2}+342 n +360\right )}-\frac {6 c^{4} \left (a \,d^{2} n^{2}+11 a \,d^{2} n +30 a \,d^{2}+20 b \,c^{2}\right ) {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}+\frac {6 n \,c^{3} \left (a \,d^{2} n^{2}+11 a \,d^{2} n +30 a \,d^{2}+20 b \,c^{2}\right ) x \,{\mathrm e}^{n \ln \left (d x +c \right )}}{d^{5} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}-\frac {3 \left (a \,d^{2} n^{2}+11 a \,d^{2} n +30 a \,d^{2}+20 b \,c^{2}\right ) c^{2} n \,x^{2} {\mathrm e}^{n \ln \left (d x +c \right )}}{d^{4} \left (n^{5}+20 n^{4}+155 n^{3}+580 n^{2}+1044 n +720\right )}\) | \(402\) |
| gosper | \(-\frac {\left (d x +c \right )^{1+n} \left (-b \,d^{5} n^{5} x^{5}-15 b \,d^{5} n^{4} x^{5}-a \,d^{5} n^{5} x^{3}+5 b c \,d^{4} n^{4} x^{4}-85 b \,d^{5} n^{3} x^{5}-17 a \,d^{5} n^{4} x^{3}+50 b c \,d^{4} n^{3} x^{4}-225 b \,d^{5} n^{2} x^{5}+3 a c \,d^{4} n^{4} x^{2}-107 a \,d^{5} n^{3} x^{3}-20 b \,c^{2} d^{3} n^{3} x^{3}+175 b c \,d^{4} n^{2} x^{4}-274 b \,d^{5} n \,x^{5}+42 a c \,d^{4} n^{3} x^{2}-307 a \,d^{5} n^{2} x^{3}-120 b \,c^{2} d^{3} n^{2} x^{3}+250 b c \,d^{4} n \,x^{4}-120 b \,d^{5} x^{5}-6 a \,c^{2} d^{3} n^{3} x +195 a c \,d^{4} n^{2} x^{2}-396 a \,d^{5} n \,x^{3}+60 b \,c^{3} d^{2} n^{2} x^{2}-220 b \,c^{2} d^{3} n \,x^{3}+120 b c \,d^{4} x^{4}-72 a \,c^{2} d^{3} n^{2} x +336 a c \,d^{4} n \,x^{2}-180 a \,d^{5} x^{3}+180 b \,c^{3} d^{2} n \,x^{2}-120 b \,c^{2} d^{3} x^{3}+6 a \,c^{3} d^{2} n^{2}-246 a \,c^{2} d^{3} n x +180 a c \,d^{4} x^{2}-120 b \,c^{4} d n x +120 b \,c^{3} d^{2} x^{2}+66 a \,c^{3} d^{2} n -180 a \,c^{2} d^{3} x -120 b \,c^{4} d x +180 a \,c^{3} d^{2}+120 b \,c^{5}\right )}{d^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\) | \(496\) |
| orering | \(-\frac {\left (d x +c \right )^{n} \left (-b \,d^{5} n^{5} x^{5}-15 b \,d^{5} n^{4} x^{5}-a \,d^{5} n^{5} x^{3}+5 b c \,d^{4} n^{4} x^{4}-85 b \,d^{5} n^{3} x^{5}-17 a \,d^{5} n^{4} x^{3}+50 b c \,d^{4} n^{3} x^{4}-225 b \,d^{5} n^{2} x^{5}+3 a c \,d^{4} n^{4} x^{2}-107 a \,d^{5} n^{3} x^{3}-20 b \,c^{2} d^{3} n^{3} x^{3}+175 b c \,d^{4} n^{2} x^{4}-274 b \,d^{5} n \,x^{5}+42 a c \,d^{4} n^{3} x^{2}-307 a \,d^{5} n^{2} x^{3}-120 b \,c^{2} d^{3} n^{2} x^{3}+250 b c \,d^{4} n \,x^{4}-120 b \,d^{5} x^{5}-6 a \,c^{2} d^{3} n^{3} x +195 a c \,d^{4} n^{2} x^{2}-396 a \,d^{5} n \,x^{3}+60 b \,c^{3} d^{2} n^{2} x^{2}-220 b \,c^{2} d^{3} n \,x^{3}+120 b c \,d^{4} x^{4}-72 a \,c^{2} d^{3} n^{2} x +336 a c \,d^{4} n \,x^{2}-180 a \,d^{5} x^{3}+180 b \,c^{3} d^{2} n \,x^{2}-120 b \,c^{2} d^{3} x^{3}+6 a \,c^{3} d^{2} n^{2}-246 a \,c^{2} d^{3} n x +180 a c \,d^{4} x^{2}-120 b \,c^{4} d n x +120 b \,c^{3} d^{2} x^{2}+66 a \,c^{3} d^{2} n -180 a \,c^{2} d^{3} x -120 b \,c^{4} d x +180 a \,c^{3} d^{2}+120 b \,c^{5}\right ) \left (d x +c \right )}{d^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )}\) | \(499\) |
| risch | \(-\frac {\left (-b \,d^{6} n^{5} x^{6}-b c \,d^{5} n^{5} x^{5}-15 b \,d^{6} n^{4} x^{6}-a \,d^{6} n^{5} x^{4}-10 b c \,d^{5} n^{4} x^{5}-85 b \,d^{6} n^{3} x^{6}-a c \,d^{5} n^{5} x^{3}-17 a \,d^{6} n^{4} x^{4}+5 b \,c^{2} d^{4} n^{4} x^{4}-35 b c \,d^{5} n^{3} x^{5}-225 b \,d^{6} n^{2} x^{6}-14 a c \,d^{5} n^{4} x^{3}-107 a \,d^{6} n^{3} x^{4}+30 b \,c^{2} d^{4} n^{3} x^{4}-50 b c \,d^{5} n^{2} x^{5}-274 b \,d^{6} n \,x^{6}+3 a \,c^{2} d^{4} n^{4} x^{2}-65 a c \,d^{5} n^{3} x^{3}-307 a \,d^{6} n^{2} x^{4}-20 b \,c^{3} d^{3} n^{3} x^{3}+55 b \,c^{2} d^{4} n^{2} x^{4}-24 b c \,d^{5} n \,x^{5}-120 b \,x^{6} d^{6}+36 a \,c^{2} d^{4} n^{3} x^{2}-112 a c \,d^{5} n^{2} x^{3}-396 a \,d^{6} n \,x^{4}-60 b \,c^{3} d^{3} n^{2} x^{3}+30 b \,c^{2} d^{4} n \,x^{4}-6 a \,c^{3} d^{3} n^{3} x +123 a \,c^{2} d^{4} n^{2} x^{2}-60 a c \,d^{5} n \,x^{3}-180 a \,d^{6} x^{4}+60 b \,c^{4} d^{2} n^{2} x^{2}-40 b \,c^{3} d^{3} n \,x^{3}-66 a \,c^{3} d^{3} n^{2} x +90 a \,c^{2} d^{4} n \,x^{2}+60 b \,c^{4} d^{2} n \,x^{2}+6 a \,c^{4} d^{2} n^{2}-180 a \,c^{3} d^{3} n x -120 b \,c^{5} d n x +66 a \,c^{4} d^{2} n +180 a \,c^{4} d^{2}+120 b \,c^{6}\right ) \left (d x +c \right )^{n}}{\left (5+n \right ) \left (6+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) d^{6}}\) | \(574\) |
| parallelrisch | \(\frac {-123 x^{2} \left (d x +c \right )^{n} a \,c^{3} d^{4} n^{2}+307 x^{4} \left (d x +c \right )^{n} a c \,d^{6} n^{2}-55 x^{4} \left (d x +c \right )^{n} b \,c^{3} d^{4} n^{2}+65 x^{3} \left (d x +c \right )^{n} a \,c^{2} d^{5} n^{3}+20 x^{3} \left (d x +c \right )^{n} b \,c^{4} d^{3} n^{3}-3 x^{2} \left (d x +c \right )^{n} a \,c^{3} d^{4} n^{4}+396 x^{4} \left (d x +c \right )^{n} a c \,d^{6} n -30 x^{4} \left (d x +c \right )^{n} b \,c^{3} d^{4} n +112 x^{3} \left (d x +c \right )^{n} a \,c^{2} d^{5} n^{2}+85 x^{6} \left (d x +c \right )^{n} b c \,d^{6} n^{3}+10 x^{5} \left (d x +c \right )^{n} b \,c^{2} d^{5} n^{4}+x^{4} \left (d x +c \right )^{n} a c \,d^{6} n^{5}+225 x^{6} \left (d x +c \right )^{n} b c \,d^{6} n^{2}+35 x^{5} \left (d x +c \right )^{n} b \,c^{2} d^{5} n^{3}+17 x^{4} \left (d x +c \right )^{n} a c \,d^{6} n^{4}-5 x^{4} \left (d x +c \right )^{n} b \,c^{3} d^{4} n^{4}+60 x^{3} \left (d x +c \right )^{n} b \,c^{4} d^{3} n^{2}-36 x^{2} \left (d x +c \right )^{n} a \,c^{3} d^{4} n^{3}+60 x^{3} \left (d x +c \right )^{n} a \,c^{2} d^{5} n -60 x^{2} \left (d x +c \right )^{n} b \,c^{5} d^{2} n^{2}+6 x \left (d x +c \right )^{n} a \,c^{4} d^{3} n^{3}-90 x^{2} \left (d x +c \right )^{n} a \,c^{3} d^{4} n +x^{3} \left (d x +c \right )^{n} a \,c^{2} d^{5} n^{5}+274 x^{6} \left (d x +c \right )^{n} b c \,d^{6} n +50 x^{5} \left (d x +c \right )^{n} b \,c^{2} d^{5} n^{2}+107 x^{4} \left (d x +c \right )^{n} a c \,d^{6} n^{3}-30 x^{4} \left (d x +c \right )^{n} b \,c^{3} d^{4} n^{3}+14 x^{3} \left (d x +c \right )^{n} a \,c^{2} d^{5} n^{4}+24 x^{5} \left (d x +c \right )^{n} b \,c^{2} d^{5} n +x^{6} \left (d x +c \right )^{n} b c \,d^{6} n^{5}+15 x^{6} \left (d x +c \right )^{n} b c \,d^{6} n^{4}+x^{5} \left (d x +c \right )^{n} b \,c^{2} d^{5} n^{5}+120 x^{6} \left (d x +c \right )^{n} b c \,d^{6}+180 x^{4} \left (d x +c \right )^{n} a c \,d^{6}-6 \left (d x +c \right )^{n} a \,c^{5} d^{2} n^{2}-66 \left (d x +c \right )^{n} a \,c^{5} d^{2} n -60 x^{2} \left (d x +c \right )^{n} b \,c^{5} d^{2} n +66 x \left (d x +c \right )^{n} a \,c^{4} d^{3} n^{2}+180 x \left (d x +c \right )^{n} a \,c^{4} d^{3} n +120 x \left (d x +c \right )^{n} b \,c^{6} d n +40 x^{3} \left (d x +c \right )^{n} b \,c^{4} d^{3} n -180 \left (d x +c \right )^{n} a \,c^{5} d^{2}-120 \left (d x +c \right )^{n} b \,c^{7}}{c \left (6+n \right ) \left (5+n \right ) \left (4+n \right ) \left (3+n \right ) \left (2+n \right ) \left (1+n \right ) d^{6}}\) | \(898\) |
Input:
int(x^3*(d*x+c)^n*(b*x^2+a),x,method=_RETURNVERBOSE)
Output:
b/(6+n)*x^6*exp(n*ln(d*x+c))+(a*d^2*n^2+11*a*d^2*n-5*b*c^2*n+30*a*d^2)/d^2 /(n^3+15*n^2+74*n+120)*x^4*exp(n*ln(d*x+c))+n*b*c/d/(n^2+11*n+30)*x^5*exp( n*ln(d*x+c))+n*c*(a*d^2*n^2+11*a*d^2*n+30*a*d^2+20*b*c^2)/d^3/(n^4+18*n^3+ 119*n^2+342*n+360)*x^3*exp(n*ln(d*x+c))-6*c^4*(a*d^2*n^2+11*a*d^2*n+30*a*d ^2+20*b*c^2)/d^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*n+720)*exp(n*ln (d*x+c))+6/d^5*n*c^3*(a*d^2*n^2+11*a*d^2*n+30*a*d^2+20*b*c^2)/(n^6+21*n^5+ 175*n^4+735*n^3+1624*n^2+1764*n+720)*x*exp(n*ln(d*x+c))-3*(a*d^2*n^2+11*a* d^2*n+30*a*d^2+20*b*c^2)*c^2/d^4*n/(n^5+20*n^4+155*n^3+580*n^2+1044*n+720) *x^2*exp(n*ln(d*x+c))
Leaf count of result is larger than twice the leaf count of optimal. 519 vs. \(2 (171) = 342\).
Time = 0.10 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.04 \[ \int x^3 (c+d x)^n \left (a+b x^2\right ) \, dx=-\frac {{\left (6 \, a c^{4} d^{2} n^{2} + 66 \, a c^{4} d^{2} n + 120 \, b c^{6} + 180 \, a c^{4} d^{2} - {\left (b d^{6} n^{5} + 15 \, b d^{6} n^{4} + 85 \, b d^{6} n^{3} + 225 \, b d^{6} n^{2} + 274 \, b d^{6} n + 120 \, b d^{6}\right )} x^{6} - {\left (b c d^{5} n^{5} + 10 \, b c d^{5} n^{4} + 35 \, b c d^{5} n^{3} + 50 \, b c d^{5} n^{2} + 24 \, b c d^{5} n\right )} x^{5} - {\left (a d^{6} n^{5} + 180 \, a d^{6} - {\left (5 \, b c^{2} d^{4} - 17 \, a d^{6}\right )} n^{4} - {\left (30 \, b c^{2} d^{4} - 107 \, a d^{6}\right )} n^{3} - {\left (55 \, b c^{2} d^{4} - 307 \, a d^{6}\right )} n^{2} - 6 \, {\left (5 \, b c^{2} d^{4} - 66 \, a d^{6}\right )} n\right )} x^{4} - {\left (a c d^{5} n^{5} + 14 \, a c d^{5} n^{4} + 5 \, {\left (4 \, b c^{3} d^{3} + 13 \, a c d^{5}\right )} n^{3} + 4 \, {\left (15 \, b c^{3} d^{3} + 28 \, a c d^{5}\right )} n^{2} + 20 \, {\left (2 \, b c^{3} d^{3} + 3 \, a c d^{5}\right )} n\right )} x^{3} + 3 \, {\left (a c^{2} d^{4} n^{4} + 12 \, a c^{2} d^{4} n^{3} + {\left (20 \, b c^{4} d^{2} + 41 \, a c^{2} d^{4}\right )} n^{2} + 10 \, {\left (2 \, b c^{4} d^{2} + 3 \, a c^{2} d^{4}\right )} n\right )} x^{2} - 6 \, {\left (a c^{3} d^{3} n^{3} + 11 \, a c^{3} d^{3} n^{2} + 10 \, {\left (2 \, b c^{5} d + 3 \, a c^{3} d^{3}\right )} n\right )} x\right )} {\left (d x + c\right )}^{n}}{d^{6} n^{6} + 21 \, d^{6} n^{5} + 175 \, d^{6} n^{4} + 735 \, d^{6} n^{3} + 1624 \, d^{6} n^{2} + 1764 \, d^{6} n + 720 \, d^{6}} \] Input:
integrate(x^3*(d*x+c)^n*(b*x^2+a),x, algorithm="fricas")
Output:
-(6*a*c^4*d^2*n^2 + 66*a*c^4*d^2*n + 120*b*c^6 + 180*a*c^4*d^2 - (b*d^6*n^ 5 + 15*b*d^6*n^4 + 85*b*d^6*n^3 + 225*b*d^6*n^2 + 274*b*d^6*n + 120*b*d^6) *x^6 - (b*c*d^5*n^5 + 10*b*c*d^5*n^4 + 35*b*c*d^5*n^3 + 50*b*c*d^5*n^2 + 2 4*b*c*d^5*n)*x^5 - (a*d^6*n^5 + 180*a*d^6 - (5*b*c^2*d^4 - 17*a*d^6)*n^4 - (30*b*c^2*d^4 - 107*a*d^6)*n^3 - (55*b*c^2*d^4 - 307*a*d^6)*n^2 - 6*(5*b* c^2*d^4 - 66*a*d^6)*n)*x^4 - (a*c*d^5*n^5 + 14*a*c*d^5*n^4 + 5*(4*b*c^3*d^ 3 + 13*a*c*d^5)*n^3 + 4*(15*b*c^3*d^3 + 28*a*c*d^5)*n^2 + 20*(2*b*c^3*d^3 + 3*a*c*d^5)*n)*x^3 + 3*(a*c^2*d^4*n^4 + 12*a*c^2*d^4*n^3 + (20*b*c^4*d^2 + 41*a*c^2*d^4)*n^2 + 10*(2*b*c^4*d^2 + 3*a*c^2*d^4)*n)*x^2 - 6*(a*c^3*d^3 *n^3 + 11*a*c^3*d^3*n^2 + 10*(2*b*c^5*d + 3*a*c^3*d^3)*n)*x)*(d*x + c)^n/( d^6*n^6 + 21*d^6*n^5 + 175*d^6*n^4 + 735*d^6*n^3 + 1624*d^6*n^2 + 1764*d^6 *n + 720*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 6999 vs. \(2 (155) = 310\).
Time = 1.83 (sec) , antiderivative size = 6999, normalized size of antiderivative = 40.93 \[ \int x^3 (c+d x)^n \left (a+b x^2\right ) \, dx=\text {Too large to display} \] Input:
integrate(x**3*(d*x+c)**n*(b*x**2+a),x)
Output:
Piecewise((c**n*(a*x**4/4 + b*x**6/6), Eq(d, 0)), (-3*a*c**3*d**2/(60*c**5 *d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c* d**10*x**4 + 60*d**11*x**5) - 15*a*c**2*d**3*x/(60*c**5*d**6 + 300*c**4*d* *7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d** 11*x**5) - 30*a*c*d**4*x**2/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d** 8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) - 30*a*d** 5*x**3/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d** 9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 60*b*c**5*log(c/d + x)/(60*c* *5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300* c*d**10*x**4 + 60*d**11*x**5) + 137*b*c**5/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x **5) + 300*b*c**4*d*x*log(c/d + x)/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c **3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 6 25*b*c**4*d*x/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c **2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 600*b*c**3*d**2*x**2*l og(c/d + x)/(60*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c** 2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 1100*b*c**3*d**2*x**2/(6 0*c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 + 300*c*d**10*x**4 + 60*d**11*x**5) + 600*b*c**2*d**3*x**3*log(c/d + x)/(60* c**5*d**6 + 300*c**4*d**7*x + 600*c**3*d**8*x**2 + 600*c**2*d**9*x**3 +...
Time = 0.04 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.67 \[ \int x^3 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} x^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} x^{2} + 6 \, c^{3} d n x - 6 \, c^{4}\right )} {\left (d x + c\right )}^{n} a}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d^{6} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} c d^{5} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c^{2} d^{4} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{3} d^{3} x^{3} - 60 \, {\left (n^{2} + n\right )} c^{4} d^{2} x^{2} + 120 \, c^{5} d n x - 120 \, c^{6}\right )} {\left (d x + c\right )}^{n} b}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} d^{6}} \] Input:
integrate(x^3*(d*x+c)^n*(b*x^2+a),x, algorithm="maxima")
Output:
((n^3 + 6*n^2 + 11*n + 6)*d^4*x^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*x^3 - 3*(n^2 + n)*c^2*d^2*x^2 + 6*c^3*d*n*x - 6*c^4)*(d*x + c)^n*a/((n^4 + 10*n^3 + 35 *n^2 + 50*n + 24)*d^4) + ((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)* d^6*x^6 + (n^5 + 10*n^4 + 35*n^3 + 50*n^2 + 24*n)*c*d^5*x^5 - 5*(n^4 + 6*n ^3 + 11*n^2 + 6*n)*c^2*d^4*x^4 + 20*(n^3 + 3*n^2 + 2*n)*c^3*d^3*x^3 - 60*( n^2 + n)*c^4*d^2*x^2 + 120*c^5*d*n*x - 120*c^6)*(d*x + c)^n*b/((n^6 + 21*n ^5 + 175*n^4 + 735*n^3 + 1624*n^2 + 1764*n + 720)*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (171) = 342\).
Time = 0.13 (sec) , antiderivative size = 882, normalized size of antiderivative = 5.16 \[ \int x^3 (c+d x)^n \left (a+b x^2\right ) \, dx =\text {Too large to display} \] Input:
integrate(x^3*(d*x+c)^n*(b*x^2+a),x, algorithm="giac")
Output:
((d*x + c)^n*b*d^6*n^5*x^6 + (d*x + c)^n*b*c*d^5*n^5*x^5 + 15*(d*x + c)^n* b*d^6*n^4*x^6 + (d*x + c)^n*a*d^6*n^5*x^4 + 10*(d*x + c)^n*b*c*d^5*n^4*x^5 + 85*(d*x + c)^n*b*d^6*n^3*x^6 + (d*x + c)^n*a*c*d^5*n^5*x^3 - 5*(d*x + c )^n*b*c^2*d^4*n^4*x^4 + 17*(d*x + c)^n*a*d^6*n^4*x^4 + 35*(d*x + c)^n*b*c* d^5*n^3*x^5 + 225*(d*x + c)^n*b*d^6*n^2*x^6 + 14*(d*x + c)^n*a*c*d^5*n^4*x ^3 - 30*(d*x + c)^n*b*c^2*d^4*n^3*x^4 + 107*(d*x + c)^n*a*d^6*n^3*x^4 + 50 *(d*x + c)^n*b*c*d^5*n^2*x^5 + 274*(d*x + c)^n*b*d^6*n*x^6 - 3*(d*x + c)^n *a*c^2*d^4*n^4*x^2 + 20*(d*x + c)^n*b*c^3*d^3*n^3*x^3 + 65*(d*x + c)^n*a*c *d^5*n^3*x^3 - 55*(d*x + c)^n*b*c^2*d^4*n^2*x^4 + 307*(d*x + c)^n*a*d^6*n^ 2*x^4 + 24*(d*x + c)^n*b*c*d^5*n*x^5 + 120*(d*x + c)^n*b*d^6*x^6 - 36*(d*x + c)^n*a*c^2*d^4*n^3*x^2 + 60*(d*x + c)^n*b*c^3*d^3*n^2*x^3 + 112*(d*x + c)^n*a*c*d^5*n^2*x^3 - 30*(d*x + c)^n*b*c^2*d^4*n*x^4 + 396*(d*x + c)^n*a* d^6*n*x^4 + 6*(d*x + c)^n*a*c^3*d^3*n^3*x - 60*(d*x + c)^n*b*c^4*d^2*n^2*x ^2 - 123*(d*x + c)^n*a*c^2*d^4*n^2*x^2 + 40*(d*x + c)^n*b*c^3*d^3*n*x^3 + 60*(d*x + c)^n*a*c*d^5*n*x^3 + 180*(d*x + c)^n*a*d^6*x^4 + 66*(d*x + c)^n* a*c^3*d^3*n^2*x - 60*(d*x + c)^n*b*c^4*d^2*n*x^2 - 90*(d*x + c)^n*a*c^2*d^ 4*n*x^2 - 6*(d*x + c)^n*a*c^4*d^2*n^2 + 120*(d*x + c)^n*b*c^5*d*n*x + 180* (d*x + c)^n*a*c^3*d^3*n*x - 66*(d*x + c)^n*a*c^4*d^2*n - 120*(d*x + c)^n*b *c^6 - 180*(d*x + c)^n*a*c^4*d^2)/(d^6*n^6 + 21*d^6*n^5 + 175*d^6*n^4 + 73 5*d^6*n^3 + 1624*d^6*n^2 + 1764*d^6*n + 720*d^6)
Time = 9.18 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.84 \[ \int x^3 (c+d x)^n \left (a+b x^2\right ) \, dx={\left (c+d\,x\right )}^n\,\left (\frac {b\,x^6\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720}-\frac {6\,c^4\,\left (20\,b\,c^2+a\,d^2\,n^2+11\,a\,d^2\,n+30\,a\,d^2\right )}{d^6\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {x^4\,\left (-5\,b\,c^2\,n+a\,d^2\,n^2+11\,a\,d^2\,n+30\,a\,d^2\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{d^2\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {6\,c^3\,n\,x\,\left (20\,b\,c^2+a\,d^2\,n^2+11\,a\,d^2\,n+30\,a\,d^2\right )}{d^5\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {c\,n\,x^3\,\left (n^2+3\,n+2\right )\,\left (20\,b\,c^2+a\,d^2\,n^2+11\,a\,d^2\,n+30\,a\,d^2\right )}{d^3\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {b\,c\,n\,x^5\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )}{d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {3\,c^2\,n\,x^2\,\left (n+1\right )\,\left (20\,b\,c^2+a\,d^2\,n^2+11\,a\,d^2\,n+30\,a\,d^2\right )}{d^4\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}\right ) \] Input:
int(x^3*(a + b*x^2)*(c + d*x)^n,x)
Output:
(c + d*x)^n*((b*x^6*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(1764 *n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720) - (6*c^4*(30*a*d^2 + 20*b*c^2 + a*d^2*n^2 + 11*a*d^2*n))/(d^6*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (x^4*(30*a*d^2 + a*d^2*n^2 + 11*a*d^2*n - 5*b*c^2*n)*(11*n + 6*n^2 + n^3 + 6))/(d^2*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (6*c^3*n*x*(30*a*d^2 + 20*b*c^2 + a*d^2* n^2 + 11*a*d^2*n))/(d^5*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (c*n*x^3*(3*n + n^2 + 2)*(30*a*d^2 + 20*b*c^2 + a*d^2*n^2 + 11*a*d^2*n))/(d^3*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (b*c*n*x^5*(50*n + 35*n^2 + 10*n^3 + n^4 + 24))/(d*(1764*n + 1624* n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (3*c^2*n*x^2*(n + 1)*(30* a*d^2 + 20*b*c^2 + a*d^2*n^2 + 11*a*d^2*n))/(d^4*(1764*n + 1624*n^2 + 735* n^3 + 175*n^4 + 21*n^5 + n^6 + 720)))
Time = 0.22 (sec) , antiderivative size = 568, normalized size of antiderivative = 3.32 \[ \int x^3 (c+d x)^n \left (a+b x^2\right ) \, dx=\frac {\left (d x +c \right )^{n} \left (b \,d^{6} n^{5} x^{6}+b c \,d^{5} n^{5} x^{5}+15 b \,d^{6} n^{4} x^{6}+a \,d^{6} n^{5} x^{4}+10 b c \,d^{5} n^{4} x^{5}+85 b \,d^{6} n^{3} x^{6}+a c \,d^{5} n^{5} x^{3}+17 a \,d^{6} n^{4} x^{4}-5 b \,c^{2} d^{4} n^{4} x^{4}+35 b c \,d^{5} n^{3} x^{5}+225 b \,d^{6} n^{2} x^{6}+14 a c \,d^{5} n^{4} x^{3}+107 a \,d^{6} n^{3} x^{4}-30 b \,c^{2} d^{4} n^{3} x^{4}+50 b c \,d^{5} n^{2} x^{5}+274 b \,d^{6} n \,x^{6}-3 a \,c^{2} d^{4} n^{4} x^{2}+65 a c \,d^{5} n^{3} x^{3}+307 a \,d^{6} n^{2} x^{4}+20 b \,c^{3} d^{3} n^{3} x^{3}-55 b \,c^{2} d^{4} n^{2} x^{4}+24 b c \,d^{5} n \,x^{5}+120 b \,d^{6} x^{6}-36 a \,c^{2} d^{4} n^{3} x^{2}+112 a c \,d^{5} n^{2} x^{3}+396 a \,d^{6} n \,x^{4}+60 b \,c^{3} d^{3} n^{2} x^{3}-30 b \,c^{2} d^{4} n \,x^{4}+6 a \,c^{3} d^{3} n^{3} x -123 a \,c^{2} d^{4} n^{2} x^{2}+60 a c \,d^{5} n \,x^{3}+180 a \,d^{6} x^{4}-60 b \,c^{4} d^{2} n^{2} x^{2}+40 b \,c^{3} d^{3} n \,x^{3}+66 a \,c^{3} d^{3} n^{2} x -90 a \,c^{2} d^{4} n \,x^{2}-60 b \,c^{4} d^{2} n \,x^{2}-6 a \,c^{4} d^{2} n^{2}+180 a \,c^{3} d^{3} n x +120 b \,c^{5} d n x -66 a \,c^{4} d^{2} n -180 a \,c^{4} d^{2}-120 b \,c^{6}\right )}{d^{6} \left (n^{6}+21 n^{5}+175 n^{4}+735 n^{3}+1624 n^{2}+1764 n +720\right )} \] Input:
int(x^3*(d*x+c)^n*(b*x^2+a),x)
Output:
((c + d*x)**n*( - 6*a*c**4*d**2*n**2 - 66*a*c**4*d**2*n - 180*a*c**4*d**2 + 6*a*c**3*d**3*n**3*x + 66*a*c**3*d**3*n**2*x + 180*a*c**3*d**3*n*x - 3*a *c**2*d**4*n**4*x**2 - 36*a*c**2*d**4*n**3*x**2 - 123*a*c**2*d**4*n**2*x** 2 - 90*a*c**2*d**4*n*x**2 + a*c*d**5*n**5*x**3 + 14*a*c*d**5*n**4*x**3 + 6 5*a*c*d**5*n**3*x**3 + 112*a*c*d**5*n**2*x**3 + 60*a*c*d**5*n*x**3 + a*d** 6*n**5*x**4 + 17*a*d**6*n**4*x**4 + 107*a*d**6*n**3*x**4 + 307*a*d**6*n**2 *x**4 + 396*a*d**6*n*x**4 + 180*a*d**6*x**4 - 120*b*c**6 + 120*b*c**5*d*n* x - 60*b*c**4*d**2*n**2*x**2 - 60*b*c**4*d**2*n*x**2 + 20*b*c**3*d**3*n**3 *x**3 + 60*b*c**3*d**3*n**2*x**3 + 40*b*c**3*d**3*n*x**3 - 5*b*c**2*d**4*n **4*x**4 - 30*b*c**2*d**4*n**3*x**4 - 55*b*c**2*d**4*n**2*x**4 - 30*b*c**2 *d**4*n*x**4 + b*c*d**5*n**5*x**5 + 10*b*c*d**5*n**4*x**5 + 35*b*c*d**5*n* *3*x**5 + 50*b*c*d**5*n**2*x**5 + 24*b*c*d**5*n*x**5 + b*d**6*n**5*x**6 + 15*b*d**6*n**4*x**6 + 85*b*d**6*n**3*x**6 + 225*b*d**6*n**2*x**6 + 274*b*d **6*n*x**6 + 120*b*d**6*x**6))/(d**6*(n**6 + 21*n**5 + 175*n**4 + 735*n**3 + 1624*n**2 + 1764*n + 720))