Integrand size = 39, antiderivative size = 206 \[ \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {2 c d (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^2 (2+p) (3+p)}+\frac {2 c^2 d^2 (d+e x)^{-2 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^3 (1+p) (2+p) (3+p)}+\frac {(d+e x)^{-2 (2+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)} \]
2*c*d*(e*x+d)^(-3-2*p)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(p+1)/(-a*e^2+c*d ^2)^2/(2+p)/(3+p)+2*c^2*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(p+1)/(-a*e^ 2+c*d^2)^3/(p+1)/(2+p)/(3+p)/((e*x+d)^(2+2*p))+(a*d*e+(a*e^2+c*d^2)*x+c*d* e*x^2)^(p+1)/(-a*e^2+c*d^2)/(3+p)/((e*x+d)^(4+2*p))
Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.64 \[ \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(d+e x)^{-2 (2+p)} ((a e+c d x) (d+e x))^{1+p} \left (a^2 e^4 \left (2+3 p+p^2\right )-2 a c d e^2 (1+p) (d (3+p)+e x)+c^2 d^2 \left (d^2 \left (6+5 p+p^2\right )+2 d e (3+p) x+2 e^2 x^2\right )\right )}{\left (c d^2-a e^2\right )^3 (1+p) (2+p) (3+p)} \]
(((a*e + c*d*x)*(d + e*x))^(1 + p)*(a^2*e^4*(2 + 3*p + p^2) - 2*a*c*d*e^2* (1 + p)*(d*(3 + p) + e*x) + c^2*d^2*(d^2*(6 + 5*p + p^2) + 2*d*e*(3 + p)*x + 2*e^2*x^2)))/((c*d^2 - a*e^2)^3*(1 + p)*(2 + p)*(3 + p)*(d + e*x)^(2*(2 + p)))
Time = 0.35 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1129, 1129, 1123}
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.
| \(\displaystyle \int (d+e x)^{-2 p-4} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, dx\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {2 c d \int (d+e x)^{-2 p-3} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^pdx}{(p+3) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {2 c d \left (\frac {c d \int (d+e x)^{-2 (p+1)} \left (c d e x^2+\left (c d^2+a e^2\right ) x+a d e\right )^pdx}{(p+2) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) \left (c d^2-a e^2\right )}\right )}{(p+3) \left (c d^2-a e^2\right )}+\frac {(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )}+\frac {2 c d \left (\frac {(d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) \left (c d^2-a e^2\right )}+\frac {c d (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) \left (c d^2-a e^2\right )^2}\right )}{(p+3) \left (c d^2-a e^2\right )}\) |
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p)/((c*d^2 - a*e^2)*(3 + p)*( d + e*x)^(2*(2 + p))) + (2*c*d*(((d + e*x)^(-3 - 2*p)*(a*d*e + (c*d^2 + a* e^2)*x + c*d*e*x^2)^(1 + p))/((c*d^2 - a*e^2)*(2 + p)) + (c*d*(a*d*e + (c* d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p))/((c*d^2 - a*e^2)^2*(1 + p)*(2 + p)*(d + e*x)^(2*(1 + p)))))/((c*d^2 - a*e^2)*(3 + p))
3.22.3.3.1 Defintions of rubi rules used
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b *e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + 2*p + 2, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) )) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d , e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 2], 0]
Time = 3.54 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.85
| method | result | size |
| gosper | \(-\frac {\left (e x +d \right )^{-3-2 p} \left (c d x +a e \right ) \left (c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e \right )^{p} \left (a^{2} e^{4} p^{2}-2 a c \,d^{2} e^{2} p^{2}-2 a c d \,e^{3} p x +c^{2} d^{4} p^{2}+2 c^{2} d^{3} e p x +2 x^{2} c^{2} d^{2} e^{2}+3 a^{2} e^{4} p -8 a c \,d^{2} e^{2} p -2 x a c d \,e^{3}+5 c^{2} d^{4} p +6 x \,c^{2} d^{3} e +2 a^{2} e^{4}-6 a c \,d^{2} e^{2}+6 c^{2} d^{4}\right )}{a^{3} e^{6} p^{3}-3 a^{2} c \,d^{2} e^{4} p^{3}+3 a \,c^{2} d^{4} e^{2} p^{3}-c^{3} d^{6} p^{3}+6 a^{3} e^{6} p^{2}-18 a^{2} c \,d^{2} e^{4} p^{2}+18 a \,c^{2} d^{4} e^{2} p^{2}-6 c^{3} d^{6} p^{2}+11 a^{3} e^{6} p -33 a^{2} c \,d^{2} e^{4} p +33 a \,c^{2} d^{4} e^{2} p -11 c^{3} d^{6} p +6 e^{6} a^{3}-18 d^{2} e^{4} a^{2} c +18 d^{4} e^{2} c^{2} a -6 c^{3} d^{6}}\) | \(381\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1846\) |
-(e*x+d)^(-3-2*p)*(c*d*x+a*e)*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^p*(a^2*e^4 *p^2-2*a*c*d^2*e^2*p^2-2*a*c*d*e^3*p*x+c^2*d^4*p^2+2*c^2*d^3*e*p*x+2*c^2*d ^2*e^2*x^2+3*a^2*e^4*p-8*a*c*d^2*e^2*p-2*a*c*d*e^3*x+5*c^2*d^4*p+6*c^2*d^3 *e*x+2*a^2*e^4-6*a*c*d^2*e^2+6*c^2*d^4)/(a^3*e^6*p^3-3*a^2*c*d^2*e^4*p^3+3 *a*c^2*d^4*e^2*p^3-c^3*d^6*p^3+6*a^3*e^6*p^2-18*a^2*c*d^2*e^4*p^2+18*a*c^2 *d^4*e^2*p^2-6*c^3*d^6*p^2+11*a^3*e^6*p-33*a^2*c*d^2*e^4*p+33*a*c^2*d^4*e^ 2*p-11*c^3*d^6*p+6*a^3*e^6-18*a^2*c*d^2*e^4+18*a*c^2*d^4*e^2-6*c^3*d^6)
Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (210) = 420\).
Time = 0.34 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.82 \[ \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {{\left (2 \, c^{3} d^{3} e^{3} x^{4} + 6 \, a c^{2} d^{5} e - 6 \, a^{2} c d^{3} e^{3} + 2 \, a^{3} d e^{5} + 2 \, {\left (4 \, c^{3} d^{4} e^{2} + {\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} p\right )} x^{3} + {\left (a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} p^{2} + {\left (12 \, c^{3} d^{5} e + {\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p^{2} + {\left (7 \, c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p\right )} x^{2} + {\left (5 \, a c^{2} d^{5} e - 8 \, a^{2} c d^{3} e^{3} + 3 \, a^{3} d e^{5}\right )} p + {\left (6 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} + {\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} p^{2} + {\left (5 \, c^{3} d^{6} - a c^{2} d^{4} e^{2} - 7 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6}\right )} p\right )} x\right )} {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4}}{6 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 6 \, a^{3} e^{6} + {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{3} + 6 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{2} + 11 \, {\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p} \]
(2*c^3*d^3*e^3*x^4 + 6*a*c^2*d^5*e - 6*a^2*c*d^3*e^3 + 2*a^3*d*e^5 + 2*(4* c^3*d^4*e^2 + (c^3*d^4*e^2 - a*c^2*d^2*e^4)*p)*x^3 + (a*c^2*d^5*e - 2*a^2* c*d^3*e^3 + a^3*d*e^5)*p^2 + (12*c^3*d^5*e + (c^3*d^5*e - 2*a*c^2*d^3*e^3 + a^2*c*d*e^5)*p^2 + (7*c^3*d^5*e - 8*a*c^2*d^3*e^3 + a^2*c*d*e^5)*p)*x^2 + (5*a*c^2*d^5*e - 8*a^2*c*d^3*e^3 + 3*a^3*d*e^5)*p + (6*c^3*d^6 + 6*a*c^2 *d^4*e^2 - 6*a^2*c*d^2*e^4 + 2*a^3*e^6 + (c^3*d^6 - a*c^2*d^4*e^2 - a^2*c* d^2*e^4 + a^3*e^6)*p^2 + (5*c^3*d^6 - a*c^2*d^4*e^2 - 7*a^2*c*d^2*e^4 + 3* a^3*e^6)*p)*x)*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4)/(6*c^3*d^6 - 18*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 - 6*a^3*e^6 + (c^3*d^ 6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*p^3 + 6*(c^3*d^6 - 3*a*c^ 2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*p^2 + 11*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*p)
\[ \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int \left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{p} \left (d + e x\right )^{- 2 p - 4}\, dx \]
\[ \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4} \,d x } \]
\[ \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\int { {\left (c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x\right )}^{p} {\left (e x + d\right )}^{-2 \, p - 4} \,d x } \]
Time = 10.48 (sec) , antiderivative size = 595, normalized size of antiderivative = 2.89 \[ \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=-{\left (c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e\right )}^p\,\left (\frac {x\,\left (a^3\,e^6\,p^2+3\,a^3\,e^6\,p+2\,a^3\,e^6-a^2\,c\,d^2\,e^4\,p^2-7\,a^2\,c\,d^2\,e^4\,p-6\,a^2\,c\,d^2\,e^4-a\,c^2\,d^4\,e^2\,p^2-a\,c^2\,d^4\,e^2\,p+6\,a\,c^2\,d^4\,e^2+c^3\,d^6\,p^2+5\,c^3\,d^6\,p+6\,c^3\,d^6\right )}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {2\,c^3\,d^3\,e^3\,x^4}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {a\,d\,e\,\left (a^2\,e^4\,p^2+3\,a^2\,e^4\,p+2\,a^2\,e^4-2\,a\,c\,d^2\,e^2\,p^2-8\,a\,c\,d^2\,e^2\,p-6\,a\,c\,d^2\,e^2+c^2\,d^4\,p^2+5\,c^2\,d^4\,p+6\,c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {c\,d\,e\,x^2\,\left (a^2\,e^4\,p^2+a^2\,e^4\,p-2\,a\,c\,d^2\,e^2\,p^2-8\,a\,c\,d^2\,e^2\,p+c^2\,d^4\,p^2+7\,c^2\,d^4\,p+12\,c^2\,d^4\right )}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}+\frac {2\,c^2\,d^2\,e^2\,x^3\,\left (4\,c\,d^2-a\,e^2\,p+c\,d^2\,p\right )}{{\left (a\,e^2-c\,d^2\right )}^3\,{\left (d+e\,x\right )}^{2\,p+4}\,\left (p^3+6\,p^2+11\,p+6\right )}\right ) \]
-(x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^p*((x*(2*a^3*e^6 + 6*c^3*d^6 + 3* a^3*e^6*p + 5*c^3*d^6*p + a^3*e^6*p^2 + c^3*d^6*p^2 + 6*a*c^2*d^4*e^2 - 6* a^2*c*d^2*e^4 - a*c^2*d^4*e^2*p - 7*a^2*c*d^2*e^4*p - a*c^2*d^4*e^2*p^2 - a^2*c*d^2*e^4*p^2))/((a*e^2 - c*d^2)^3*(d + e*x)^(2*p + 4)*(11*p + 6*p^2 + p^3 + 6)) + (2*c^3*d^3*e^3*x^4)/((a*e^2 - c*d^2)^3*(d + e*x)^(2*p + 4)*(1 1*p + 6*p^2 + p^3 + 6)) + (a*d*e*(2*a^2*e^4 + 6*c^2*d^4 + 3*a^2*e^4*p + 5* c^2*d^4*p + a^2*e^4*p^2 + c^2*d^4*p^2 - 6*a*c*d^2*e^2 - 8*a*c*d^2*e^2*p - 2*a*c*d^2*e^2*p^2))/((a*e^2 - c*d^2)^3*(d + e*x)^(2*p + 4)*(11*p + 6*p^2 + p^3 + 6)) + (c*d*e*x^2*(12*c^2*d^4 + a^2*e^4*p + 7*c^2*d^4*p + a^2*e^4*p^ 2 + c^2*d^4*p^2 - 8*a*c*d^2*e^2*p - 2*a*c*d^2*e^2*p^2))/((a*e^2 - c*d^2)^3 *(d + e*x)^(2*p + 4)*(11*p + 6*p^2 + p^3 + 6)) + (2*c^2*d^2*e^2*x^3*(4*c*d ^2 - a*e^2*p + c*d^2*p))/((a*e^2 - c*d^2)^3*(d + e*x)^(2*p + 4)*(11*p + 6* p^2 + p^3 + 6)))