Integrand size = 39, antiderivative size = 233 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt {d+e x}}+\frac {4 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 c d} \]
32/315*(-a*e^2+c*d^2)^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^4/d^4/(e *x+d)^(3/2)+2/9*(e*x+d)^(3/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c/d+ 16/105*(-a*e^2+c*d^2)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/c^3/d^3/(e *x+d)^(1/2)+4/21*(-a*e^2+c*d^2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*(e *x+d)^(1/2)/c^2/d^2
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.56 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (-16 a^3 e^6+24 a^2 c d e^4 (3 d+e x)-6 a c^2 d^2 e^2 \left (21 d^2+18 d e x+5 e^2 x^2\right )+c^3 d^3 \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )\right )}{315 c^4 d^4 (d+e x)^{3/2}} \]
(2*((a*e + c*d*x)*(d + e*x))^(3/2)*(-16*a^3*e^6 + 24*a^2*c*d*e^4*(3*d + e* x) - 6*a*c^2*d^2*e^2*(21*d^2 + 18*d*e*x + 5*e^2*x^2) + c^3*d^3*(105*d^3 + 189*d^2*e*x + 135*d*e^2*x^2 + 35*e^3*x^3)))/(315*c^4*d^4*(d + e*x)^(3/2))
Time = 0.40 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1128, 1128, 1128, 1122}
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.
| \(\displaystyle \int (d+e x)^{5/2} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \, dx\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int (d+e x)^{3/2} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{3 d}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \int \sqrt {d+e x} \sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}dx}{7 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d}\right )}{3 d}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d}\) |
| \(\Big \downarrow \) 1128 |
| \(\displaystyle \frac {2 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \int \frac {\sqrt {c d e x^2+\left (c d^2+a e^2\right ) x+a d e}}{\sqrt {d+e x}}dx}{5 d}+\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}}\right )}{7 d}+\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d}\right )}{3 d}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d}\) |
| \(\Big \downarrow \) 1122 |
| \(\displaystyle \frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d}+\frac {2 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \sqrt {d+e x} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{7 c d}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (\frac {2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d \sqrt {d+e x}}+\frac {4 \left (d^2-\frac {a e^2}{c}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{15 c d^2 (d+e x)^{3/2}}\right )}{7 d}\right )}{3 d}\) |
(2*(d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(9*c*d) + (2*(d^2 - (a*e^2)/c)*((2*Sqrt[d + e*x]*(a*d*e + (c*d^2 + a*e^2)*x + c*d* e*x^2)^(3/2))/(7*c*d) + (4*(d^2 - (a*e^2)/c)*((4*(d^2 - (a*e^2)/c)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(15*c*d^2*(d + e*x)^(3/2)) + (2*(a *d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*c*d*Sqrt[d + e*x])))/(7*d) ))/(3*d)
3.21.28.3.1 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 - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[m + p, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1))) 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] && IGtQ[Simplify[m + p], 0]
Time = 2.46 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.68
| method | result | size |
| default | \(-\frac {2 \left (c d x +a e \right ) \left (-35 x^{3} c^{3} d^{3} e^{3}+30 x^{2} a \,c^{2} d^{2} e^{4}-135 x^{2} c^{3} d^{4} e^{2}-24 x \,a^{2} c d \,e^{5}+108 x a \,c^{2} d^{3} e^{3}-189 x \,c^{3} d^{5} e +16 e^{6} a^{3}-72 d^{2} e^{4} a^{2} c +126 d^{4} e^{2} c^{2} a -105 c^{3} d^{6}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{315 c^{4} d^{4} \sqrt {e x +d}}\) | \(158\) |
| gosper | \(-\frac {2 \left (c d x +a e \right ) \left (-35 x^{3} c^{3} d^{3} e^{3}+30 x^{2} a \,c^{2} d^{2} e^{4}-135 x^{2} c^{3} d^{4} e^{2}-24 x \,a^{2} c d \,e^{5}+108 x a \,c^{2} d^{3} e^{3}-189 x \,c^{3} d^{5} e +16 e^{6} a^{3}-72 d^{2} e^{4} a^{2} c +126 d^{4} e^{2} c^{2} a -105 c^{3} d^{6}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 c^{4} d^{4} \sqrt {e x +d}}\) | \(168\) |
-2/315*(c*d*x+a*e)*(-35*c^3*d^3*e^3*x^3+30*a*c^2*d^2*e^4*x^2-135*c^3*d^4*e ^2*x^2-24*a^2*c*d*e^5*x+108*a*c^2*d^3*e^3*x-189*c^3*d^5*e*x+16*a^3*e^6-72* a^2*c*d^2*e^4+126*a*c^2*d^4*e^2-105*c^3*d^6)*((c*d*x+a*e)*(e*x+d))^(1/2)/c ^4/d^4/(e*x+d)^(1/2)
Time = 0.37 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.99 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \, {\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]
2/315*(35*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 126*a^2*c^2*d^4*e^3 + 72*a^3 *c*d^2*e^5 - 16*a^4*e^7 + 5*(27*c^4*d^5*e^2 + a*c^3*d^3*e^4)*x^3 + 3*(63*c ^4*d^6*e + 9*a*c^3*d^4*e^3 - 2*a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 + 63*a* c^3*d^5*e^2 - 36*a^2*c^2*d^3*e^4 + 8*a^3*c*d*e^6)*x)*sqrt(c*d*e*x^2 + a*d* e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)/(c^4*d^4*e*x + c^4*d^5)
Timed out. \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.91 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \, {\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]
2/315*(35*c^4*d^4*e^3*x^4 + 105*a*c^3*d^6*e - 126*a^2*c^2*d^4*e^3 + 72*a^3 *c*d^2*e^5 - 16*a^4*e^7 + 5*(27*c^4*d^5*e^2 + a*c^3*d^3*e^4)*x^3 + 3*(63*c ^4*d^6*e + 9*a*c^3*d^4*e^3 - 2*a^2*c^2*d^2*e^5)*x^2 + (105*c^4*d^7 + 63*a* c^3*d^5*e^2 - 36*a^2*c^2*d^3*e^4 + 8*a^3*c*d*e^6)*x)*sqrt(c*d*x + a*e)*(e* x + d)/(c^4*d^4*e*x + c^4*d^5)
Leaf count of result is larger than twice the leaf count of optimal. 750 vs. \(2 (209) = 418\).
Time = 0.30 (sec) , antiderivative size = 750, normalized size of antiderivative = 3.22 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (\frac {105 \, d^{3} {\left (\frac {\sqrt {-c d^{2} e + a e^{3}} c d^{2} - \sqrt {-c d^{2} e + a e^{3}} a e^{2}}{c d} + \frac {{\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}}}{c d e}\right )} {\left | e \right |}}{e^{2}} + 9 \, d {\left (\frac {15 \, \sqrt {-c d^{2} e + a e^{3}} c^{3} d^{6} - 3 \, \sqrt {-c d^{2} e + a e^{3}} a c^{2} d^{4} e^{2} - 4 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c d^{2} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} e^{6}}{c^{3} d^{3} e^{2}} + \frac {35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{2} e^{6} - 42 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a e^{3} + 15 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}}}{c^{3} d^{3} e^{5}}\right )} {\left | e \right |} - e {\left (\frac {35 \, \sqrt {-c d^{2} e + a e^{3}} c^{4} d^{8} - 5 \, \sqrt {-c d^{2} e + a e^{3}} a c^{3} d^{6} e^{2} - 6 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{2} d^{4} e^{4} - 8 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c d^{2} e^{6} - 16 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} e^{8}}{c^{4} d^{4} e^{3}} + \frac {105 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{3} e^{9} - 189 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{2} e^{6} + 135 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a e^{3} - 35 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}}}{c^{4} d^{4} e^{7}}\right )} {\left | e \right |} - \frac {63 \, d^{2} {\left (\frac {3 \, \sqrt {-c d^{2} e + a e^{3}} c^{2} d^{4} - \sqrt {-c d^{2} e + a e^{3}} a c d^{2} e^{2} - 2 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} e^{4}}{c^{2} d^{2}} + \frac {5 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a e^{3} - 3 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}}}{c^{2} d^{2} e^{2}}\right )} {\left | e \right |}}{e^{2}}\right )}}{315 \, e} \]
2/315*(105*d^3*((sqrt(-c*d^2*e + a*e^3)*c*d^2 - sqrt(-c*d^2*e + a*e^3)*a*e ^2)/(c*d) + ((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)/(c*d*e))*abs(e)/e^2 + 9*d*((15*sqrt(-c*d^2*e + a*e^3)*c^3*d^6 - 3*sqrt(-c*d^2*e + a*e^3)*a*c^2 *d^4*e^2 - 4*sqrt(-c*d^2*e + a*e^3)*a^2*c*d^2*e^4 - 8*sqrt(-c*d^2*e + a*e^ 3)*a^3*e^6)/(c^3*d^3*e^2) + (35*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)* a^2*e^6 - 42*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2)*a*e^3 + 15*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2))/(c^3*d^3*e^5))*abs(e) - e*((35*sqrt(-c* d^2*e + a*e^3)*c^4*d^8 - 5*sqrt(-c*d^2*e + a*e^3)*a*c^3*d^6*e^2 - 6*sqrt(- c*d^2*e + a*e^3)*a^2*c^2*d^4*e^4 - 8*sqrt(-c*d^2*e + a*e^3)*a^3*c*d^2*e^6 - 16*sqrt(-c*d^2*e + a*e^3)*a^4*e^8)/(c^4*d^4*e^3) + (105*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(3/2)*a^3*e^9 - 189*((e*x + d)*c*d*e - c*d^2*e + a*e^3 )^(5/2)*a^2*e^6 + 135*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(7/2)*a*e^3 - 35 *((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(9/2))/(c^4*d^4*e^7))*abs(e) - 63*d^2 *((3*sqrt(-c*d^2*e + a*e^3)*c^2*d^4 - sqrt(-c*d^2*e + a*e^3)*a*c*d^2*e^2 - 2*sqrt(-c*d^2*e + a*e^3)*a^2*e^4)/(c^2*d^2) + (5*((e*x + d)*c*d*e - c*d^2 *e + a*e^3)^(3/2)*a*e^3 - 3*((e*x + d)*c*d*e - c*d^2*e + a*e^3)^(5/2))/(c^ 2*d^2*e^2))*abs(e)/e^2)/e
Time = 10.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.10 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e^2\,x^4\,\sqrt {d+e\,x}}{9}-\frac {\sqrt {d+e\,x}\,\left (32\,a^4\,e^7-144\,a^3\,c\,d^2\,e^5+252\,a^2\,c^2\,d^4\,e^3-210\,a\,c^3\,d^6\,e\right )}{315\,c^4\,d^4\,e}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (-2\,a^2\,e^4+9\,a\,c\,d^2\,e^2+63\,c^2\,d^4\right )}{105\,c^2\,d^2}+\frac {x\,\sqrt {d+e\,x}\,\left (16\,a^3\,c\,d\,e^6-72\,a^2\,c^2\,d^3\,e^4+126\,a\,c^3\,d^5\,e^2+210\,c^4\,d^7\right )}{315\,c^4\,d^4\,e}+\frac {2\,e\,x^3\,\left (27\,c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}}{63\,c\,d}\right )}{x+\frac {d}{e}} \]
((x*(a*e^2 + c*d^2) + a*d*e + c*d*e*x^2)^(1/2)*((2*e^2*x^4*(d + e*x)^(1/2) )/9 - ((d + e*x)^(1/2)*(32*a^4*e^7 - 144*a^3*c*d^2*e^5 + 252*a^2*c^2*d^4*e ^3 - 210*a*c^3*d^6*e))/(315*c^4*d^4*e) + (2*x^2*(d + e*x)^(1/2)*(63*c^2*d^ 4 - 2*a^2*e^4 + 9*a*c*d^2*e^2))/(105*c^2*d^2) + (x*(d + e*x)^(1/2)*(210*c^ 4*d^7 + 126*a*c^3*d^5*e^2 - 72*a^2*c^2*d^3*e^4 + 16*a^3*c*d*e^6))/(315*c^4 *d^4*e) + (2*e*x^3*(a*e^2 + 27*c*d^2)*(d + e*x)^(1/2))/(63*c*d)))/(x + d/e )