Integrand size = 39, antiderivative size = 233 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {32 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt {d+e x}}+\frac {4 \left (c d^2-a e^2\right ) \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{9 c d} \]
[Out]
Time = 0.12 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662} \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {32 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{315 c^4 d^4 (d+e x)^{3/2}}+\frac {16 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{105 c^3 d^3 \sqrt {d+e x}}+\frac {4 \sqrt {d+e x} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{21 c^2 d^2}+\frac {2 (d+e x)^{3/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{9 c d} \]
[In]
[Out]
Rule 662
Rule 670
Rubi steps \begin{align*} \text {integral}& = \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}+\frac {\left (2 \left (d^2-\frac {a e^2}{c}\right )\right ) \int (d+e x)^{3/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{3 d} \\ & = \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}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )^2\right ) \int \sqrt {d+e x} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx}{21 d^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}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^3\right ) \int \frac {\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt {d+e x}} \, dx}{105 d^3} \\ & = \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} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.56 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (-16 a^3 e^6+24 a^2 c d e^4 (3 d+e x)-6 a c^2 d^2 e^2 \left (21 d^2+18 d e x+5 e^2 x^2\right )+c^3 d^3 \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )\right )}{315 c^4 d^4 (d+e x)^{3/2}} \]
[In]
[Out]
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 | \(-\frac {2 \left (c d x +a e \right ) \left (-35 x^{3} c^{3} d^{3} e^{3}+30 x^{2} a \,c^{2} d^{2} e^{4}-135 x^{2} c^{3} d^{4} e^{2}-24 x \,a^{2} c d \,e^{5}+108 x a \,c^{2} d^{3} e^{3}-189 x \,c^{3} d^{5} e +16 e^{6} a^{3}-72 d^{2} e^{4} a^{2} c +126 d^{4} e^{2} c^{2} a -105 c^{3} d^{6}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{315 c^{4} d^{4} \sqrt {e x +d}}\) | \(168\) |
[In]
[Out]
none
Time = 0.37 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.99 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \, {\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]
[In]
[Out]
Timed out. \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Timed out} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.91 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (35 \, c^{4} d^{4} e^{3} x^{4} + 105 \, a c^{3} d^{6} e - 126 \, a^{2} c^{2} d^{4} e^{3} + 72 \, a^{3} c d^{2} e^{5} - 16 \, a^{4} e^{7} + 5 \, {\left (27 \, c^{4} d^{5} e^{2} + a c^{3} d^{3} e^{4}\right )} x^{3} + 3 \, {\left (63 \, c^{4} d^{6} e + 9 \, a c^{3} d^{4} e^{3} - 2 \, a^{2} c^{2} d^{2} e^{5}\right )} x^{2} + {\left (105 \, c^{4} d^{7} + 63 \, a c^{3} d^{5} e^{2} - 36 \, a^{2} c^{2} d^{3} e^{4} + 8 \, a^{3} c d e^{6}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{315 \, {\left (c^{4} d^{4} e x + c^{4} d^{5}\right )}} \]
[In]
[Out]
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} \]
[In]
[Out]
Time = 10.30 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.10 \[ \int (d+e x)^{5/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e^2\,x^4\,\sqrt {d+e\,x}}{9}-\frac {\sqrt {d+e\,x}\,\left (32\,a^4\,e^7-144\,a^3\,c\,d^2\,e^5+252\,a^2\,c^2\,d^4\,e^3-210\,a\,c^3\,d^6\,e\right )}{315\,c^4\,d^4\,e}+\frac {2\,x^2\,\sqrt {d+e\,x}\,\left (-2\,a^2\,e^4+9\,a\,c\,d^2\,e^2+63\,c^2\,d^4\right )}{105\,c^2\,d^2}+\frac {x\,\sqrt {d+e\,x}\,\left (16\,a^3\,c\,d\,e^6-72\,a^2\,c^2\,d^3\,e^4+126\,a\,c^3\,d^5\,e^2+210\,c^4\,d^7\right )}{315\,c^4\,d^4\,e}+\frac {2\,e\,x^3\,\left (27\,c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}}{63\,c\,d}\right )}{x+\frac {d}{e}} \]
[In]
[Out]