Integrand size = 39, antiderivative size = 295 \[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \left (c d^2-a e^2\right )^2 \sqrt {d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac {16 \left (c d^2-a e^2\right ) (d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d} \]
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Time = 0.17 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {670, 662} \[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {256 \left (c d^2-a e^2\right )^4 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \left (c d^2-a e^2\right )^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \sqrt {d+e x} \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{231 c^3 d^3}+\frac {16 (d+e x)^{3/2} \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{11 c d} \]
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Rule 662
Rule 670
Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac {\left (8 \left (d^2-\frac {a e^2}{c}\right )\right ) \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}{11 d} \\ & = \frac {16 \left (c d^2-a e^2\right ) (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}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac {\left (16 \left (d^2-\frac {a e^2}{c}\right )^2\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}{33 d^2} \\ & = \frac {32 \left (c d^2-a e^2\right )^2 \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}}{231 c^3 d^3}+\frac {16 \left (c d^2-a e^2\right ) (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}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac {\left (64 \left (d^2-\frac {a e^2}{c}\right )^3\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}{231 d^3} \\ & = \frac {128 \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}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \left (c d^2-a e^2\right )^2 \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}}{231 c^3 d^3}+\frac {16 \left (c d^2-a e^2\right ) (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}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d}+\frac {\left (128 \left (d^2-\frac {a e^2}{c}\right )^4\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}{1155 d^4} \\ & = \frac {256 \left (c d^2-a e^2\right )^4 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{3465 c^5 d^5 (d+e x)^{3/2}}+\frac {128 \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}}{1155 c^4 d^4 \sqrt {d+e x}}+\frac {32 \left (c d^2-a e^2\right )^2 \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}}{231 c^3 d^3}+\frac {16 \left (c d^2-a e^2\right ) (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}}{99 c^2 d^2}+\frac {2 (d+e x)^{5/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{11 c d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.63 \[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 ((a e+c d x) (d+e x))^{3/2} \left (128 a^4 e^8-64 a^3 c d e^6 (11 d+3 e x)+48 a^2 c^2 d^2 e^4 \left (33 d^2+22 d e x+5 e^2 x^2\right )-8 a c^3 d^3 e^2 \left (231 d^3+297 d^2 e x+165 d e^2 x^2+35 e^3 x^3\right )+c^4 d^4 \left (1155 d^4+2772 d^3 e x+2970 d^2 e^2 x^2+1540 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 c^5 d^5 (d+e x)^{3/2}} \]
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Time = 2.90 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.79
method | result | size |
default | \(\frac {2 \left (c d x +a e \right ) \left (315 c^{4} d^{4} e^{4} x^{4}-280 a \,c^{3} d^{3} e^{5} x^{3}+1540 c^{4} d^{5} e^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{6} x^{2}-1320 a \,c^{3} d^{4} e^{4} x^{2}+2970 c^{4} d^{6} e^{2} x^{2}-192 a^{3} c d \,e^{7} x +1056 a^{2} c^{2} d^{3} e^{5} x -2376 a \,c^{3} d^{5} e^{3} x +2772 c^{4} d^{7} e x +128 a^{4} e^{8}-704 a^{3} c \,d^{2} e^{6}+1584 a^{2} c^{2} d^{4} e^{4}-1848 a \,c^{3} d^{6} e^{2}+1155 c^{4} d^{8}\right ) \sqrt {\left (c d x +a e \right ) \left (e x +d \right )}}{3465 c^{5} d^{5} \sqrt {e x +d}}\) | \(233\) |
gosper | \(\frac {2 \left (c d x +a e \right ) \left (315 c^{4} d^{4} e^{4} x^{4}-280 a \,c^{3} d^{3} e^{5} x^{3}+1540 c^{4} d^{5} e^{3} x^{3}+240 a^{2} c^{2} d^{2} e^{6} x^{2}-1320 a \,c^{3} d^{4} e^{4} x^{2}+2970 c^{4} d^{6} e^{2} x^{2}-192 a^{3} c d \,e^{7} x +1056 a^{2} c^{2} d^{3} e^{5} x -2376 a \,c^{3} d^{5} e^{3} x +2772 c^{4} d^{7} e x +128 a^{4} e^{8}-704 a^{3} c \,d^{2} e^{6}+1584 a^{2} c^{2} d^{4} e^{4}-1848 a \,c^{3} d^{6} e^{2}+1155 c^{4} d^{8}\right ) \sqrt {c d e \,x^{2}+a \,e^{2} x +c \,d^{2} x +a d e}}{3465 c^{5} d^{5} \sqrt {e x +d}}\) | \(243\) |
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Time = 0.46 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.07 \[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (315 \, c^{5} d^{5} e^{4} x^{5} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \, {\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} c d e^{8}\right )} x\right )} \sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x} \sqrt {e x + d}}{3465 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
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Timed out. \[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.00 \[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (315 \, c^{5} d^{5} e^{4} x^{5} + 1155 \, a c^{4} d^{8} e - 1848 \, a^{2} c^{3} d^{6} e^{3} + 1584 \, a^{3} c^{2} d^{4} e^{5} - 704 \, a^{4} c d^{2} e^{7} + 128 \, a^{5} e^{9} + 35 \, {\left (44 \, c^{5} d^{6} e^{3} + a c^{4} d^{4} e^{5}\right )} x^{4} + 10 \, {\left (297 \, c^{5} d^{7} e^{2} + 22 \, a c^{4} d^{5} e^{4} - 4 \, a^{2} c^{3} d^{3} e^{6}\right )} x^{3} + 6 \, {\left (462 \, c^{5} d^{8} e + 99 \, a c^{4} d^{6} e^{3} - 44 \, a^{2} c^{3} d^{4} e^{5} + 8 \, a^{3} c^{2} d^{2} e^{7}\right )} x^{2} + {\left (1155 \, c^{5} d^{9} + 924 \, a c^{4} d^{7} e^{2} - 792 \, a^{2} c^{3} d^{5} e^{4} + 352 \, a^{3} c^{2} d^{3} e^{6} - 64 \, a^{4} c d e^{8}\right )} x\right )} \sqrt {c d x + a e} {\left (e x + d\right )}}{3465 \, {\left (c^{5} d^{5} e x + c^{5} d^{6}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1092 vs. \(2 (265) = 530\).
Time = 0.32 (sec) , antiderivative size = 1092, normalized size of antiderivative = 3.70 \[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {2 \, {\left (\frac {1155 \, d^{4} {\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}} + 198 \, d^{2} {\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 |} - 44 \, d 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 |} + e^{2} {\left (\frac {315 \, \sqrt {-c d^{2} e + a e^{3}} c^{5} d^{10} - 35 \, \sqrt {-c d^{2} e + a e^{3}} a c^{4} d^{8} e^{2} - 40 \, \sqrt {-c d^{2} e + a e^{3}} a^{2} c^{3} d^{6} e^{4} - 48 \, \sqrt {-c d^{2} e + a e^{3}} a^{3} c^{2} d^{4} e^{6} - 64 \, \sqrt {-c d^{2} e + a e^{3}} a^{4} c d^{2} e^{8} - 128 \, \sqrt {-c d^{2} e + a e^{3}} a^{5} e^{10}}{c^{5} d^{5} e^{4}} + \frac {1155 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {3}{2}} a^{4} e^{12} - 2772 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {5}{2}} a^{3} e^{9} + 2970 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {7}{2}} a^{2} e^{6} - 1540 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {9}{2}} a e^{3} + 315 \, {\left ({\left (e x + d\right )} c d e - c d^{2} e + a e^{3}\right )}^{\frac {11}{2}}}{c^{5} d^{5} e^{9}}\right )} {\left | e \right |} - \frac {924 \, d^{3} {\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 )}}{3465 \, e} \]
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Time = 10.59 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.17 \[ \int (d+e x)^{7/2} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx=\frac {\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}\,\left (\frac {2\,e^3\,x^5\,\sqrt {d+e\,x}}{11}+\frac {4\,x^2\,\sqrt {d+e\,x}\,\left (8\,a^3\,e^6-44\,a^2\,c\,d^2\,e^4+99\,a\,c^2\,d^4\,e^2+462\,c^3\,d^6\right )}{1155\,c^3\,d^3}+\frac {\sqrt {d+e\,x}\,\left (256\,a^5\,e^9-1408\,a^4\,c\,d^2\,e^7+3168\,a^3\,c^2\,d^4\,e^5-3696\,a^2\,c^3\,d^6\,e^3+2310\,a\,c^4\,d^8\,e\right )}{3465\,c^5\,d^5\,e}+\frac {2\,e^2\,x^4\,\left (44\,c\,d^2+a\,e^2\right )\,\sqrt {d+e\,x}}{99\,c\,d}+\frac {x\,\sqrt {d+e\,x}\,\left (-128\,a^4\,c\,d\,e^8+704\,a^3\,c^2\,d^3\,e^6-1584\,a^2\,c^3\,d^5\,e^4+1848\,a\,c^4\,d^7\,e^2+2310\,c^5\,d^9\right )}{3465\,c^5\,d^5\,e}+\frac {4\,e\,x^3\,\sqrt {d+e\,x}\,\left (-4\,a^2\,e^4+22\,a\,c\,d^2\,e^2+297\,c^2\,d^4\right )}{693\,c^2\,d^2}\right )}{x+\frac {d}{e}} \]
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