Integrand size = 28, antiderivative size = 129 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (b d-a e)^4 (d+e x)^{3/2}}{3 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{5/2}}{5 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{9/2}}{9 e^5}+\frac {2 b^4 (d+e x)^{11/2}}{11 e^5} \]
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Time = 0.03 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {27, 45} \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {8 b^3 (d+e x)^{9/2} (b d-a e)}{9 e^5}+\frac {12 b^2 (d+e x)^{7/2} (b d-a e)^2}{7 e^5}-\frac {8 b (d+e x)^{5/2} (b d-a e)^3}{5 e^5}+\frac {2 (d+e x)^{3/2} (b d-a e)^4}{3 e^5}+\frac {2 b^4 (d+e x)^{11/2}}{11 e^5} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int (a+b x)^4 \sqrt {d+e x} \, dx \\ & = \int \left (\frac {(-b d+a e)^4 \sqrt {d+e x}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{3/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{5/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{7/2}}{e^4}+\frac {b^4 (d+e x)^{9/2}}{e^4}\right ) \, dx \\ & = \frac {2 (b d-a e)^4 (d+e x)^{3/2}}{3 e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{5/2}}{5 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{7/2}}{7 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{9/2}}{9 e^5}+\frac {2 b^4 (d+e x)^{11/2}}{11 e^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{3/2} \left (1155 a^4 e^4+924 a^3 b e^3 (-2 d+3 e x)+198 a^2 b^2 e^2 \left (8 d^2-12 d e x+15 e^2 x^2\right )+44 a b^3 e \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )\right )}{3465 e^5} \]
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Time = 2.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.11
method | result | size |
pseudoelliptic | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (\left (\frac {3}{11} b^{4} x^{4}+\frac {4}{3} a \,b^{3} x^{3}+\frac {18}{7} a^{2} b^{2} x^{2}+\frac {12}{5} a^{3} b x +a^{4}\right ) e^{4}-\frac {8 b \left (\frac {5}{33} b^{3} x^{3}+\frac {5}{7} a \,b^{2} x^{2}+\frac {9}{7} a^{2} b x +a^{3}\right ) d \,e^{3}}{5}+\frac {48 \left (\frac {5}{33} b^{2} x^{2}+\frac {2}{3} a b x +a^{2}\right ) b^{2} d^{2} e^{2}}{35}-\frac {64 b^{3} d^{3} \left (\frac {3 b x}{11}+a \right ) e}{105}+\frac {128 b^{4} d^{4}}{1155}\right )}{3 e^{5}}\) | \(143\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(167\) |
default | \(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}}{e^{5}}\) | \(167\) |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 b^{4} x^{4} e^{4}+1540 x^{3} a \,b^{3} e^{4}-280 x^{3} b^{4} d \,e^{3}+2970 x^{2} a^{2} b^{2} e^{4}-1320 x^{2} a \,b^{3} d \,e^{3}+240 x^{2} b^{4} d^{2} e^{2}+2772 x \,a^{3} b \,e^{4}-2376 x \,a^{2} b^{2} d \,e^{3}+1056 x a \,b^{3} d^{2} e^{2}-192 x \,b^{4} d^{3} e +1155 e^{4} a^{4}-1848 b \,e^{3} d \,a^{3}+1584 b^{2} e^{2} d^{2} a^{2}-704 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right )}{3465 e^{5}}\) | \(186\) |
trager | \(\frac {2 \left (315 b^{4} e^{5} x^{5}+1540 a \,b^{3} e^{5} x^{4}+35 b^{4} d \,e^{4} x^{4}+2970 a^{2} b^{2} e^{5} x^{3}+220 a \,b^{3} d \,e^{4} x^{3}-40 d^{2} e^{3} b^{4} x^{3}+2772 a^{3} b \,e^{5} x^{2}+594 a^{2} b^{2} d \,e^{4} x^{2}-264 a \,b^{3} d^{2} e^{3} x^{2}+48 b^{4} d^{3} e^{2} x^{2}+1155 e^{5} a^{4} x +924 a^{3} b d \,e^{4} x -792 a^{2} b^{2} d^{2} e^{3} x +352 a \,b^{3} d^{3} e^{2} x -64 b^{4} d^{4} e x +1155 d \,e^{4} a^{4}-1848 a^{3} b \,d^{2} e^{3}+1584 a^{2} b^{2} d^{3} e^{2}-704 a \,b^{3} d^{4} e +128 b^{4} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(257\) |
risch | \(\frac {2 \left (315 b^{4} e^{5} x^{5}+1540 a \,b^{3} e^{5} x^{4}+35 b^{4} d \,e^{4} x^{4}+2970 a^{2} b^{2} e^{5} x^{3}+220 a \,b^{3} d \,e^{4} x^{3}-40 d^{2} e^{3} b^{4} x^{3}+2772 a^{3} b \,e^{5} x^{2}+594 a^{2} b^{2} d \,e^{4} x^{2}-264 a \,b^{3} d^{2} e^{3} x^{2}+48 b^{4} d^{3} e^{2} x^{2}+1155 e^{5} a^{4} x +924 a^{3} b d \,e^{4} x -792 a^{2} b^{2} d^{2} e^{3} x +352 a \,b^{3} d^{3} e^{2} x -64 b^{4} d^{4} e x +1155 d \,e^{4} a^{4}-1848 a^{3} b \,d^{2} e^{3}+1584 a^{2} b^{2} d^{3} e^{2}-704 a \,b^{3} d^{4} e +128 b^{4} d^{5}\right ) \sqrt {e x +d}}{3465 e^{5}}\) | \(257\) |
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (109) = 218\).
Time = 0.26 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.90 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (315 \, b^{4} e^{5} x^{5} + 128 \, b^{4} d^{5} - 704 \, a b^{3} d^{4} e + 1584 \, a^{2} b^{2} d^{3} e^{2} - 1848 \, a^{3} b d^{2} e^{3} + 1155 \, a^{4} d e^{4} + 35 \, {\left (b^{4} d e^{4} + 44 \, a b^{3} e^{5}\right )} x^{4} - 10 \, {\left (4 \, b^{4} d^{2} e^{3} - 22 \, a b^{3} d e^{4} - 297 \, a^{2} b^{2} e^{5}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{3} e^{2} - 44 \, a b^{3} d^{2} e^{3} + 99 \, a^{2} b^{2} d e^{4} + 462 \, a^{3} b e^{5}\right )} x^{2} - {\left (64 \, b^{4} d^{4} e - 352 \, a b^{3} d^{3} e^{2} + 792 \, a^{2} b^{2} d^{2} e^{3} - 924 \, a^{3} b d e^{4} - 1155 \, a^{4} e^{5}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (119) = 238\).
Time = 1.03 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.11 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (4 a b^{3} e - 4 b^{4} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (4 a^{3} b e^{3} - 12 a^{2} b^{2} d e^{2} + 12 a b^{3} d^{2} e - 4 b^{4} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}\right )}{3 e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\sqrt {d} \left (a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.40 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} b^{4} - 1540 \, {\left (b^{4} d - a b^{3} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 2970 \, {\left (b^{4} d^{2} - 2 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 2772 \, {\left (b^{4} d^{3} - 3 \, a b^{3} d^{2} e + 3 \, a^{2} b^{2} d e^{2} - a^{3} b e^{3}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{4} d^{4} - 4 \, a b^{3} d^{3} e + 6 \, a^{2} b^{2} d^{2} e^{2} - 4 \, a^{3} b d e^{3} + a^{4} e^{4}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (109) = 218\).
Time = 0.29 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.64 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (3465 \, \sqrt {e x + d} a^{4} d + 1155 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{4} + \frac {4620 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{3} b d}{e} + \frac {1386 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} b^{2} d}{e^{2}} + \frac {924 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{3} b}{e} + \frac {396 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b^{3} d}{e^{3}} + \frac {594 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{2} b^{2}}{e^{2}} + \frac {11 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{4} d}{e^{4}} + \frac {44 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a b^{3}}{e^{3}} + \frac {5 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} b^{4}}{e^{4}}\right )}}{3465 \, e} \]
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Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^4\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5} \]
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