Integrand size = 28, antiderivative size = 127 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 (b d-a e)^4 \sqrt {d+e x}}{e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{3/2}}{3 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{5/2}}{5 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{7/2}}{7 e^5}+\frac {2 b^4 (d+e x)^{9/2}}{9 e^5} \]
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Time = 0.03 (sec) , antiderivative size = 127, 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} \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=-\frac {8 b^3 (d+e x)^{7/2} (b d-a e)}{7 e^5}+\frac {12 b^2 (d+e x)^{5/2} (b d-a e)^2}{5 e^5}-\frac {8 b (d+e x)^{3/2} (b d-a e)^3}{3 e^5}+\frac {2 \sqrt {d+e x} (b d-a e)^4}{e^5}+\frac {2 b^4 (d+e x)^{9/2}}{9 e^5} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^4}{\sqrt {d+e x}} \, dx \\ & = \int \left (\frac {(-b d+a e)^4}{e^4 \sqrt {d+e x}}-\frac {4 b (b d-a e)^3 \sqrt {d+e x}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{3/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{5/2}}{e^4}+\frac {b^4 (d+e x)^{7/2}}{e^4}\right ) \, dx \\ & = \frac {2 (b d-a e)^4 \sqrt {d+e x}}{e^5}-\frac {8 b (b d-a e)^3 (d+e x)^{3/2}}{3 e^5}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{5/2}}{5 e^5}-\frac {8 b^3 (b d-a e) (d+e x)^{7/2}}{7 e^5}+\frac {2 b^4 (d+e x)^{9/2}}{9 e^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (315 a^4 e^4+420 a^3 b e^3 (-2 d+e x)+126 a^2 b^2 e^2 \left (8 d^2-4 d e x+3 e^2 x^2\right )+36 a b^3 e \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+b^4 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )\right )}{315 e^5} \]
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Time = 2.29 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.13
method | result | size |
pseudoelliptic | \(\frac {2 \sqrt {e x +d}\, \left (\left (\frac {1}{9} b^{4} x^{4}+\frac {4}{7} a \,b^{3} x^{3}+\frac {6}{5} a^{2} b^{2} x^{2}+\frac {4}{3} a^{3} b x +a^{4}\right ) e^{4}-\frac {8 \left (\frac {1}{21} b^{3} x^{3}+\frac {9}{35} a \,b^{2} x^{2}+\frac {3}{5} a^{2} b x +a^{3}\right ) b d \,e^{3}}{3}+\frac {16 \left (\frac {1}{21} b^{2} x^{2}+\frac {2}{7} a b x +a^{2}\right ) b^{2} d^{2} e^{2}}{5}-\frac {64 b^{3} d^{3} \left (\frac {b x}{9}+a \right ) e}{35}+\frac {128 b^{4} d^{4}}{315}\right )}{e^{5}}\) | \(143\) |
derivativedivides | \(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \sqrt {e x +d}}{e^{5}}\) | \(166\) |
default | \(\frac {\frac {2 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {4 \left (2 a e b -2 b^{2} d \right ) b^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a e b -2 b^{2} d \right )^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a e b -2 b^{2} d \right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \sqrt {e x +d}}{e^{5}}\) | \(166\) |
gosper | \(\frac {2 \left (35 b^{4} x^{4} e^{4}+180 x^{3} a \,b^{3} e^{4}-40 x^{3} b^{4} d \,e^{3}+378 x^{2} a^{2} b^{2} e^{4}-216 x^{2} a \,b^{3} d \,e^{3}+48 x^{2} b^{4} d^{2} e^{2}+420 x \,a^{3} b \,e^{4}-504 x \,a^{2} b^{2} d \,e^{3}+288 x a \,b^{3} d^{2} e^{2}-64 x \,b^{4} d^{3} e +315 e^{4} a^{4}-840 b \,e^{3} d \,a^{3}+1008 b^{2} e^{2} d^{2} a^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\) | \(186\) |
trager | \(\frac {2 \left (35 b^{4} x^{4} e^{4}+180 x^{3} a \,b^{3} e^{4}-40 x^{3} b^{4} d \,e^{3}+378 x^{2} a^{2} b^{2} e^{4}-216 x^{2} a \,b^{3} d \,e^{3}+48 x^{2} b^{4} d^{2} e^{2}+420 x \,a^{3} b \,e^{4}-504 x \,a^{2} b^{2} d \,e^{3}+288 x a \,b^{3} d^{2} e^{2}-64 x \,b^{4} d^{3} e +315 e^{4} a^{4}-840 b \,e^{3} d \,a^{3}+1008 b^{2} e^{2} d^{2} a^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\) | \(186\) |
risch | \(\frac {2 \left (35 b^{4} x^{4} e^{4}+180 x^{3} a \,b^{3} e^{4}-40 x^{3} b^{4} d \,e^{3}+378 x^{2} a^{2} b^{2} e^{4}-216 x^{2} a \,b^{3} d \,e^{3}+48 x^{2} b^{4} d^{2} e^{2}+420 x \,a^{3} b \,e^{4}-504 x \,a^{2} b^{2} d \,e^{3}+288 x a \,b^{3} d^{2} e^{2}-64 x \,b^{4} d^{3} e +315 e^{4} a^{4}-840 b \,e^{3} d \,a^{3}+1008 b^{2} e^{2} d^{2} a^{2}-576 a \,b^{3} d^{3} e +128 b^{4} d^{4}\right ) \sqrt {e x +d}}{315 e^{5}}\) | \(186\) |
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Time = 0.27 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.43 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (35 \, b^{4} e^{4} x^{4} + 128 \, b^{4} d^{4} - 576 \, a b^{3} d^{3} e + 1008 \, a^{2} b^{2} d^{2} e^{2} - 840 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - 20 \, {\left (2 \, b^{4} d e^{3} - 9 \, a b^{3} e^{4}\right )} x^{3} + 6 \, {\left (8 \, b^{4} d^{2} e^{2} - 36 \, a b^{3} d e^{3} + 63 \, a^{2} b^{2} e^{4}\right )} x^{2} - 4 \, {\left (16 \, b^{4} d^{3} e - 72 \, a b^{3} d^{2} e^{2} + 126 \, a^{2} b^{2} d e^{3} - 105 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{315 \, e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (117) = 234\).
Time = 1.01 (sec) , antiderivative size = 270, normalized size of antiderivative = 2.13 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (4 a b^{3} e - 4 b^{4} d\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (6 a^{2} b^{2} e^{2} - 12 a b^{3} d e + 6 b^{4} d^{2}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (4 a^{3} b e^{3} - 12 a^{2} b^{2} d e^{2} + 12 a b^{3} d^{2} e - 4 b^{4} d^{3}\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (a^{4} e^{4} - 4 a^{3} b d e^{3} + 6 a^{2} b^{2} d^{2} e^{2} - 4 a b^{3} d^{3} e + b^{4} d^{4}\right )}{e^{4}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac {b^{4} x^{5}}{5}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (109) = 218\).
Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.94 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} a^{4} + 42 \, {\left (\frac {10 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a b}{e} + \frac {{\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} b^{2}}{e^{2}}\right )} a^{2} + \frac {84 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} b^{2}}{e^{2}} + \frac {36 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b^{3}}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{4}}{e^{4}}\right )}}{315 \, e} \]
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Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.61 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (315 \, \sqrt {e x + d} a^{4} + \frac {420 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{3} b}{e} + \frac {126 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{2} b^{2}}{e^{2}} + \frac {36 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a b^{3}}{e^{3}} + \frac {{\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} b^{4}}{e^{4}}\right )}}{315 \, e} \]
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Time = 0.04 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a^2+2 a b x+b^2 x^2\right )^2}{\sqrt {d+e x}} \, dx=\frac {2\,b^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}-\frac {\left (8\,b^4\,d-8\,a\,b^3\,e\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^4\,\sqrt {d+e\,x}}{e^5}+\frac {12\,b^2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {8\,b\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5} \]
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