Integrand size = 30, antiderivative size = 320 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^6 (a+b x)} \]
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Time = 0.07 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {660, 45} \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^3}{9 e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^4}{7 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^5}{5 e^6 (a+b x)}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2}}{15 e^6 (a+b x)}-\frac {10 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)}{13 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^2}{11 e^6 (a+b x)} \]
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Rule 45
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^5 (d+e x)^{3/2} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^5 (b d-a e)^5 (d+e x)^{3/2}}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{5/2}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{7/2}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{9/2}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{11/2}}{e^5}+\frac {b^{10} (d+e x)^{13/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {2 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^6 (a+b x)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.73 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (9009 a^5 e^5+6435 a^4 b e^4 (-2 d+5 e x)+1430 a^3 b^2 e^3 \left (8 d^2-20 d e x+35 e^2 x^2\right )+390 a^2 b^3 e^2 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+15 a b^4 e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b^5 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )}{45045 e^6 (a+b x)} \]
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Time = 2.31 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 x^{5} e^{5} b^{5}+17325 x^{4} a \,b^{4} e^{5}-2310 x^{4} b^{5} d \,e^{4}+40950 x^{3} a^{2} b^{3} e^{5}-12600 x^{3} a \,b^{4} d \,e^{4}+1680 x^{3} b^{5} d^{2} e^{3}+50050 x^{2} a^{3} b^{2} e^{5}-27300 x^{2} a^{2} b^{3} d \,e^{4}+8400 x^{2} a \,b^{4} d^{2} e^{3}-1120 x^{2} b^{5} d^{3} e^{2}+32175 a^{4} b \,e^{5} x -28600 a^{3} b^{2} d \,e^{4} x +15600 x \,a^{2} b^{3} d^{2} e^{3}-4800 x a \,b^{4} d^{3} e^{2}+640 b^{5} d^{4} e x +9009 a^{5} e^{5}-12870 a^{4} b d \,e^{4}+11440 a^{3} b^{2} d^{2} e^{3}-6240 a^{2} b^{3} d^{3} e^{2}+1920 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
default | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 x^{5} e^{5} b^{5}+17325 x^{4} a \,b^{4} e^{5}-2310 x^{4} b^{5} d \,e^{4}+40950 x^{3} a^{2} b^{3} e^{5}-12600 x^{3} a \,b^{4} d \,e^{4}+1680 x^{3} b^{5} d^{2} e^{3}+50050 x^{2} a^{3} b^{2} e^{5}-27300 x^{2} a^{2} b^{3} d \,e^{4}+8400 x^{2} a \,b^{4} d^{2} e^{3}-1120 x^{2} b^{5} d^{3} e^{2}+32175 a^{4} b \,e^{5} x -28600 a^{3} b^{2} d \,e^{4} x +15600 x \,a^{2} b^{3} d^{2} e^{3}-4800 x a \,b^{4} d^{3} e^{2}+640 b^{5} d^{4} e x +9009 a^{5} e^{5}-12870 a^{4} b d \,e^{4}+11440 a^{3} b^{2} d^{2} e^{3}-6240 a^{2} b^{3} d^{3} e^{2}+1920 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 b^{5} x^{7} e^{7}+17325 a \,b^{4} e^{7} x^{6}+3696 b^{5} d \,e^{6} x^{6}+40950 a^{2} b^{3} e^{7} x^{5}+22050 a \,b^{4} d \,e^{6} x^{5}+63 b^{5} d^{2} e^{5} x^{5}+50050 a^{3} b^{2} e^{7} x^{4}+54600 a^{2} b^{3} d \,e^{6} x^{4}+525 a \,b^{4} d^{2} e^{5} x^{4}-70 b^{5} d^{3} e^{4} x^{4}+32175 a^{4} b \,e^{7} x^{3}+71500 a^{3} b^{2} d \,e^{6} x^{3}+1950 a^{2} b^{3} d^{2} e^{5} x^{3}-600 a \,b^{4} d^{3} e^{4} x^{3}+80 b^{5} d^{4} e^{3} x^{3}+9009 a^{5} e^{7} x^{2}+51480 a^{4} b d \,e^{6} x^{2}+4290 a^{3} b^{2} d^{2} e^{5} x^{2}-2340 a^{2} b^{3} d^{3} e^{4} x^{2}+720 a \,b^{4} d^{4} e^{3} x^{2}-96 b^{5} d^{5} e^{2} x^{2}+18018 a^{5} d \,e^{6} x +6435 a^{4} b \,d^{2} e^{5} x -5720 a^{3} b^{2} d^{3} e^{4} x +3120 a^{2} b^{3} d^{4} e^{3} x -960 a \,b^{4} d^{5} e^{2} x +128 b^{5} d^{6} e x +9009 a^{5} d^{2} e^{5}-12870 a^{4} b \,d^{3} e^{4}+11440 a^{3} b^{2} d^{4} e^{3}-6240 a^{2} b^{3} d^{5} e^{2}+1920 a \,b^{4} d^{6} e -256 b^{5} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{6}}\) | \(469\) |
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Time = 0.40 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.31 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
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\[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (d + e x\right )^{\frac {3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]
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Time = 0.25 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.31 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (230) = 460\).
Time = 0.36 (sec) , antiderivative size = 1192, normalized size of antiderivative = 3.72 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]
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