Integrand size = 30, antiderivative size = 320 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}-\frac {2 b^4 (b d-a e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^6 (a+b x)} \]
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Time = 0.08 (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} \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {20 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (b d-a e)^3}{11 e^6 (a+b x)}+\frac {10 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e)^4}{9 e^6 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^5}{7 e^6 (a+b x)}+\frac {2 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{17/2}}{17 e^6 (a+b x)}-\frac {2 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{15/2} (b d-a e)}{3 e^6 (a+b x)}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2} (b d-a e)^2}{13 e^6 (a+b x)} \]
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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)^{5/2} \, 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)^{5/2}}{e^5}+\frac {5 b^6 (b d-a e)^4 (d+e x)^{7/2}}{e^5}-\frac {10 b^7 (b d-a e)^3 (d+e x)^{9/2}}{e^5}+\frac {10 b^8 (b d-a e)^2 (d+e x)^{11/2}}{e^5}-\frac {5 b^9 (b d-a e) (d+e x)^{13/2}}{e^5}+\frac {b^{10} (d+e x)^{15/2}}{e^5}\right ) \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = -\frac {2 (b d-a e)^5 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (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^2 (b d-a e)^3 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (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^4 (b d-a e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^6 (a+b x)} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.73 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (21879 a^5 e^5+12155 a^4 b e^4 (-2 d+7 e x)+2210 a^3 b^2 e^3 \left (8 d^2-28 d e x+63 e^2 x^2\right )+510 a^2 b^3 e^2 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+17 a b^4 e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+b^5 \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )\right )}{153153 e^6 (a+b x)} \]
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Time = 2.30 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.90
method | result | size |
gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (9009 x^{5} e^{5} b^{5}+51051 x^{4} a \,b^{4} e^{5}-6006 x^{4} b^{5} d \,e^{4}+117810 x^{3} a^{2} b^{3} e^{5}-31416 x^{3} a \,b^{4} d \,e^{4}+3696 x^{3} b^{5} d^{2} e^{3}+139230 x^{2} a^{3} b^{2} e^{5}-64260 x^{2} a^{2} b^{3} d \,e^{4}+17136 x^{2} a \,b^{4} d^{2} e^{3}-2016 x^{2} b^{5} d^{3} e^{2}+85085 a^{4} b \,e^{5} x -61880 a^{3} b^{2} d \,e^{4} x +28560 x \,a^{2} b^{3} d^{2} e^{3}-7616 x a \,b^{4} d^{3} e^{2}+896 b^{5} d^{4} e x +21879 a^{5} e^{5}-24310 a^{4} b d \,e^{4}+17680 a^{3} b^{2} d^{2} e^{3}-8160 a^{2} b^{3} d^{3} e^{2}+2176 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{153153 e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
default | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (9009 x^{5} e^{5} b^{5}+51051 x^{4} a \,b^{4} e^{5}-6006 x^{4} b^{5} d \,e^{4}+117810 x^{3} a^{2} b^{3} e^{5}-31416 x^{3} a \,b^{4} d \,e^{4}+3696 x^{3} b^{5} d^{2} e^{3}+139230 x^{2} a^{3} b^{2} e^{5}-64260 x^{2} a^{2} b^{3} d \,e^{4}+17136 x^{2} a \,b^{4} d^{2} e^{3}-2016 x^{2} b^{5} d^{3} e^{2}+85085 a^{4} b \,e^{5} x -61880 a^{3} b^{2} d \,e^{4} x +28560 x \,a^{2} b^{3} d^{2} e^{3}-7616 x a \,b^{4} d^{3} e^{2}+896 b^{5} d^{4} e x +21879 a^{5} e^{5}-24310 a^{4} b d \,e^{4}+17680 a^{3} b^{2} d^{2} e^{3}-8160 a^{2} b^{3} d^{3} e^{2}+2176 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{153153 e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (9009 b^{5} e^{8} x^{8}+51051 a \,b^{4} e^{8} x^{7}+21021 b^{5} d \,e^{7} x^{7}+117810 a^{2} b^{3} e^{8} x^{6}+121737 a \,b^{4} d \,e^{7} x^{6}+12705 b^{5} d^{2} e^{6} x^{6}+139230 a^{3} b^{2} e^{8} x^{5}+289170 a^{2} b^{3} d \,e^{7} x^{5}+76041 a \,b^{4} d^{2} e^{6} x^{5}+63 b^{5} d^{3} e^{5} x^{5}+85085 a^{4} b \,e^{8} x^{4}+355810 a^{3} b^{2} d \,e^{7} x^{4}+189210 a^{2} b^{3} d^{2} e^{6} x^{4}+595 a \,b^{4} d^{3} e^{5} x^{4}-70 b^{5} d^{4} e^{4} x^{4}+21879 a^{5} e^{8} x^{3}+230945 a^{4} b d \,e^{7} x^{3}+249730 a^{3} b^{2} d^{2} e^{6} x^{3}+2550 a^{2} b^{3} d^{3} e^{5} x^{3}-680 a \,b^{4} d^{4} e^{4} x^{3}+80 b^{5} d^{5} e^{3} x^{3}+65637 a^{5} d \,e^{7} x^{2}+182325 a^{4} b \,d^{2} e^{6} x^{2}+6630 a^{3} b^{2} d^{3} e^{5} x^{2}-3060 a^{2} b^{3} d^{4} e^{4} x^{2}+816 a \,b^{4} d^{5} e^{3} x^{2}-96 b^{5} d^{6} e^{2} x^{2}+65637 a^{5} d^{2} e^{6} x +12155 a^{4} b \,d^{3} e^{5} x -8840 a^{3} b^{2} d^{4} e^{4} x +4080 a^{2} b^{3} d^{5} e^{3} x -1088 a \,b^{4} d^{6} e^{2} x +128 b^{5} d^{7} e x +21879 a^{5} d^{3} e^{5}-24310 a^{4} b \,d^{4} e^{4}+17680 a^{3} b^{2} d^{5} e^{3}-8160 a^{2} b^{3} d^{6} e^{2}+2176 a \,b^{4} d^{7} e -256 b^{5} d^{8}\right ) \sqrt {e x +d}}{153153 \left (b x +a \right ) e^{6}}\) | \(561\) |
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Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (230) = 460\).
Time = 0.39 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.55 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \]
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\[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (d + e x\right )^{\frac {5}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (230) = 460\).
Time = 0.21 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.55 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1743 vs. \(2 (230) = 460\).
Time = 0.38 (sec) , antiderivative size = 1743, normalized size of antiderivative = 5.45 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]
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