Integrand size = 33, antiderivative size = 362 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=-\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{16 e^7 (a+b x) (d+e x)^{16}}+\frac {2 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{15}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {5 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}} \]
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Time = 0.14 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {784, 21, 45} \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=-\frac {15 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^4}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^5}{5 e^7 (a+b x) (d+e x)^{15}}-\frac {\sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^6}{16 e^7 (a+b x) (d+e x)^{16}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}}+\frac {6 b^5 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {5 b^4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {20 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^3}{13 e^7 (a+b x) (d+e x)^{13}} \]
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Rule 21
Rule 45
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x) \left (a b+b^2 x\right )^5}{(d+e x)^{17}} \, dx}{b^4 \left (a b+b^2 x\right )} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \frac {(a+b x)^6}{(d+e x)^{17}} \, dx}{a b+b^2 x} \\ & = \frac {\left (b \sqrt {a^2+2 a b x+b^2 x^2}\right ) \int \left (\frac {(-b d+a e)^6}{e^6 (d+e x)^{17}}-\frac {6 b (b d-a e)^5}{e^6 (d+e x)^{16}}+\frac {15 b^2 (b d-a e)^4}{e^6 (d+e x)^{15}}-\frac {20 b^3 (b d-a e)^3}{e^6 (d+e x)^{14}}+\frac {15 b^4 (b d-a e)^2}{e^6 (d+e x)^{13}}-\frac {6 b^5 (b d-a e)}{e^6 (d+e x)^{12}}+\frac {b^6}{e^6 (d+e x)^{11}}\right ) \, dx}{a b+b^2 x} \\ & = -\frac {(b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{16 e^7 (a+b x) (d+e x)^{16}}+\frac {2 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{15}}-\frac {15 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{14 e^7 (a+b x) (d+e x)^{14}}+\frac {20 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13}}-\frac {5 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{4 e^7 (a+b x) (d+e x)^{12}}+\frac {6 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11}}-\frac {b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{10 e^7 (a+b x) (d+e x)^{10}} \\ \end{align*}
Time = 1.08 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.81 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=-\frac {\sqrt {(a+b x)^2} \left (5005 a^6 e^6+2002 a^5 b e^5 (d+16 e x)+715 a^4 b^2 e^4 \left (d^2+16 d e x+120 e^2 x^2\right )+220 a^3 b^3 e^3 \left (d^3+16 d^2 e x+120 d e^2 x^2+560 e^3 x^3\right )+55 a^2 b^4 e^2 \left (d^4+16 d^3 e x+120 d^2 e^2 x^2+560 d e^3 x^3+1820 e^4 x^4\right )+10 a b^5 e \left (d^5+16 d^4 e x+120 d^3 e^2 x^2+560 d^2 e^3 x^3+1820 d e^4 x^4+4368 e^5 x^5\right )+b^6 \left (d^6+16 d^5 e x+120 d^4 e^2 x^2+560 d^3 e^3 x^3+1820 d^2 e^4 x^4+4368 d e^5 x^5+8008 e^6 x^6\right )\right )}{80080 e^7 (a+b x) (d+e x)^{16}} \]
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Time = 86.88 (sec) , antiderivative size = 351, normalized size of antiderivative = 0.97
method | result | size |
risch | \(\frac {\sqrt {\left (b x +a \right )^{2}}\, \left (-\frac {5005 e^{6} a^{6}+2002 b d \,e^{5} a^{5}+715 b^{2} d^{2} e^{4} a^{4}+220 b^{3} d^{3} e^{3} a^{3}+55 b^{4} d^{4} e^{2} a^{2}+10 b^{5} d^{5} e a +b^{6} d^{6}}{80080 e^{7}}-\frac {b \left (2002 e^{5} a^{5}+715 b d \,e^{4} a^{4}+220 b^{2} d^{2} e^{3} a^{3}+55 b^{3} d^{3} e^{2} a^{2}+10 b^{4} d^{4} e a +b^{5} d^{5}\right ) x}{5005 e^{6}}-\frac {3 b^{2} \left (715 e^{4} a^{4}+220 b d \,e^{3} a^{3}+55 b^{2} d^{2} e^{2} a^{2}+10 b^{3} d^{3} e a +b^{4} d^{4}\right ) x^{2}}{2002 e^{5}}-\frac {b^{3} \left (220 a^{3} e^{3}+55 a^{2} b d \,e^{2}+10 a \,b^{2} d^{2} e +b^{3} d^{3}\right ) x^{3}}{143 e^{4}}-\frac {b^{4} \left (55 e^{2} a^{2}+10 a b d e +b^{2} d^{2}\right ) x^{4}}{44 e^{3}}-\frac {3 b^{5} \left (10 a e +b d \right ) x^{5}}{55 e^{2}}-\frac {b^{6} x^{6}}{10 e}\right )}{\left (b x +a \right ) \left (e x +d \right )^{16}}\) | \(351\) |
gosper | \(-\frac {\left (8008 b^{6} e^{6} x^{6}+43680 a \,b^{5} e^{6} x^{5}+4368 b^{6} d \,e^{5} x^{5}+100100 a^{2} b^{4} e^{6} x^{4}+18200 a \,b^{5} d \,e^{5} x^{4}+1820 b^{6} d^{2} e^{4} x^{4}+123200 a^{3} b^{3} e^{6} x^{3}+30800 a^{2} b^{4} d \,e^{5} x^{3}+5600 a \,b^{5} d^{2} e^{4} x^{3}+560 b^{6} d^{3} e^{3} x^{3}+85800 a^{4} b^{2} e^{6} x^{2}+26400 a^{3} b^{3} d \,e^{5} x^{2}+6600 a^{2} b^{4} d^{2} e^{4} x^{2}+1200 a \,b^{5} d^{3} e^{3} x^{2}+120 b^{6} d^{4} e^{2} x^{2}+32032 a^{5} b \,e^{6} x +11440 a^{4} b^{2} d \,e^{5} x +3520 a^{3} b^{3} d^{2} e^{4} x +880 a^{2} b^{4} d^{3} e^{3} x +160 a \,b^{5} d^{4} e^{2} x +16 b^{6} d^{5} e x +5005 e^{6} a^{6}+2002 b d \,e^{5} a^{5}+715 b^{2} d^{2} e^{4} a^{4}+220 b^{3} d^{3} e^{3} a^{3}+55 b^{4} d^{4} e^{2} a^{2}+10 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{80080 e^{7} \left (e x +d \right )^{16} \left (b x +a \right )^{5}}\) | \(392\) |
default | \(-\frac {\left (8008 b^{6} e^{6} x^{6}+43680 a \,b^{5} e^{6} x^{5}+4368 b^{6} d \,e^{5} x^{5}+100100 a^{2} b^{4} e^{6} x^{4}+18200 a \,b^{5} d \,e^{5} x^{4}+1820 b^{6} d^{2} e^{4} x^{4}+123200 a^{3} b^{3} e^{6} x^{3}+30800 a^{2} b^{4} d \,e^{5} x^{3}+5600 a \,b^{5} d^{2} e^{4} x^{3}+560 b^{6} d^{3} e^{3} x^{3}+85800 a^{4} b^{2} e^{6} x^{2}+26400 a^{3} b^{3} d \,e^{5} x^{2}+6600 a^{2} b^{4} d^{2} e^{4} x^{2}+1200 a \,b^{5} d^{3} e^{3} x^{2}+120 b^{6} d^{4} e^{2} x^{2}+32032 a^{5} b \,e^{6} x +11440 a^{4} b^{2} d \,e^{5} x +3520 a^{3} b^{3} d^{2} e^{4} x +880 a^{2} b^{4} d^{3} e^{3} x +160 a \,b^{5} d^{4} e^{2} x +16 b^{6} d^{5} e x +5005 e^{6} a^{6}+2002 b d \,e^{5} a^{5}+715 b^{2} d^{2} e^{4} a^{4}+220 b^{3} d^{3} e^{3} a^{3}+55 b^{4} d^{4} e^{2} a^{2}+10 b^{5} d^{5} e a +b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{80080 e^{7} \left (e x +d \right )^{16} \left (b x +a \right )^{5}}\) | \(392\) |
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Time = 0.37 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=-\frac {8008 \, b^{6} e^{6} x^{6} + b^{6} d^{6} + 10 \, a b^{5} d^{5} e + 55 \, a^{2} b^{4} d^{4} e^{2} + 220 \, a^{3} b^{3} d^{3} e^{3} + 715 \, a^{4} b^{2} d^{2} e^{4} + 2002 \, a^{5} b d e^{5} + 5005 \, a^{6} e^{6} + 4368 \, {\left (b^{6} d e^{5} + 10 \, a b^{5} e^{6}\right )} x^{5} + 1820 \, {\left (b^{6} d^{2} e^{4} + 10 \, a b^{5} d e^{5} + 55 \, a^{2} b^{4} e^{6}\right )} x^{4} + 560 \, {\left (b^{6} d^{3} e^{3} + 10 \, a b^{5} d^{2} e^{4} + 55 \, a^{2} b^{4} d e^{5} + 220 \, a^{3} b^{3} e^{6}\right )} x^{3} + 120 \, {\left (b^{6} d^{4} e^{2} + 10 \, a b^{5} d^{3} e^{3} + 55 \, a^{2} b^{4} d^{2} e^{4} + 220 \, a^{3} b^{3} d e^{5} + 715 \, a^{4} b^{2} e^{6}\right )} x^{2} + 16 \, {\left (b^{6} d^{5} e + 10 \, a b^{5} d^{4} e^{2} + 55 \, a^{2} b^{4} d^{3} e^{3} + 220 \, a^{3} b^{3} d^{2} e^{4} + 715 \, a^{4} b^{2} d e^{5} + 2002 \, a^{5} b e^{6}\right )} x}{80080 \, {\left (e^{23} x^{16} + 16 \, d e^{22} x^{15} + 120 \, d^{2} e^{21} x^{14} + 560 \, d^{3} e^{20} x^{13} + 1820 \, d^{4} e^{19} x^{12} + 4368 \, d^{5} e^{18} x^{11} + 8008 \, d^{6} e^{17} x^{10} + 11440 \, d^{7} e^{16} x^{9} + 12870 \, d^{8} e^{15} x^{8} + 11440 \, d^{9} e^{14} x^{7} + 8008 \, d^{10} e^{13} x^{6} + 4368 \, d^{11} e^{12} x^{5} + 1820 \, d^{12} e^{11} x^{4} + 560 \, d^{13} e^{10} x^{3} + 120 \, d^{14} e^{9} x^{2} + 16 \, d^{15} e^{8} x + d^{16} e^{7}\right )}} \]
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Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=\text {Exception raised: HeuristicGCDFailed} \]
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Exception generated. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (271) = 542\).
Time = 0.28 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.92 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=\frac {b^{16} \mathrm {sgn}\left (b x + a\right )}{80080 \, {\left (b^{10} d^{10} e^{7} - 10 \, a b^{9} d^{9} e^{8} + 45 \, a^{2} b^{8} d^{8} e^{9} - 120 \, a^{3} b^{7} d^{7} e^{10} + 210 \, a^{4} b^{6} d^{6} e^{11} - 252 \, a^{5} b^{5} d^{5} e^{12} + 210 \, a^{6} b^{4} d^{4} e^{13} - 120 \, a^{7} b^{3} d^{3} e^{14} + 45 \, a^{8} b^{2} d^{2} e^{15} - 10 \, a^{9} b d e^{16} + a^{10} e^{17}\right )}} - \frac {8008 \, b^{6} e^{6} x^{6} \mathrm {sgn}\left (b x + a\right ) + 4368 \, b^{6} d e^{5} x^{5} \mathrm {sgn}\left (b x + a\right ) + 43680 \, a b^{5} e^{6} x^{5} \mathrm {sgn}\left (b x + a\right ) + 1820 \, b^{6} d^{2} e^{4} x^{4} \mathrm {sgn}\left (b x + a\right ) + 18200 \, a b^{5} d e^{5} x^{4} \mathrm {sgn}\left (b x + a\right ) + 100100 \, a^{2} b^{4} e^{6} x^{4} \mathrm {sgn}\left (b x + a\right ) + 560 \, b^{6} d^{3} e^{3} x^{3} \mathrm {sgn}\left (b x + a\right ) + 5600 \, a b^{5} d^{2} e^{4} x^{3} \mathrm {sgn}\left (b x + a\right ) + 30800 \, a^{2} b^{4} d e^{5} x^{3} \mathrm {sgn}\left (b x + a\right ) + 123200 \, a^{3} b^{3} e^{6} x^{3} \mathrm {sgn}\left (b x + a\right ) + 120 \, b^{6} d^{4} e^{2} x^{2} \mathrm {sgn}\left (b x + a\right ) + 1200 \, a b^{5} d^{3} e^{3} x^{2} \mathrm {sgn}\left (b x + a\right ) + 6600 \, a^{2} b^{4} d^{2} e^{4} x^{2} \mathrm {sgn}\left (b x + a\right ) + 26400 \, a^{3} b^{3} d e^{5} x^{2} \mathrm {sgn}\left (b x + a\right ) + 85800 \, a^{4} b^{2} e^{6} x^{2} \mathrm {sgn}\left (b x + a\right ) + 16 \, b^{6} d^{5} e x \mathrm {sgn}\left (b x + a\right ) + 160 \, a b^{5} d^{4} e^{2} x \mathrm {sgn}\left (b x + a\right ) + 880 \, a^{2} b^{4} d^{3} e^{3} x \mathrm {sgn}\left (b x + a\right ) + 3520 \, a^{3} b^{3} d^{2} e^{4} x \mathrm {sgn}\left (b x + a\right ) + 11440 \, a^{4} b^{2} d e^{5} x \mathrm {sgn}\left (b x + a\right ) + 32032 \, a^{5} b e^{6} x \mathrm {sgn}\left (b x + a\right ) + b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 10 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 55 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 220 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 715 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 2002 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 5005 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )}{80080 \, {\left (e x + d\right )}^{16} e^{7}} \]
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Time = 11.20 (sec) , antiderivative size = 1010, normalized size of antiderivative = 2.79 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{17}} \, dx=\frac {\left (\frac {-6\,a^5\,b\,e^5+15\,a^4\,b^2\,d\,e^4-20\,a^3\,b^3\,d^2\,e^3+15\,a^2\,b^4\,d^3\,e^2-6\,a\,b^5\,d^4\,e+b^6\,d^5}{15\,e^7}+\frac {d\,\left (\frac {15\,a^4\,b^2\,e^5-20\,a^3\,b^3\,d\,e^4+15\,a^2\,b^4\,d^2\,e^3-6\,a\,b^5\,d^3\,e^2+b^6\,d^4\,e}{15\,e^7}-\frac {d\,\left (\frac {20\,a^3\,b^3\,e^5-15\,a^2\,b^4\,d\,e^4+6\,a\,b^5\,d^2\,e^3-b^6\,d^3\,e^2}{15\,e^7}-\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{15\,e^3}-\frac {b^5\,\left (6\,a\,e-b\,d\right )}{15\,e^3}\right )}{e}+\frac {b^4\,\left (15\,a^2\,e^2-6\,a\,b\,d\,e+b^2\,d^2\right )}{15\,e^4}\right )}{e}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{15}}-\frac {\left (\frac {15\,a^2\,b^4\,e^2-24\,a\,b^5\,d\,e+10\,b^6\,d^2}{12\,e^7}+\frac {d\,\left (\frac {b^6\,d}{12\,e^6}-\frac {b^5\,\left (3\,a\,e-2\,b\,d\right )}{6\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{12}}-\frac {\left (\frac {a^6}{16\,e}-\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {d\,\left (\frac {3\,a\,b^5}{8\,e}-\frac {b^6\,d}{16\,e^2}\right )}{e}-\frac {15\,a^2\,b^4}{16\,e}\right )}{e}+\frac {5\,a^3\,b^3}{4\,e}\right )}{e}-\frac {15\,a^4\,b^2}{16\,e}\right )}{e}+\frac {3\,a^5\,b}{8\,e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{16}}-\frac {\left (\frac {15\,a^4\,b^2\,e^4-40\,a^3\,b^3\,d\,e^3+45\,a^2\,b^4\,d^2\,e^2-24\,a\,b^5\,d^3\,e+5\,b^6\,d^4}{14\,e^7}+\frac {d\,\left (\frac {-20\,a^3\,b^3\,e^4+30\,a^2\,b^4\,d\,e^3-18\,a\,b^5\,d^2\,e^2+4\,b^6\,d^3\,e}{14\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{14\,e^4}-\frac {b^5\,\left (3\,a\,e-b\,d\right )}{7\,e^4}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-4\,a\,b\,d\,e+b^2\,d^2\right )}{14\,e^5}\right )}{e}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{14}}+\frac {\left (\frac {5\,b^6\,d-6\,a\,b^5\,e}{11\,e^7}+\frac {b^6\,d}{11\,e^7}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{11}}+\frac {\left (\frac {-20\,a^3\,b^3\,e^3+45\,a^2\,b^4\,d\,e^2-36\,a\,b^5\,d^2\,e+10\,b^6\,d^3}{13\,e^7}+\frac {d\,\left (\frac {d\,\left (\frac {b^6\,d}{13\,e^5}-\frac {3\,b^5\,\left (2\,a\,e-b\,d\right )}{13\,e^5}\right )}{e}+\frac {3\,b^4\,\left (5\,a^2\,e^2-6\,a\,b\,d\,e+2\,b^2\,d^2\right )}{13\,e^6}\right )}{e}\right )\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{13}}-\frac {b^6\,\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}}{10\,e^7\,\left (a+b\,x\right )\,{\left (d+e\,x\right )}^{10}} \]
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