Integrand size = 28, antiderivative size = 253 \[ \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}-\frac {9009 e^5 (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}} \]
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Time = 0.13 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {9009 e^5 (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}}+\frac {9009 e^5 \sqrt {d+e x} (b d-a e)^2}{128 b^8}+\frac {3003 e^5 (d+e x)^{3/2} (b d-a e)}{128 b^7}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6} \]
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Rule 27
Rule 43
Rule 52
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{15/2}}{(a+b x)^6} \, dx \\ & = -\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {(3 e) \int \frac {(d+e x)^{13/2}}{(a+b x)^5} \, dx}{2 b} \\ & = -\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (39 e^2\right ) \int \frac {(d+e x)^{11/2}}{(a+b x)^4} \, dx}{16 b^2} \\ & = -\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (143 e^3\right ) \int \frac {(d+e x)^{9/2}}{(a+b x)^3} \, dx}{32 b^3} \\ & = -\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (1287 e^4\right ) \int \frac {(d+e x)^{7/2}}{(a+b x)^2} \, dx}{128 b^4} \\ & = -\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5\right ) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{256 b^5} \\ & = \frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)\right ) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{256 b^6} \\ & = \frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)^2\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{256 b^7} \\ & = \frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^5 (b d-a e)^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^8} \\ & = \frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}+\frac {\left (9009 e^4 (b d-a e)^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{128 b^8} \\ & = \frac {9009 e^5 (b d-a e)^2 \sqrt {d+e x}}{128 b^8}+\frac {3003 e^5 (b d-a e) (d+e x)^{3/2}}{128 b^7}+\frac {9009 e^5 (d+e x)^{5/2}}{640 b^6}-\frac {1287 e^4 (d+e x)^{7/2}}{128 b^5 (a+b x)}-\frac {143 e^3 (d+e x)^{9/2}}{64 b^4 (a+b x)^2}-\frac {13 e^2 (d+e x)^{11/2}}{16 b^3 (a+b x)^3}-\frac {3 e (d+e x)^{13/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{15/2}}{5 b (a+b x)^5}-\frac {9009 e^5 (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{17/2}} \\ \end{align*}
Time = 1.93 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.72 \[ \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (-45045 a^7 e^7+105105 a^6 b e^6 (d-2 e x)-3003 a^5 b^2 e^5 \left (23 d^2-164 d e x+128 e^2 x^2\right )+2145 a^4 b^3 e^4 \left (3 d^3-152 d^2 e x+422 d e^2 x^2-158 e^3 x^3\right )+715 a^3 b^4 e^3 \left (2 d^4+44 d^3 e x-846 d^2 e^2 x^2+1124 d e^3 x^3-193 e^4 x^4\right )+65 a^2 b^5 e^2 \left (8 d^5+106 d^4 e x+938 d^3 e^2 x^2-8368 d^2 e^3 x^3+5089 d e^4 x^4-256 e^5 x^5\right )+5 a b^6 e \left (48 d^6+496 d^5 e x+2618 d^4 e^2 x^2+11620 d^3 e^3 x^3-45677 d^2 e^4 x^4+8192 d e^5 x^5+256 e^6 x^6\right )+b^7 \left (128 d^7+1136 d^6 e x+4648 d^5 e^2 x^2+12110 d^4 e^3 x^3+26635 d^3 e^4 x^4-29696 d^2 e^5 x^5-3072 d e^6 x^6-256 e^7 x^7\right )\right )}{640 b^8 (a+b x)^5}-\frac {9009 e^5 (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{17/2}} \]
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Time = 3.51 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.35
method | result | size |
risch | \(\frac {2 e^{5} \left (x^{2} b^{2} e^{2}-10 x a b \,e^{2}+12 b^{2} d e x +105 a^{2} e^{2}-220 a b d e +116 b^{2} d^{2}\right ) \sqrt {e x +d}}{5 b^{8}}-\frac {\left (2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) e^{5} \left (\frac {-\frac {5327 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}-\frac {9443 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {1001}{10} a^{2} b^{2} e^{2}+\frac {1001}{5} a \,b^{3} d e -\frac {1001}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {7837}{128} a^{3} b \,e^{3}+\frac {23511}{128} a^{2} b^{2} d \,e^{2}-\frac {23511}{128} a \,b^{3} d^{2} e +\frac {7837}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {3633}{256} e^{4} a^{4}+\frac {3633}{64} b \,e^{3} d \,a^{3}-\frac {10899}{128} b^{2} e^{2} d^{2} a^{2}+\frac {3633}{64} a \,b^{3} d^{3} e -\frac {3633}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {9009 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{8}}\) | \(342\) |
pseudoelliptic | \(-\frac {9009 \left (e^{5} \left (b x +a \right )^{5} \left (a e -b d \right )^{3} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )-\sqrt {e x +d}\, \left (\left (\frac {256}{45045} e^{7} x^{7}-\frac {128}{45045} d^{7}+\frac {1024}{15015} e^{6} x^{6} d +\frac {29696}{45045} d^{2} e^{5} x^{5}-\frac {761}{1287} d^{3} e^{4} x^{4}-\frac {346}{1287} d^{4} e^{3} x^{3}-\frac {664}{6435} e^{2} x^{2} d^{5}-\frac {1136}{45045} x \,d^{6} e \right ) b^{7}-\frac {16 \left (\frac {16}{3} e^{6} x^{6}+\frac {512}{3} d \,e^{5} x^{5}-\frac {45677}{48} d^{2} e^{4} x^{4}+\frac {2905}{12} x^{3} d^{3} e^{3}+\frac {1309}{24} d^{4} e^{2} x^{2}+\frac {31}{3} d^{5} e x +d^{6}\right ) e a \,b^{6}}{3003}-\frac {8 \left (-32 e^{5} x^{5}+\frac {5089}{8} x^{4} d \,e^{4}-1046 d^{2} e^{3} x^{3}+\frac {469}{4} d^{3} e^{2} x^{2}+\frac {53}{4} d^{4} e x +d^{5}\right ) e^{2} a^{2} b^{5}}{693}-\frac {2 \left (-\frac {193}{2} e^{4} x^{4}+562 d \,e^{3} x^{3}-423 d^{2} e^{2} x^{2}+22 d^{3} e x +d^{4}\right ) e^{3} a^{3} b^{4}}{63}-\frac {\left (-\frac {158}{3} e^{3} x^{3}+\frac {422}{3} d \,e^{2} x^{2}-\frac {152}{3} d^{2} e x +d^{3}\right ) e^{4} a^{4} b^{3}}{7}+\frac {23 e^{5} a^{5} \left (\frac {128}{23} x^{2} e^{2}-\frac {164}{23} d e x +d^{2}\right ) b^{2}}{15}-\frac {7 a^{6} e^{6} \left (-2 e x +d \right ) b}{3}+a^{7} e^{7}\right ) \sqrt {\left (a e -b d \right ) b}\right )}{128 \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} b^{8}}\) | \(437\) |
derivativedivides | \(2 e^{5} \left (\frac {\frac {b^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-2 a b e \left (e x +d \right )^{\frac {3}{2}}+2 b^{2} d \left (e x +d \right )^{\frac {3}{2}}+21 a^{2} e^{2} \sqrt {e x +d}-42 a b d e \sqrt {e x +d}+21 b^{2} d^{2} \sqrt {e x +d}}{b^{8}}-\frac {\frac {\left (-\frac {5327}{256} a^{3} b^{4} e^{3}+\frac {15981}{256} a^{2} b^{5} d \,e^{2}-\frac {15981}{256} a \,b^{6} d^{2} e +\frac {5327}{256} d^{3} b^{7}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {9443 b^{3} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {1001}{10} a^{5} b^{2} e^{5}+\frac {1001}{2} a^{4} b^{3} d \,e^{4}-1001 a^{3} b^{4} d^{2} e^{3}+1001 a^{2} b^{5} d^{3} e^{2}-\frac {1001}{2} a \,b^{6} d^{4} e +\frac {1001}{10} d^{5} b^{7}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {7837}{128} a^{6} b \,e^{6}+\frac {23511}{64} a^{5} b^{2} d \,e^{5}-\frac {117555}{128} a^{4} b^{3} d^{2} e^{4}+\frac {39185}{32} a^{3} b^{4} d^{3} e^{3}-\frac {117555}{128} a^{2} b^{5} d^{4} e^{2}+\frac {23511}{64} a \,b^{6} d^{5} e -\frac {7837}{128} d^{6} b^{7}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {3633}{256} a^{7} e^{7}+\frac {25431}{256} a^{6} b d \,e^{6}-\frac {76293}{256} a^{5} b^{2} d^{2} e^{5}+\frac {127155}{256} b^{3} d^{3} e^{4} a^{4}-\frac {127155}{256} b^{4} d^{4} e^{3} a^{3}+\frac {76293}{256} b^{5} d^{5} e^{2} a^{2}-\frac {25431}{256} b^{6} d^{6} e a +\frac {3633}{256} b^{7} d^{7}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {9009 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}}{b^{8}}\right )\) | \(569\) |
default | \(2 e^{5} \left (\frac {\frac {b^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-2 a b e \left (e x +d \right )^{\frac {3}{2}}+2 b^{2} d \left (e x +d \right )^{\frac {3}{2}}+21 a^{2} e^{2} \sqrt {e x +d}-42 a b d e \sqrt {e x +d}+21 b^{2} d^{2} \sqrt {e x +d}}{b^{8}}-\frac {\frac {\left (-\frac {5327}{256} a^{3} b^{4} e^{3}+\frac {15981}{256} a^{2} b^{5} d \,e^{2}-\frac {15981}{256} a \,b^{6} d^{2} e +\frac {5327}{256} d^{3} b^{7}\right ) \left (e x +d \right )^{\frac {9}{2}}-\frac {9443 b^{3} \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (-\frac {1001}{10} a^{5} b^{2} e^{5}+\frac {1001}{2} a^{4} b^{3} d \,e^{4}-1001 a^{3} b^{4} d^{2} e^{3}+1001 a^{2} b^{5} d^{3} e^{2}-\frac {1001}{2} a \,b^{6} d^{4} e +\frac {1001}{10} d^{5} b^{7}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (-\frac {7837}{128} a^{6} b \,e^{6}+\frac {23511}{64} a^{5} b^{2} d \,e^{5}-\frac {117555}{128} a^{4} b^{3} d^{2} e^{4}+\frac {39185}{32} a^{3} b^{4} d^{3} e^{3}-\frac {117555}{128} a^{2} b^{5} d^{4} e^{2}+\frac {23511}{64} a \,b^{6} d^{5} e -\frac {7837}{128} d^{6} b^{7}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {3633}{256} a^{7} e^{7}+\frac {25431}{256} a^{6} b d \,e^{6}-\frac {76293}{256} a^{5} b^{2} d^{2} e^{5}+\frac {127155}{256} b^{3} d^{3} e^{4} a^{4}-\frac {127155}{256} b^{4} d^{4} e^{3} a^{3}+\frac {76293}{256} b^{5} d^{5} e^{2} a^{2}-\frac {25431}{256} b^{6} d^{6} e a +\frac {3633}{256} b^{7} d^{7}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {9009 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}}{b^{8}}\right )\) | \(569\) |
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Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (209) = 418\).
Time = 0.36 (sec) , antiderivative size = 1606, normalized size of antiderivative = 6.35 \[ \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (209) = 418\).
Time = 0.30 (sec) , antiderivative size = 784, normalized size of antiderivative = 3.10 \[ \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {9009 \, {\left (b^{3} d^{3} e^{5} - 3 \, a b^{2} d^{2} e^{6} + 3 \, a^{2} b d e^{7} - a^{3} e^{8}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{8}} - \frac {26635 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{7} d^{3} e^{5} - 94430 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{7} d^{4} e^{5} + 128128 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{7} d^{5} e^{5} - 78370 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{7} d^{6} e^{5} + 18165 \, \sqrt {e x + d} b^{7} d^{7} e^{5} - 79905 \, {\left (e x + d\right )}^{\frac {9}{2}} a b^{6} d^{2} e^{6} + 377720 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{6} d^{3} e^{6} - 640640 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{6} d^{4} e^{6} + 470220 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{6} d^{5} e^{6} - 127155 \, \sqrt {e x + d} a b^{6} d^{6} e^{6} + 79905 \, {\left (e x + d\right )}^{\frac {9}{2}} a^{2} b^{5} d e^{7} - 566580 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{2} b^{5} d^{2} e^{7} + 1281280 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{5} d^{3} e^{7} - 1175550 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{5} d^{4} e^{7} + 381465 \, \sqrt {e x + d} a^{2} b^{5} d^{5} e^{7} - 26635 \, {\left (e x + d\right )}^{\frac {9}{2}} a^{3} b^{4} e^{8} + 377720 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{3} b^{4} d e^{8} - 1281280 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{3} b^{4} d^{2} e^{8} + 1567400 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{4} d^{3} e^{8} - 635775 \, \sqrt {e x + d} a^{3} b^{4} d^{4} e^{8} - 94430 \, {\left (e x + d\right )}^{\frac {7}{2}} a^{4} b^{3} e^{9} + 640640 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{4} b^{3} d e^{9} - 1175550 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{4} b^{3} d^{2} e^{9} + 635775 \, \sqrt {e x + d} a^{4} b^{3} d^{3} e^{9} - 128128 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{5} b^{2} e^{10} + 470220 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{5} b^{2} d e^{10} - 381465 \, \sqrt {e x + d} a^{5} b^{2} d^{2} e^{10} - 78370 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{6} b e^{11} + 127155 \, \sqrt {e x + d} a^{6} b d e^{11} - 18165 \, \sqrt {e x + d} a^{7} e^{12}}{640 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5} b^{8}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {5}{2}} b^{24} e^{5} + 10 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{24} d e^{5} + 105 \, \sqrt {e x + d} b^{24} d^{2} e^{5} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{23} e^{6} - 210 \, \sqrt {e x + d} a b^{23} d e^{6} + 105 \, \sqrt {e x + d} a^{2} b^{22} e^{7}\right )}}{5 \, b^{30}} \]
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Time = 9.74 (sec) , antiderivative size = 846, normalized size of antiderivative = 3.34 \[ \int \frac {(d+e x)^{15/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left (\frac {2\,e^5\,{\left (6\,b^6\,d-6\,a\,b^5\,e\right )}^2}{b^{18}}-\frac {30\,e^5\,{\left (a\,e-b\,d\right )}^2}{b^8}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (\frac {3633\,a^7\,e^{12}}{128}-\frac {25431\,a^6\,b\,d\,e^{11}}{128}+\frac {76293\,a^5\,b^2\,d^2\,e^{10}}{128}-\frac {127155\,a^4\,b^3\,d^3\,e^9}{128}+\frac {127155\,a^3\,b^4\,d^4\,e^8}{128}-\frac {76293\,a^2\,b^5\,d^5\,e^7}{128}+\frac {25431\,a\,b^6\,d^6\,e^6}{128}-\frac {3633\,b^7\,d^7\,e^5}{128}\right )+{\left (d+e\,x\right )}^{5/2}\,\left (\frac {1001\,a^5\,b^2\,e^{10}}{5}-1001\,a^4\,b^3\,d\,e^9+2002\,a^3\,b^4\,d^2\,e^8-2002\,a^2\,b^5\,d^3\,e^7+1001\,a\,b^6\,d^4\,e^6-\frac {1001\,b^7\,d^5\,e^5}{5}\right )+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {7837\,a^6\,b\,e^{11}}{64}-\frac {23511\,a^5\,b^2\,d\,e^{10}}{32}+\frac {117555\,a^4\,b^3\,d^2\,e^9}{64}-\frac {39185\,a^3\,b^4\,d^3\,e^8}{16}+\frac {117555\,a^2\,b^5\,d^4\,e^7}{64}-\frac {23511\,a\,b^6\,d^5\,e^6}{32}+\frac {7837\,b^7\,d^6\,e^5}{64}\right )+{\left (d+e\,x\right )}^{9/2}\,\left (\frac {5327\,a^3\,b^4\,e^8}{128}-\frac {15981\,a^2\,b^5\,d\,e^7}{128}+\frac {15981\,a\,b^6\,d^2\,e^6}{128}-\frac {5327\,b^7\,d^3\,e^5}{128}\right )+{\left (d+e\,x\right )}^{7/2}\,\left (\frac {9443\,a^4\,b^3\,e^9}{64}-\frac {9443\,a^3\,b^4\,d\,e^8}{16}+\frac {28329\,a^2\,b^5\,d^2\,e^7}{32}-\frac {9443\,a\,b^6\,d^3\,e^6}{16}+\frac {9443\,b^7\,d^4\,e^5}{64}\right )}{\left (d+e\,x\right )\,\left (5\,a^4\,b^9\,e^4-20\,a^3\,b^{10}\,d\,e^3+30\,a^2\,b^{11}\,d^2\,e^2-20\,a\,b^{12}\,d^3\,e+5\,b^{13}\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^{10}\,e^3+30\,a^2\,b^{11}\,d\,e^2-30\,a\,b^{12}\,d^2\,e+10\,b^{13}\,d^3\right )+b^{13}\,{\left (d+e\,x\right )}^5-\left (5\,b^{13}\,d-5\,a\,b^{12}\,e\right )\,{\left (d+e\,x\right )}^4-b^{13}\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^{11}\,e^2-20\,a\,b^{12}\,d\,e+10\,b^{13}\,d^2\right )+a^5\,b^8\,e^5-5\,a^4\,b^9\,d\,e^4-10\,a^2\,b^{11}\,d^3\,e^2+10\,a^3\,b^{10}\,d^2\,e^3+5\,a\,b^{12}\,d^4\,e}+\frac {2\,e^5\,{\left (d+e\,x\right )}^{5/2}}{5\,b^6}+\frac {2\,e^5\,\left (6\,b^6\,d-6\,a\,b^5\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^{12}}-\frac {9009\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^5\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^8-3\,a^2\,b\,d\,e^7+3\,a\,b^2\,d^2\,e^6-b^3\,d^3\,e^5}\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{128\,b^{17/2}} \]
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