Integrand size = 28, antiderivative size = 239 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {693 \sqrt {b} e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}} \]
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Time = 0.09 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {693 \sqrt {b} e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}}-\frac {693 e^5}{128 \sqrt {d+e x} (b d-a e)^6}-\frac {231 e^4}{128 (a+b x) \sqrt {d+e x} (b d-a e)^5}+\frac {231 e^3}{320 (a+b x)^2 \sqrt {d+e x} (b d-a e)^4}-\frac {33 e^2}{80 (a+b x)^3 \sqrt {d+e x} (b d-a e)^3}+\frac {11 e}{40 (a+b x)^4 \sqrt {d+e x} (b d-a e)^2}-\frac {1}{5 (a+b x)^5 \sqrt {d+e x} (b d-a e)} \]
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Rule 27
Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(a+b x)^6 (d+e x)^{3/2}} \, dx \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}-\frac {(11 e) \int \frac {1}{(a+b x)^5 (d+e x)^{3/2}} \, dx}{10 (b d-a e)} \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}+\frac {\left (99 e^2\right ) \int \frac {1}{(a+b x)^4 (d+e x)^{3/2}} \, dx}{80 (b d-a e)^2} \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}-\frac {\left (231 e^3\right ) \int \frac {1}{(a+b x)^3 (d+e x)^{3/2}} \, dx}{160 (b d-a e)^3} \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}+\frac {\left (231 e^4\right ) \int \frac {1}{(a+b x)^2 (d+e x)^{3/2}} \, dx}{128 (b d-a e)^4} \\ & = -\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 e^5\right ) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{256 (b d-a e)^5} \\ & = -\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 b e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 (b d-a e)^6} \\ & = -\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}-\frac {\left (693 b e^4\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 d-a e)^6} \\ & = -\frac {693 e^5}{128 (b d-a e)^6 \sqrt {d+e x}}-\frac {1}{5 (b d-a e) (a+b x)^5 \sqrt {d+e x}}+\frac {11 e}{40 (b d-a e)^2 (a+b x)^4 \sqrt {d+e x}}-\frac {33 e^2}{80 (b d-a e)^3 (a+b x)^3 \sqrt {d+e x}}+\frac {231 e^3}{320 (b d-a e)^4 (a+b x)^2 \sqrt {d+e x}}-\frac {231 e^4}{128 (b d-a e)^5 (a+b x) \sqrt {d+e x}}+\frac {693 \sqrt {b} e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 (b d-a e)^{13/2}} \\ \end{align*}
Time = 1.48 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.19 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {1}{640} \left (\frac {-1280 a^5 e^5-5 a^4 b e^4 (843 d+2123 e x)-10 a^3 b^2 e^3 \left (-359 d^2+968 d e x+2607 e^2 x^2\right )-2 a^2 b^3 e^2 \left (1124 d^3-2013 d^2 e x+5247 d e^2 x^2+14784 e^3 x^3\right )-2 a b^4 e \left (-408 d^4+616 d^3 e x-1089 d^2 e^2 x^2+2772 d e^3 x^3+8085 e^4 x^4\right )-b^5 \left (128 d^5-176 d^4 e x+264 d^3 e^2 x^2-462 d^2 e^3 x^3+1155 d e^4 x^4+3465 e^5 x^5\right )}{(b d-a e)^6 (a+b x)^5 \sqrt {d+e x}}-\frac {3465 \sqrt {b} e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{13/2}}\right ) \]
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Time = 2.50 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(2 e^{5} \left (-\frac {1}{\left (a e -b d \right )^{6} \sqrt {e x +d}}-\frac {b \left (\frac {\frac {437 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {977 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (\frac {131}{10} a^{2} b^{2} e^{2}-\frac {131}{5} a \,b^{3} d e +\frac {131}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {1327}{128} a^{3} b \,e^{3}-\frac {3981}{128} a^{2} b^{2} d \,e^{2}+\frac {3981}{128} a \,b^{3} d^{2} e -\frac {1327}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {843}{256} e^{4} a^{4}-\frac {843}{64} b \,e^{3} d \,a^{3}+\frac {2529}{128} b^{2} e^{2} d^{2} a^{2}-\frac {843}{64} a \,b^{3} d^{3} e +\frac {843}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {693 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{6}}\right )\) | \(270\) |
default | \(2 e^{5} \left (-\frac {1}{\left (a e -b d \right )^{6} \sqrt {e x +d}}-\frac {b \left (\frac {\frac {437 b^{4} \left (e x +d \right )^{\frac {9}{2}}}{256}+\frac {977 \left (a e -b d \right ) b^{3} \left (e x +d \right )^{\frac {7}{2}}}{128}+\left (\frac {131}{10} a^{2} b^{2} e^{2}-\frac {131}{5} a \,b^{3} d e +\frac {131}{10} b^{4} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}+\left (\frac {1327}{128} a^{3} b \,e^{3}-\frac {3981}{128} a^{2} b^{2} d \,e^{2}+\frac {3981}{128} a \,b^{3} d^{2} e -\frac {1327}{128} b^{4} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (\frac {843}{256} e^{4} a^{4}-\frac {843}{64} b \,e^{3} d \,a^{3}+\frac {2529}{128} b^{2} e^{2} d^{2} a^{2}-\frac {843}{64} a \,b^{3} d^{3} e +\frac {843}{256} b^{4} d^{4}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {693 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{6}}\right )\) | \(270\) |
pseudoelliptic | \(-\frac {693 \left (b \,e^{5} \sqrt {e x +d}\, \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )+\frac {256 \left (\left (\frac {693}{256} e^{5} x^{5}+\frac {231}{256} x^{4} d \,e^{4}-\frac {231}{640} d^{2} e^{3} x^{3}+\frac {33}{160} d^{3} e^{2} x^{2}-\frac {11}{80} d^{4} e x +\frac {1}{10} d^{5}\right ) b^{5}-\frac {51 \left (-\frac {2695}{136} e^{4} x^{4}-\frac {231}{34} d \,e^{3} x^{3}+\frac {363}{136} d^{2} e^{2} x^{2}-\frac {77}{51} d^{3} e x +d^{4}\right ) e a \,b^{4}}{80}+\frac {281 e^{2} \left (\frac {3696}{281} e^{3} x^{3}+\frac {5247}{1124} d \,e^{2} x^{2}-\frac {2013}{1124} d^{2} e x +d^{3}\right ) a^{2} b^{3}}{160}-\frac {359 \left (-\frac {2607}{359} x^{2} e^{2}-\frac {968}{359} d e x +d^{2}\right ) e^{3} a^{3} b^{2}}{128}+\frac {843 \left (\frac {2123 e x}{843}+d \right ) e^{4} a^{4} b}{256}+a^{5} e^{5}\right ) \sqrt {\left (a e -b d \right ) b}}{693}\right )}{128 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} \left (a e -b d \right )^{6}}\) | \(287\) |
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Leaf count of result is larger than twice the leaf count of optimal. 1152 vs. \(2 (203) = 406\).
Time = 1.02 (sec) , antiderivative size = 2314, normalized size of antiderivative = 9.68 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (203) = 406\).
Time = 0.30 (sec) , antiderivative size = 579, normalized size of antiderivative = 2.42 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {693 \, b e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {-b^{2} d + a b e}} - \frac {2 \, e^{5}}{{\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} \sqrt {e x + d}} - \frac {2185 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{5} e^{5} - 9770 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} d e^{5} + 16768 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d^{2} e^{5} - 13270 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{3} e^{5} + 4215 \, \sqrt {e x + d} b^{5} d^{4} e^{5} + 9770 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{4} e^{6} - 33536 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} d e^{6} + 39810 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d^{2} e^{6} - 16860 \, \sqrt {e x + d} a b^{4} d^{3} e^{6} + 16768 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{3} e^{7} - 39810 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} d e^{7} + 25290 \, \sqrt {e x + d} a^{2} b^{3} d^{2} e^{7} + 13270 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b^{2} e^{8} - 16860 \, \sqrt {e x + d} a^{3} b^{2} d e^{8} + 4215 \, \sqrt {e x + d} a^{4} b e^{9}}{640 \, {\left (b^{6} d^{6} - 6 \, a b^{5} d^{5} e + 15 \, a^{2} b^{4} d^{4} e^{2} - 20 \, a^{3} b^{3} d^{3} e^{3} + 15 \, a^{4} b^{2} d^{2} e^{4} - 6 \, a^{5} b d e^{5} + a^{6} e^{6}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]
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Time = 10.10 (sec) , antiderivative size = 515, normalized size of antiderivative = 2.15 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\frac {2\,e^5}{a\,e-b\,d}+\frac {2607\,b^2\,e^5\,{\left (d+e\,x\right )}^2}{64\,{\left (a\,e-b\,d\right )}^3}+\frac {231\,b^3\,e^5\,{\left (d+e\,x\right )}^3}{5\,{\left (a\,e-b\,d\right )}^4}+\frac {1617\,b^4\,e^5\,{\left (d+e\,x\right )}^4}{64\,{\left (a\,e-b\,d\right )}^5}+\frac {693\,b^5\,e^5\,{\left (d+e\,x\right )}^5}{128\,{\left (a\,e-b\,d\right )}^6}+\frac {2123\,b\,e^5\,\left (d+e\,x\right )}{128\,{\left (a\,e-b\,d\right )}^2}}{\sqrt {d+e\,x}\,\left (a^5\,e^5-5\,a^4\,b\,d\,e^4+10\,a^3\,b^2\,d^2\,e^3-10\,a^2\,b^3\,d^3\,e^2+5\,a\,b^4\,d^4\,e-b^5\,d^5\right )-{\left (d+e\,x\right )}^{5/2}\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+{\left (d+e\,x\right )}^{3/2}\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )+b^5\,{\left (d+e\,x\right )}^{11/2}-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{9/2}+{\left (d+e\,x\right )}^{7/2}\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )}-\frac {693\,\sqrt {b}\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^6\,e^6-6\,a^5\,b\,d\,e^5+15\,a^4\,b^2\,d^2\,e^4-20\,a^3\,b^3\,d^3\,e^3+15\,a^2\,b^4\,d^4\,e^2-6\,a\,b^5\,d^5\,e+b^6\,d^6\right )}{{\left (a\,e-b\,d\right )}^{13/2}}\right )}{128\,{\left (a\,e-b\,d\right )}^{13/2}} \]
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