Integrand size = 28, antiderivative size = 178 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}-\frac {63 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} \sqrt {b d-a e}} \]
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Time = 0.06 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 43, 65, 214} \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {63 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} \sqrt {b d-a e}}-\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5} \]
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Rule 27
Rule 43
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{9/2}}{(a+b x)^6} \, dx \\ & = -\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {(9 e) \int \frac {(d+e x)^{7/2}}{(a+b x)^5} \, dx}{10 b} \\ & = -\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (63 e^2\right ) \int \frac {(d+e x)^{5/2}}{(a+b x)^4} \, dx}{80 b^2} \\ & = -\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (21 e^3\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^3} \, dx}{32 b^3} \\ & = -\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (63 e^4\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^2} \, dx}{128 b^4} \\ & = -\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (63 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^5} \\ & = -\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}+\frac {\left (63 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^5} \\ & = -\frac {63 e^4 \sqrt {d+e x}}{128 b^5 (a+b x)}-\frac {21 e^3 (d+e x)^{3/2}}{64 b^4 (a+b x)^2}-\frac {21 e^2 (d+e x)^{5/2}}{80 b^3 (a+b x)^3}-\frac {9 e (d+e x)^{7/2}}{40 b^2 (a+b x)^4}-\frac {(d+e x)^{9/2}}{5 b (a+b x)^5}-\frac {63 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{11/2} \sqrt {b d-a e}} \\ \end{align*}
Time = 1.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.19 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (315 a^4 e^4+210 a^3 b e^3 (d+7 e x)+42 a^2 b^2 e^2 \left (4 d^2+23 d e x+64 e^2 x^2\right )+6 a b^3 e \left (24 d^3+128 d^2 e x+289 d e^2 x^2+395 e^3 x^3\right )+b^4 \left (128 d^4+656 d^3 e x+1368 d^2 e^2 x^2+1490 d e^3 x^3+965 e^4 x^4\right )\right )}{640 b^5 (a+b x)^5}+\frac {63 e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{11/2} \sqrt {-b d+a e}} \]
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Time = 3.53 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.19
method | result | size |
pseudoelliptic | \(\frac {\frac {63 e^{5} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}-\frac {63 \left (\left (\frac {193}{63} e^{4} x^{4}+\frac {298}{63} d \,e^{3} x^{3}+\frac {152}{35} d^{2} e^{2} x^{2}+\frac {656}{315} d^{3} e x +\frac {128}{315} d^{4}\right ) b^{4}+\frac {16 \left (\frac {395}{24} e^{3} x^{3}+\frac {289}{24} d \,e^{2} x^{2}+\frac {16}{3} d^{2} e x +d^{3}\right ) e a \,b^{3}}{35}+\frac {8 e^{2} \left (16 x^{2} e^{2}+\frac {23}{4} d e x +d^{2}\right ) a^{2} b^{2}}{15}+\frac {2 a^{3} e^{3} \left (7 e x +d \right ) b}{3}+e^{4} a^{4}\right ) \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}}{128}}{b^{5} \left (b x +a \right )^{5} \sqrt {\left (a e -b d \right ) b}}\) | \(211\) |
derivativedivides | \(2 e^{5} \left (\frac {-\frac {193 \left (e x +d \right )^{\frac {9}{2}}}{256 b}-\frac {237 \left (a e -b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{128 b^{2}}-\frac {21 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{10 b^{3}}-\frac {147 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{4}}-\frac {63 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {e x +d}}{256 b^{5}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {63 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{5} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(239\) |
default | \(2 e^{5} \left (\frac {-\frac {193 \left (e x +d \right )^{\frac {9}{2}}}{256 b}-\frac {237 \left (a e -b d \right ) \left (e x +d \right )^{\frac {7}{2}}}{128 b^{2}}-\frac {21 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {5}{2}}}{10 b^{3}}-\frac {147 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{4}}-\frac {63 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \sqrt {e x +d}}{256 b^{5}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {63 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{5} \sqrt {\left (a e -b d \right ) b}}\right )\) | \(239\) |
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Leaf count of result is larger than twice the leaf count of optimal. 495 vs. \(2 (146) = 292\).
Time = 0.36 (sec) , antiderivative size = 1003, normalized size of antiderivative = 5.63 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\left [\frac {315 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (128 \, b^{6} d^{5} + 16 \, a b^{5} d^{4} e + 24 \, a^{2} b^{4} d^{3} e^{2} + 42 \, a^{3} b^{3} d^{2} e^{3} + 105 \, a^{4} b^{2} d e^{4} - 315 \, a^{5} b e^{5} + 965 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{4} + 10 \, {\left (149 \, b^{6} d^{2} e^{3} + 88 \, a b^{5} d e^{4} - 237 \, a^{2} b^{4} e^{5}\right )} x^{3} + 6 \, {\left (228 \, b^{6} d^{3} e^{2} + 61 \, a b^{5} d^{2} e^{3} + 159 \, a^{2} b^{4} d e^{4} - 448 \, a^{3} b^{3} e^{5}\right )} x^{2} + 2 \, {\left (328 \, b^{6} d^{4} e + 56 \, a b^{5} d^{3} e^{2} + 99 \, a^{2} b^{4} d^{2} e^{3} + 252 \, a^{3} b^{3} d e^{4} - 735 \, a^{4} b^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{1280 \, {\left (a^{5} b^{7} d - a^{6} b^{6} e + {\left (b^{12} d - a b^{11} e\right )} x^{5} + 5 \, {\left (a b^{11} d - a^{2} b^{10} e\right )} x^{4} + 10 \, {\left (a^{2} b^{10} d - a^{3} b^{9} e\right )} x^{3} + 10 \, {\left (a^{3} b^{9} d - a^{4} b^{8} e\right )} x^{2} + 5 \, {\left (a^{4} b^{8} d - a^{5} b^{7} e\right )} x\right )}}, \frac {315 \, {\left (b^{5} e^{5} x^{5} + 5 \, a b^{4} e^{5} x^{4} + 10 \, a^{2} b^{3} e^{5} x^{3} + 10 \, a^{3} b^{2} e^{5} x^{2} + 5 \, a^{4} b e^{5} x + a^{5} e^{5}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (128 \, b^{6} d^{5} + 16 \, a b^{5} d^{4} e + 24 \, a^{2} b^{4} d^{3} e^{2} + 42 \, a^{3} b^{3} d^{2} e^{3} + 105 \, a^{4} b^{2} d e^{4} - 315 \, a^{5} b e^{5} + 965 \, {\left (b^{6} d e^{4} - a b^{5} e^{5}\right )} x^{4} + 10 \, {\left (149 \, b^{6} d^{2} e^{3} + 88 \, a b^{5} d e^{4} - 237 \, a^{2} b^{4} e^{5}\right )} x^{3} + 6 \, {\left (228 \, b^{6} d^{3} e^{2} + 61 \, a b^{5} d^{2} e^{3} + 159 \, a^{2} b^{4} d e^{4} - 448 \, a^{3} b^{3} e^{5}\right )} x^{2} + 2 \, {\left (328 \, b^{6} d^{4} e + 56 \, a b^{5} d^{3} e^{2} + 99 \, a^{2} b^{4} d^{2} e^{3} + 252 \, a^{3} b^{3} d e^{4} - 735 \, a^{4} b^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{640 \, {\left (a^{5} b^{7} d - a^{6} b^{6} e + {\left (b^{12} d - a b^{11} e\right )} x^{5} + 5 \, {\left (a b^{11} d - a^{2} b^{10} e\right )} x^{4} + 10 \, {\left (a^{2} b^{10} d - a^{3} b^{9} e\right )} x^{3} + 10 \, {\left (a^{3} b^{9} d - a^{4} b^{8} e\right )} x^{2} + 5 \, {\left (a^{4} b^{8} d - a^{5} b^{7} e\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (146) = 292\).
Time = 0.31 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63 \, e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, \sqrt {-b^{2} d + a b e} b^{5}} - \frac {965 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 2370 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{5} + 2688 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} - 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} + 315 \, \sqrt {e x + d} b^{4} d^{4} e^{5} + 2370 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{6} - 5376 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} + 4410 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} - 1260 \, \sqrt {e x + d} a b^{3} d^{3} e^{6} + 2688 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} - 4410 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} + 1890 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{7} + 1470 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{8} - 1260 \, \sqrt {e x + d} a^{3} b d e^{8} + 315 \, \sqrt {e x + d} a^{4} e^{9}}{640 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5} b^{5}} \]
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Time = 9.56 (sec) , antiderivative size = 480, normalized size of antiderivative = 2.70 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {63\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{11/2}\,\sqrt {a\,e-b\,d}}-\frac {\frac {193\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,b}+\frac {63\,e^5\,\sqrt {d+e\,x}\,\left (a^4\,e^4-4\,a^3\,b\,d\,e^3+6\,a^2\,b^2\,d^2\,e^2-4\,a\,b^3\,d^3\,e+b^4\,d^4\right )}{128\,b^5}+\frac {21\,e^5\,{\left (d+e\,x\right )}^{5/2}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{5\,b^3}+\frac {147\,e^5\,{\left (d+e\,x\right )}^{3/2}\,\left (a^3\,e^3-3\,a^2\,b\,d\,e^2+3\,a\,b^2\,d^2\,e-b^3\,d^3\right )}{64\,b^4}+\frac {237\,e^5\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{64\,b^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \]
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