Integrand size = 28, antiderivative size = 198 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {e^2 \sqrt {d+e x}}{16 b^3 (a+b x)^3}-\frac {e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) (a+b x)^2}+\frac {3 e^4 \sqrt {d+e x}}{128 b^3 (b d-a e)^2 (a+b x)}-\frac {e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{5/2}}{5 b (a+b x)^5}-\frac {3 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 44, 65, 214} \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {3 e^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{5/2}}+\frac {3 e^4 \sqrt {d+e x}}{128 b^3 (a+b x) (b d-a e)^2}-\frac {e^3 \sqrt {d+e x}}{64 b^3 (a+b x)^2 (b d-a e)}-\frac {e^2 \sqrt {d+e x}}{16 b^3 (a+b x)^3}-\frac {e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{5/2}}{5 b (a+b x)^5} \]
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Rule 27
Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \frac {(d+e x)^{5/2}}{(a+b x)^6} \, dx \\ & = -\frac {(d+e x)^{5/2}}{5 b (a+b x)^5}+\frac {e \int \frac {(d+e x)^{3/2}}{(a+b x)^5} \, dx}{2 b} \\ & = -\frac {e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 e^2\right ) \int \frac {\sqrt {d+e x}}{(a+b x)^4} \, dx}{16 b^2} \\ & = -\frac {e^2 \sqrt {d+e x}}{16 b^3 (a+b x)^3}-\frac {e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{5/2}}{5 b (a+b x)^5}+\frac {e^3 \int \frac {1}{(a+b x)^3 \sqrt {d+e x}} \, dx}{32 b^3} \\ & = -\frac {e^2 \sqrt {d+e x}}{16 b^3 (a+b x)^3}-\frac {e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) (a+b x)^2}-\frac {e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{5/2}}{5 b (a+b x)^5}-\frac {\left (3 e^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {d+e x}} \, dx}{128 b^3 (b d-a e)} \\ & = -\frac {e^2 \sqrt {d+e x}}{16 b^3 (a+b x)^3}-\frac {e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) (a+b x)^2}+\frac {3 e^4 \sqrt {d+e x}}{128 b^3 (b d-a e)^2 (a+b x)}-\frac {e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 e^5\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{256 b^3 (b d-a e)^2} \\ & = -\frac {e^2 \sqrt {d+e x}}{16 b^3 (a+b x)^3}-\frac {e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) (a+b x)^2}+\frac {3 e^4 \sqrt {d+e x}}{128 b^3 (b d-a e)^2 (a+b x)}-\frac {e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{5/2}}{5 b (a+b x)^5}+\frac {\left (3 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^3 (b d-a e)^2} \\ & = -\frac {e^2 \sqrt {d+e x}}{16 b^3 (a+b x)^3}-\frac {e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) (a+b x)^2}+\frac {3 e^4 \sqrt {d+e x}}{128 b^3 (b d-a e)^2 (a+b x)}-\frac {e (d+e x)^{3/2}}{8 b^2 (a+b x)^4}-\frac {(d+e x)^{5/2}}{5 b (a+b x)^5}-\frac {3 e^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{128 b^{7/2} (b d-a e)^{5/2}} \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.12 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=-\frac {\sqrt {d+e x} \left (15 a^4 e^4+10 a^3 b e^3 (d+7 e x)+2 a^2 b^2 e^2 \left (4 d^2+23 d e x+64 e^2 x^2\right )-2 a b^3 e \left (88 d^3+256 d^2 e x+233 d e^2 x^2+35 e^3 x^3\right )+b^4 \left (128 d^4+336 d^3 e x+248 d^2 e^2 x^2+10 d e^3 x^3-15 e^4 x^4\right )\right )}{640 b^3 (b d-a e)^2 (a+b x)^5}+\frac {3 e^5 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{128 b^{7/2} (-b d+a e)^{5/2}} \]
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Time = 3.70 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.04
method | result | size |
derivativedivides | \(2 e^{5} \left (\frac {\frac {3 b \left (e x +d \right )^{\frac {9}{2}}}{256 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {7 \left (e x +d \right )^{\frac {7}{2}}}{128 \left (a e -b d \right )}-\frac {\left (e x +d \right )^{\frac {5}{2}}}{10 b}-\frac {7 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{2}}-\frac {3 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}}{256 b^{3}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(205\) |
default | \(2 e^{5} \left (\frac {\frac {3 b \left (e x +d \right )^{\frac {9}{2}}}{256 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right )}+\frac {7 \left (e x +d \right )^{\frac {7}{2}}}{128 \left (a e -b d \right )}-\frac {\left (e x +d \right )^{\frac {5}{2}}}{10 b}-\frac {7 \left (a e -b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{128 b^{2}}-\frac {3 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {e x +d}}{256 b^{3}}}{\left (b \left (e x +d \right )+a e -b d \right )^{5}}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{256 b^{3} \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \sqrt {\left (a e -b d \right ) b}}\right )\) | \(205\) |
pseudoelliptic | \(\frac {-\frac {3 \sqrt {\left (a e -b d \right ) b}\, \left (\left (-b^{4} x^{4}-\frac {14}{3} a \,b^{3} x^{3}+\frac {128}{15} a^{2} b^{2} x^{2}+\frac {14}{3} a^{3} b x +a^{4}\right ) e^{4}+\frac {2 b \left (b^{3} x^{3}-\frac {233}{5} a \,b^{2} x^{2}+\frac {23}{5} a^{2} b x +a^{3}\right ) d \,e^{3}}{3}+\frac {8 b^{2} d^{2} \left (31 b^{2} x^{2}-64 a b x +a^{2}\right ) e^{2}}{15}-\frac {176 \left (-\frac {21 b x}{11}+a \right ) b^{3} d^{3} e}{15}+\frac {128 b^{4} d^{4}}{15}\right ) \sqrt {e x +d}}{128}+\frac {3 e^{5} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128}}{\sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )^{5} \left (a e -b d \right )^{2} b^{3}}\) | \(219\) |
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Leaf count of result is larger than twice the leaf count of optimal. 662 vs. \(2 (166) = 332\).
Time = 0.32 (sec) , antiderivative size = 1337, normalized size of antiderivative = 6.75 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 380 vs. \(2 (166) = 332\).
Time = 0.29 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.92 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3 \, e^{5} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{128 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \sqrt {-b^{2} d + a b e}} + \frac {15 \, {\left (e x + d\right )}^{\frac {9}{2}} b^{4} e^{5} - 70 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{5} - 128 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{5} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{5} - 15 \, \sqrt {e x + d} b^{4} d^{4} e^{5} + 70 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{6} + 256 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{6} - 210 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{6} + 60 \, \sqrt {e x + d} a b^{3} d^{3} e^{6} - 128 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{7} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{7} - 90 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{7} - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{8} + 60 \, \sqrt {e x + d} a^{3} b d e^{8} - 15 \, \sqrt {e x + d} a^{4} e^{9}}{640 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{5}} \]
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Time = 0.17 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.08 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^3} \, dx=\frac {3\,e^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}}{\sqrt {a\,e-b\,d}}\right )}{128\,b^{7/2}\,{\left (a\,e-b\,d\right )}^{5/2}}-\frac {\frac {e^5\,{\left (d+e\,x\right )}^{5/2}}{5\,b}-\frac {7\,e^5\,{\left (d+e\,x\right )}^{7/2}}{64\,\left (a\,e-b\,d\right )}+\frac {3\,e^5\,\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{128\,b^3}+\frac {7\,e^5\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{3/2}}{64\,b^2}-\frac {3\,b\,e^5\,{\left (d+e\,x\right )}^{9/2}}{128\,{\left (a\,e-b\,d\right )}^2}}{\left (d+e\,x\right )\,\left (5\,a^4\,b\,e^4-20\,a^3\,b^2\,d\,e^3+30\,a^2\,b^3\,d^2\,e^2-20\,a\,b^4\,d^3\,e+5\,b^5\,d^4\right )-{\left (d+e\,x\right )}^2\,\left (-10\,a^3\,b^2\,e^3+30\,a^2\,b^3\,d\,e^2-30\,a\,b^4\,d^2\,e+10\,b^5\,d^3\right )+b^5\,{\left (d+e\,x\right )}^5-\left (5\,b^5\,d-5\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^4+a^5\,e^5-b^5\,d^5+{\left (d+e\,x\right )}^3\,\left (10\,a^2\,b^3\,e^2-20\,a\,b^4\,d\,e+10\,b^5\,d^2\right )-10\,a^2\,b^3\,d^3\,e^2+10\,a^3\,b^2\,d^2\,e^3+5\,a\,b^4\,d^4\,e-5\,a^4\,b\,d\,e^4} \]
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