Integrand size = 28, antiderivative size = 145 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {35 e^3 \sqrt {d+e x}}{8 b^4}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}-\frac {35 e^3 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2}} \]
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Time = 0.05 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {35 e^3 \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2}}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {35 e^3 \sqrt {d+e x}}{8 b^4} \]
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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)^{7/2}}{(a+b x)^4} \, dx \\ & = -\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{(a+b x)^3} \, dx}{6 b} \\ & = -\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {\left (35 e^2\right ) \int \frac {(d+e x)^{3/2}}{(a+b x)^2} \, dx}{24 b^2} \\ & = -\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {\left (35 e^3\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{16 b^3} \\ & = \frac {35 e^3 \sqrt {d+e x}}{8 b^4}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {\left (35 e^3 (b d-a e)\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{16 b^4} \\ & = \frac {35 e^3 \sqrt {d+e x}}{8 b^4}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}+\frac {\left (35 e^2 (b d-a e)\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{8 b^4} \\ & = \frac {35 e^3 \sqrt {d+e x}}{8 b^4}-\frac {35 e^2 (d+e x)^{3/2}}{24 b^3 (a+b x)}-\frac {7 e (d+e x)^{5/2}}{12 b^2 (a+b x)^2}-\frac {(d+e x)^{7/2}}{3 b (a+b x)^3}-\frac {35 e^3 \sqrt {b d-a e} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{8 b^{9/2}} \\ \end{align*}
Time = 0.59 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.10 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=-\frac {\sqrt {d+e x} \left (-105 a^3 e^3+35 a^2 b e^2 (d-8 e x)+7 a b^2 e \left (2 d^2+14 d e x-33 e^2 x^2\right )+b^3 \left (8 d^3+38 d^2 e x+87 d e^2 x^2-48 e^3 x^3\right )\right )}{24 b^4 (a+b x)^3}-\frac {35 e^3 \sqrt {-b d+a e} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{8 b^{9/2}} \]
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Time = 2.88 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {2 e^{3} \sqrt {e x +d}}{b^{4}}-\frac {\left (2 a e -2 b d \right ) e^{3} \left (\frac {-\frac {29 b^{2} \left (e x +d \right )^{\frac {5}{2}}}{16}-\frac {17 \left (a e -b d \right ) b \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} a^{2} e^{2}+\frac {19}{8} a b d e -\frac {19}{16} b^{2} d^{2}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {35 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{4}}\) | \(151\) |
pseudoelliptic | \(-\frac {35 \left (e^{3} \left (b x +a \right )^{3} \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )-\sqrt {e x +d}\, \left (\frac {\left (16 e^{3} x^{3}-29 d \,e^{2} x^{2}-\frac {38}{3} d^{2} e x -\frac {8}{3} d^{3}\right ) b^{3}}{35}-\frac {2 e \left (-\frac {33}{2} x^{2} e^{2}+7 d e x +d^{2}\right ) a \,b^{2}}{15}-\frac {a^{2} e^{2} \left (-8 e x +d \right ) b}{3}+a^{3} e^{3}\right ) \sqrt {\left (a e -b d \right ) b}\right )}{8 \sqrt {\left (a e -b d \right ) b}\, b^{4} \left (b x +a \right )^{3}}\) | \(170\) |
derivativedivides | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{b^{4}}-\frac {\frac {\left (-\frac {29}{16} a \,b^{2} e +\frac {29}{16} b^{3} d \right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {17 b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} a^{3} e^{3}+\frac {57}{16} a^{2} b d \,e^{2}-\frac {57}{16} a \,b^{2} d^{2} e +\frac {19}{16} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {35 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}}{b^{4}}\right )\) | \(185\) |
default | \(2 e^{3} \left (\frac {\sqrt {e x +d}}{b^{4}}-\frac {\frac {\left (-\frac {29}{16} a \,b^{2} e +\frac {29}{16} b^{3} d \right ) \left (e x +d \right )^{\frac {5}{2}}-\frac {17 b \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}}{6}+\left (-\frac {19}{16} a^{3} e^{3}+\frac {57}{16} a^{2} b d \,e^{2}-\frac {57}{16} a \,b^{2} d^{2} e +\frac {19}{16} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{3}}+\frac {35 \left (a e -b d \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{16 \sqrt {\left (a e -b d \right ) b}}}{b^{4}}\right )\) | \(185\) |
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Leaf count of result is larger than twice the leaf count of optimal. 244 vs. \(2 (117) = 234\).
Time = 0.28 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.43 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\left [\frac {105 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \, {\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \, {\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{48 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}, -\frac {105 \, {\left (b^{3} e^{3} x^{3} + 3 \, a b^{2} e^{3} x^{2} + 3 \, a^{2} b e^{3} x + a^{3} e^{3}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (48 \, b^{3} e^{3} x^{3} - 8 \, b^{3} d^{3} - 14 \, a b^{2} d^{2} e - 35 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} - 3 \, {\left (29 \, b^{3} d e^{2} - 77 \, a b^{2} e^{3}\right )} x^{2} - 2 \, {\left (19 \, b^{3} d^{2} e + 49 \, a b^{2} d e^{2} - 140 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{24 \, {\left (b^{7} x^{3} + 3 \, a b^{6} x^{2} + 3 \, a^{2} b^{5} x + a^{3} b^{4}\right )}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (117) = 234\).
Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.69 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {2 \, \sqrt {e x + d} e^{3}}{b^{4}} + \frac {35 \, {\left (b d e^{3} - a e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{8 \, \sqrt {-b^{2} d + a b e} b^{4}} - \frac {87 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} d e^{3} - 136 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{3} + 57 \, \sqrt {e x + d} b^{3} d^{3} e^{3} - 87 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{2} e^{4} + 272 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} d e^{4} - 171 \, \sqrt {e x + d} a b^{2} d^{2} e^{4} - 136 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b e^{5} + 171 \, \sqrt {e x + d} a^{2} b d e^{5} - 57 \, \sqrt {e x + d} a^{3} e^{6}}{24 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{3} b^{4}} \]
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Time = 9.47 (sec) , antiderivative size = 303, normalized size of antiderivative = 2.09 \[ \int \frac {(d+e x)^{7/2}}{\left (a^2+2 a b x+b^2 x^2\right )^2} \, dx=\frac {\sqrt {d+e\,x}\,\left (\frac {19\,a^3\,e^6}{8}-\frac {57\,a^2\,b\,d\,e^5}{8}+\frac {57\,a\,b^2\,d^2\,e^4}{8}-\frac {19\,b^3\,d^3\,e^3}{8}\right )+\left (\frac {29\,a\,b^2\,e^4}{8}-\frac {29\,b^3\,d\,e^3}{8}\right )\,{\left (d+e\,x\right )}^{5/2}+{\left (d+e\,x\right )}^{3/2}\,\left (\frac {17\,a^2\,b\,e^5}{3}-\frac {34\,a\,b^2\,d\,e^4}{3}+\frac {17\,b^3\,d^2\,e^3}{3}\right )}{b^7\,{\left (d+e\,x\right )}^3-\left (3\,b^7\,d-3\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^2+\left (d+e\,x\right )\,\left (3\,a^2\,b^5\,e^2-6\,a\,b^6\,d\,e+3\,b^7\,d^2\right )-b^7\,d^3+a^3\,b^4\,e^3-3\,a^2\,b^5\,d\,e^2+3\,a\,b^6\,d^2\,e}+\frac {2\,e^3\,\sqrt {d+e\,x}}{b^4}-\frac {35\,e^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,e^3\,\sqrt {a\,e-b\,d}\,\sqrt {d+e\,x}}{a\,e^4-b\,d\,e^3}\right )\,\sqrt {a\,e-b\,d}}{8\,b^{9/2}} \]
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