Integrand size = 28, antiderivative size = 137 \[ \int \frac {(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {7 e (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}-\frac {7 e (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}} \]
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Time = 0.05 (sec) , antiderivative size = 137, 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} \[ \int \frac {(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {7 e (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}}+\frac {7 e \sqrt {d+e x} (b d-a e)^2}{b^4}+\frac {7 e (d+e x)^{3/2} (b d-a e)}{3 b^3}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {7 e (d+e x)^{5/2}}{5 b^2} \]
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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)^2} \, dx \\ & = -\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {(7 e) \int \frac {(d+e x)^{5/2}}{a+b x} \, dx}{2 b} \\ & = \frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {(7 e (b d-a e)) \int \frac {(d+e x)^{3/2}}{a+b x} \, dx}{2 b^2} \\ & = \frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {\left (7 e (b d-a e)^2\right ) \int \frac {\sqrt {d+e x}}{a+b x} \, dx}{2 b^3} \\ & = \frac {7 e (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {\left (7 e (b d-a e)^3\right ) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 b^4} \\ & = \frac {7 e (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}+\frac {\left (7 (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 )}{b^4} \\ & = \frac {7 e (b d-a e)^2 \sqrt {d+e x}}{b^4}+\frac {7 e (b d-a e) (d+e x)^{3/2}}{3 b^3}+\frac {7 e (d+e x)^{5/2}}{5 b^2}-\frac {(d+e x)^{7/2}}{b (a+b x)}-\frac {7 e (b d-a e)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.15 \[ \int \frac {(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=-\frac {\sqrt {d+e x} \left (-105 a^3 e^3+35 a^2 b e^2 (7 d-2 e x)+7 a b^2 e \left (-23 d^2+24 d e x+2 e^2 x^2\right )+b^3 \left (15 d^3-116 d^2 e x-32 d e^2 x^2-6 e^3 x^3\right )\right )}{15 b^4 (a+b x)}-\frac {7 e (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{b^{9/2}} \]
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Time = 2.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.22
method | result | size |
pseudoelliptic | \(-\frac {7 \left (e \left (a e -b d \right )^{3} \left (b x +a \right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )-\sqrt {e x +d}\, \left (\left (\frac {2}{35} e^{3} x^{3}+\frac {32}{105} d \,e^{2} x^{2}+\frac {116}{105} d^{2} e x -\frac {1}{7} d^{3}\right ) b^{3}+\frac {23 \left (-\frac {2}{23} x^{2} e^{2}-\frac {24}{23} d e x +d^{2}\right ) e a \,b^{2}}{15}-\frac {7 \left (-\frac {2 e x}{7}+d \right ) e^{2} a^{2} b}{3}+a^{3} e^{3}\right ) \sqrt {\left (a e -b d \right ) b}\right )}{\sqrt {\left (a e -b d \right ) b}\, b^{4} \left (b x +a \right )}\) | \(167\) |
risch | \(\frac {2 e \left (3 x^{2} b^{2} e^{2}-10 x a b \,e^{2}+16 b^{2} d e x +45 a^{2} e^{2}-100 a b d e +58 b^{2} d^{2}\right ) \sqrt {e x +d}}{15 b^{4}}-\frac {\left (2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) e \left (-\frac {\sqrt {e x +d}}{2 \left (b \left (e x +d \right )+a e -b d \right )}+\frac {7 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{b^{4}}\) | \(171\) |
derivativedivides | \(2 e \left (\frac {\frac {b^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 a b e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a^{2} e^{2} \sqrt {e x +d}-6 a b d e \sqrt {e x +d}+3 b^{2} d^{2} \sqrt {e x +d}}{b^{4}}-\frac {\frac {\left (-\frac {1}{2} a^{3} e^{3}+\frac {3}{2} a^{2} b d \,e^{2}-\frac {3}{2} a \,b^{2} d^{2} e +\frac {1}{2} b^{3} d^{3}\right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {7 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}}{b^{4}}\right )\) | \(230\) |
default | \(2 e \left (\frac {\frac {b^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 a b e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} d \left (e x +d \right )^{\frac {3}{2}}}{3}+3 a^{2} e^{2} \sqrt {e x +d}-6 a b d e \sqrt {e x +d}+3 b^{2} d^{2} \sqrt {e x +d}}{b^{4}}-\frac {\frac {\left (-\frac {1}{2} a^{3} e^{3}+\frac {3}{2} a^{2} b d \,e^{2}-\frac {3}{2} a \,b^{2} d^{2} e +\frac {1}{2} b^{3} d^{3}\right ) \sqrt {e x +d}}{b \left (e x +d \right )+a e -b d}+\frac {7 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}}{b^{4}}\right )\) | \(230\) |
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Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (115) = 230\).
Time = 0.28 (sec) , antiderivative size = 486, normalized size of antiderivative = 3.55 \[ \int \frac {(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left [\frac {105 \, {\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \, {\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{30 \, {\left (b^{5} x + a b^{4}\right )}}, -\frac {105 \, {\left (a b^{2} d^{2} e - 2 \, a^{2} b d e^{2} + a^{3} e^{3} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (6 \, b^{3} e^{3} x^{3} - 15 \, b^{3} d^{3} + 161 \, a b^{2} d^{2} e - 245 \, a^{2} b d e^{2} + 105 \, a^{3} e^{3} + 2 \, {\left (16 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + 2 \, {\left (58 \, b^{3} d^{2} e - 84 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (b^{5} x + a b^{4}\right )}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (115) = 230\).
Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.97 \[ \int \frac {(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\frac {7 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} - \frac {\sqrt {e x + d} b^{3} d^{3} e - 3 \, \sqrt {e x + d} a b^{2} d^{2} e^{2} + 3 \, \sqrt {e x + d} a^{2} b d e^{3} - \sqrt {e x + d} a^{3} e^{4}}{{\left ({\left (e x + d\right )} b - b d + a e\right )} b^{4}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{8} e + 10 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{8} d e + 45 \, \sqrt {e x + d} b^{8} d^{2} e - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{7} e^{2} - 90 \, \sqrt {e x + d} a b^{7} d e^{2} + 45 \, \sqrt {e x + d} a^{2} b^{6} e^{3}\right )}}{15 \, b^{10}} \]
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Time = 9.41 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.72 \[ \int \frac {(d+e x)^{7/2}}{a^2+2 a b x+b^2 x^2} \, dx=\left (\frac {2\,e\,{\left (2\,b^2\,d-2\,a\,b\,e\right )}^2}{b^6}-\frac {2\,e\,{\left (a\,e-b\,d\right )}^2}{b^4}\right )\,\sqrt {d+e\,x}+\frac {\sqrt {d+e\,x}\,\left (a^3\,e^4-3\,a^2\,b\,d\,e^3+3\,a\,b^2\,d^2\,e^2-b^3\,d^3\,e\right )}{b^5\,\left (d+e\,x\right )-b^5\,d+a\,b^4\,e}+\frac {2\,e\,{\left (d+e\,x\right )}^{5/2}}{5\,b^2}+\frac {2\,e\,\left (2\,b^2\,d-2\,a\,b\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,b^4}-\frac {7\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,e\,{\left (a\,e-b\,d\right )}^{5/2}\,\sqrt {d+e\,x}}{a^3\,e^4-3\,a^2\,b\,d\,e^3+3\,a\,b^2\,d^2\,e^2-b^3\,d^3\,e}\right )\,{\left (a\,e-b\,d\right )}^{5/2}}{b^{9/2}} \]
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