Integrand size = 28, antiderivative size = 99 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {3 e}{(b d-a e)^2 \sqrt {d+e x}}-\frac {1}{(b d-a e) (a+b x) \sqrt {d+e x}}+\frac {3 \sqrt {b} e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}} \]
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Time = 0.04 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {27, 44, 53, 65, 214} \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\frac {3 \sqrt {b} e \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}}-\frac {3 e}{\sqrt {d+e x} (b d-a e)^2}-\frac {1}{(a+b x) \sqrt {d+e x} (b d-a e)} \]
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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)^2 (d+e x)^{3/2}} \, dx \\ & = -\frac {1}{(b d-a e) (a+b x) \sqrt {d+e x}}-\frac {(3 e) \int \frac {1}{(a+b x) (d+e x)^{3/2}} \, dx}{2 (b d-a e)} \\ & = -\frac {3 e}{(b d-a e)^2 \sqrt {d+e x}}-\frac {1}{(b d-a e) (a+b x) \sqrt {d+e x}}-\frac {(3 b e) \int \frac {1}{(a+b x) \sqrt {d+e x}} \, dx}{2 (b d-a e)^2} \\ & = -\frac {3 e}{(b d-a e)^2 \sqrt {d+e x}}-\frac {1}{(b d-a e) (a+b x) \sqrt {d+e x}}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{a-\frac {b d}{e}+\frac {b x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{(b d-a e)^2} \\ & = -\frac {3 e}{(b d-a e)^2 \sqrt {d+e x}}-\frac {1}{(b d-a e) (a+b x) \sqrt {d+e x}}+\frac {3 \sqrt {b} e \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{(b d-a e)^{5/2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {2 a e+b (d+3 e x)}{(b d-a e)^2 (a+b x) \sqrt {d+e x}}-\frac {3 \sqrt {b} e \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{5/2}} \]
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Time = 2.54 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.79
method | result | size |
pseudoelliptic | \(\frac {-\frac {b \sqrt {e x +d}}{b x +a}-\frac {3 e b \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{\sqrt {\left (a e -b d \right ) b}}-\frac {2 e}{\sqrt {e x +d}}}{\left (a e -b d \right )^{2}}\) | \(78\) |
derivativedivides | \(2 e \left (-\frac {1}{\left (a e -b d \right )^{2} \sqrt {e x +d}}-\frac {b \left (\frac {\sqrt {e x +d}}{2 b \left (e x +d \right )+2 a e -2 b d}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2}}\right )\) | \(100\) |
default | \(2 e \left (-\frac {1}{\left (a e -b d \right )^{2} \sqrt {e x +d}}-\frac {b \left (\frac {\sqrt {e x +d}}{2 b \left (e x +d \right )+2 a e -2 b d}+\frac {3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{2 \sqrt {\left (a e -b d \right ) b}}\right )}{\left (a e -b d \right )^{2}}\right )\) | \(100\) |
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Leaf count of result is larger than twice the leaf count of optimal. 206 vs. \(2 (85) = 170\).
Time = 0.39 (sec) , antiderivative size = 423, normalized size of antiderivative = 4.27 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\left [\frac {3 \, {\left (b e^{2} x^{2} + a d e + {\left (b d e + a e^{2}\right )} x\right )} \sqrt {\frac {b}{b d - a e}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, {\left (b d - a e\right )} \sqrt {e x + d} \sqrt {\frac {b}{b d - a e}}}{b x + a}\right ) - 2 \, {\left (3 \, b e x + b d + 2 \, a e\right )} \sqrt {e x + d}}{2 \, {\left (a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} + {\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x\right )}}, \frac {3 \, {\left (b e^{2} x^{2} + a d e + {\left (b d e + a e^{2}\right )} x\right )} \sqrt {-\frac {b}{b d - a e}} \arctan \left (-\frac {{\left (b d - a e\right )} \sqrt {e x + d} \sqrt {-\frac {b}{b d - a e}}}{b e x + b d}\right ) - {\left (3 \, b e x + b d + 2 \, a e\right )} \sqrt {e x + d}}{a b^{2} d^{3} - 2 \, a^{2} b d^{2} e + a^{3} d e^{2} + {\left (b^{3} d^{2} e - 2 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} x^{2} + {\left (b^{3} d^{3} - a b^{2} d^{2} e - a^{2} b d e^{2} + a^{3} e^{3}\right )} x}\right ] \]
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\[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\int \frac {1}{\left (a + b x\right )^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.44 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {3 \, b e \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e}} - \frac {3 \, {\left (e x + d\right )} b e - 2 \, b d e + 2 \, a e^{2}}{{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} {\left ({\left (e x + d\right )}^{\frac {3}{2}} b - \sqrt {e x + d} b d + \sqrt {e x + d} a e\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.24 \[ \int \frac {1}{(d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )} \, dx=-\frac {\frac {2\,e}{a\,e-b\,d}+\frac {3\,b\,e\,\left (d+e\,x\right )}{{\left (a\,e-b\,d\right )}^2}}{b\,{\left (d+e\,x\right )}^{3/2}+\left (a\,e-b\,d\right )\,\sqrt {d+e\,x}}-\frac {3\,\sqrt {b}\,e\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {d+e\,x}\,\left (a^2\,e^2-2\,a\,b\,d\,e+b^2\,d^2\right )}{{\left (a\,e-b\,d\right )}^{5/2}}\right )}{{\left (a\,e-b\,d\right )}^{5/2}} \]
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