Integrand size = 30, antiderivative size = 172 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 e \sqrt {d+e x}}{4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^2 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 44, 65, 214} \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {3 e^2 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{5/2}}+\frac {3 e \sqrt {d+e x}}{4 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^2}-\frac {\sqrt {d+e x}}{2 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)} \]
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Rule 44
Rule 65
Rule 214
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^3 \sqrt {d+e x}} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (3 b e \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{4 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3 e \sqrt {d+e x}}{4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{8 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3 e \sqrt {d+e x}}{4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (3 e \left (a b+b^2 x\right )\right ) \text {Subst}\left (\int \frac {1}{a b-\frac {b^2 d}{e}+\frac {b^2 x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {3 e \sqrt {d+e x}}{4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\sqrt {d+e x}}{2 (b d-a e) (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e^2 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 \sqrt {b} (b d-a e)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {\sqrt {b} \sqrt {-b d+a e} \sqrt {d+e x} (-2 b d+5 a e+3 b e x)+3 e^2 (a+b x)^2 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{4 \sqrt {b} (-b d+a e)^{5/2} (a+b x) \sqrt {(a+b x)^2}} \]
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Time = 2.76 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.18
method | result | size |
default | \(\frac {\left (3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{2} e^{2} x^{2}+6 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a b \,e^{2} x +3 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, b e x +3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} e^{2}+5 \sqrt {e x +d}\, a e \sqrt {\left (a e -b d \right ) b}-2 \sqrt {e x +d}\, d b \sqrt {\left (a e -b d \right ) b}\right ) \left (b x +a \right )}{4 \sqrt {\left (a e -b d \right ) b}\, \left (a e -b d \right )^{2} \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}\) | \(203\) |
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Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (119) = 238\).
Time = 0.32 (sec) , antiderivative size = 549, normalized size of antiderivative = 3.19 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) - 2 \, {\left (2 \, b^{3} d^{2} - 7 \, a b^{2} d e + 5 \, a^{2} b e^{2} - 3 \, {\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{8 \, {\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3} + {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{2} + 2 \, {\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}\right )} \sqrt {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) - {\left (2 \, b^{3} d^{2} - 7 \, a b^{2} d e + 5 \, a^{2} b e^{2} - 3 \, {\left (b^{3} d e - a b^{2} e^{2}\right )} x\right )} \sqrt {e x + d}}{4 \, {\left (a^{2} b^{4} d^{3} - 3 \, a^{3} b^{3} d^{2} e + 3 \, a^{4} b^{2} d e^{2} - a^{5} b e^{3} + {\left (b^{6} d^{3} - 3 \, a b^{5} d^{2} e + 3 \, a^{2} b^{4} d e^{2} - a^{3} b^{3} e^{3}\right )} x^{2} + 2 \, {\left (a b^{5} d^{3} - 3 \, a^{2} b^{4} d^{2} e + 3 \, a^{3} b^{3} d e^{2} - a^{4} b^{2} e^{3}\right )} x\right )}}\right ] \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}} \sqrt {e x + d}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {3 \, e^{2} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, {\left (b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} + \frac {3 \, {\left (e x + d\right )}^{\frac {3}{2}} b e^{2} - 5 \, \sqrt {e x + d} b d e^{2} + 5 \, \sqrt {e x + d} a e^{3}}{4 \, {\left (b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2}} \]
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Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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