Integrand size = 30, antiderivative size = 161 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (b d-a e) (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{3/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.06 (sec) , antiderivative size = 161, 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, 52, 65, 214} \[ \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (a+b x) (b d-a e)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) \sqrt {d+e x} (b d-a e)}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 52
Rule 65
Rule 214
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a b+b^2 x\right ) \int \frac {(d+e x)^{3/2}}{a b+b^2 x} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e) (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right )^2 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e) (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right )^2 \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 )}{b^4 e \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e) (a+b x) \sqrt {d+e x}}{b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2}}{3 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{3/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.60 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x} (4 b d-3 a e+b e x)+3 (-b d+a e)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{3 b^{5/2} \sqrt {(a+b x)^2}} \]
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Time = 2.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.75
method | result | size |
risch | \(-\frac {2 \left (-b e x +3 a e -4 b d \right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{3 b^{2} \left (b x +a \right )}+\frac {2 \left (a^{2} e^{2}-2 a b d e +b^{2} d^{2}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{2} \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )}\) | \(120\) |
default | \(\frac {2 \left (b x +a \right ) \left (\sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} b +3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} e^{2}-6 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a b d e +3 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{2} d^{2}-3 \sqrt {e x +d}\, a e \sqrt {\left (a e -b d \right ) b}+3 \sqrt {e x +d}\, d b \sqrt {\left (a e -b d \right ) b}\right )}{3 \sqrt {\left (b x +a \right )^{2}}\, b^{2} \sqrt {\left (a e -b d \right ) b}}\) | \(188\) |
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Time = 0.37 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [-\frac {3 \, {\left (b d - a e\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (b e x + 4 \, b d - 3 \, a e\right )} \sqrt {e x + d}}{3 \, b^{2}}, -\frac {2 \, {\left (3 \, {\left (b d - a e\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (b e x + 4 \, b d - 3 \, a e\right )} \sqrt {e x + d}\right )}}{3 \, b^{2}}\right ] \]
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\[ \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\sqrt {\left (a + b x\right )^{2}}}\, dx \]
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\[ \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {{\left (b x + a\right )}^{2}}} \,d x } \]
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Time = 0.28 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 \, {\left (b^{2} d^{2} \mathrm {sgn}\left (b x + a\right ) - 2 \, a b d e \mathrm {sgn}\left (b x + a\right ) + a^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{2}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} b^{2} \mathrm {sgn}\left (b x + a\right ) + 3 \, \sqrt {e x + d} b^{2} d \mathrm {sgn}\left (b x + a\right ) - 3 \, \sqrt {e x + d} a b e \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, b^{3}} \]
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Timed out. \[ \int \frac {(d+e x)^{3/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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