Integrand size = 30, antiderivative size = 212 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{5/2}}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{5/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 52, 65, 214} \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (a+b x) (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2} (b d-a e)}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{5/2}}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) \sqrt {d+e x} (b d-a e)^2}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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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)^{5/2}}{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)^{5/2}}{5 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 {(d+e x)^{3/2}}{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) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{5/2}}{5 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 {\sqrt {d+e x}}{a b+b^2 x} \, dx}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{5/2}}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right )^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{5/2}}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right )^3 \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^6 e \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{3/2}}{3 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{5/2}}{5 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{5/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.60 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x} \left (15 a^2 e^2-5 a b e (7 d+e x)+b^2 \left (23 d^2+11 d e x+3 e^2 x^2\right )\right )-15 (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{15 b^{7/2} \sqrt {(a+b x)^2}} \]
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Time = 2.12 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.81
method | result | size |
risch | \(\frac {2 \left (3 x^{2} b^{2} e^{2}-5 x a b \,e^{2}+11 b^{2} d e x +15 a^{2} e^{2}-35 a b d e +23 b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{15 b^{3} \left (b x +a \right )}-\frac {2 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{3} \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )}\) | \(171\) |
default | \(\frac {2 \left (b x +a \right ) \left (3 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{2}-5 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a b e +5 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{2} d -15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} e^{3}+45 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b d \,e^{2}-45 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{2} d^{2} e +15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{3} d^{3}+15 \sqrt {e x +d}\, a^{2} e^{2} \sqrt {\left (a e -b d \right ) b}-30 \sqrt {e x +d}\, a d e b \sqrt {\left (a e -b d \right ) b}+15 \sqrt {e x +d}\, d^{2} b^{2} \sqrt {\left (a e -b d \right ) b}\right )}{15 \sqrt {\left (b x +a \right )^{2}}\, b^{3} \sqrt {\left (a e -b d \right ) b}}\) | \(309\) |
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Time = 0.45 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.37 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [\frac {15 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (3 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 35 \, a b d e + 15 \, a^{2} e^{2} + {\left (11 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt {e x + d}}{15 \, b^{3}}, -\frac {2 \, {\left (15 \, {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (3 \, b^{2} e^{2} x^{2} + 23 \, b^{2} d^{2} - 35 \, a b d e + 15 \, a^{2} e^{2} + {\left (11 \, b^{2} d e - 5 \, a b e^{2}\right )} x\right )} \sqrt {e x + d}\right )}}{15 \, b^{3}}\right ] \]
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\[ \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {5}{2}}}{\sqrt {\left (a + b x\right )^{2}}}\, dx \]
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\[ \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {{\left (b x + a\right )}^{2}}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.09 \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 \, {\left (b^{3} d^{3} \mathrm {sgn}\left (b x + a\right ) - 3 \, a b^{2} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 3 \, a^{2} b d e^{2} \mathrm {sgn}\left (b x + a\right ) - a^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{3}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} \mathrm {sgn}\left (b x + a\right ) + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {e x + d} b^{4} d^{2} \mathrm {sgn}\left (b x + a\right ) - 5 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} e \mathrm {sgn}\left (b x + a\right ) - 30 \, \sqrt {e x + d} a b^{3} d e \mathrm {sgn}\left (b x + a\right ) + 15 \, \sqrt {e x + d} a^{2} b^{2} e^{2} \mathrm {sgn}\left (b x + a\right )\right )}}{15 \, b^{5}} \]
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Timed out. \[ \int \frac {(d+e x)^{5/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{5/2}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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