Integrand size = 30, antiderivative size = 263 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e) (a+b x) (d+e x)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{7/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {660, 52, 65, 214} \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=-\frac {2 (a+b x) (b d-a e)^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{5/2} (b d-a e)}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) \sqrt {d+e x} (b d-a e)^3}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{3/2} (b d-a e)^2}{3 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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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)^{7/2}}{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)^{7/2}}{7 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)^{5/2}}{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)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 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 {(d+e x)^{3/2}}{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) (d+e x)^{3/2}}{3 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)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 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 {\sqrt {d+e x}}{a b+b^2 x} \, dx}{b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 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)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (\left (b^2 d-a b e\right )^4 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{b^8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 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)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (2 \left (b^2 d-a b e\right )^4 \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^8 e \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {2 (b d-a e)^3 (a+b x) \sqrt {d+e x}}{b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (b d-a e)^2 (a+b x) (d+e x)^{3/2}}{3 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)^{5/2}}{5 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {2 (a+b x) (d+e x)^{7/2}}{7 b \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {2 (b d-a e)^{7/2} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{b^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.64 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 (a+b x) \left (\sqrt {b} \sqrt {d+e x} \left (-105 a^3 e^3+35 a^2 b e^2 (10 d+e x)-7 a b^2 e \left (58 d^2+16 d e x+3 e^2 x^2\right )+b^3 \left (176 d^3+122 d^2 e x+66 d e^2 x^2+15 e^3 x^3\right )\right )+105 (-b d+a e)^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{105 b^{9/2} \sqrt {(a+b x)^2}} \]
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Time = 2.39 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {2 \left (-15 e^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} e^{3}-66 x^{2} b^{3} d \,e^{2}-35 a^{2} b \,e^{3} x +112 x a \,b^{2} d \,e^{2}-122 b^{3} d^{2} e x +105 a^{3} e^{3}-350 a^{2} b d \,e^{2}+406 a \,b^{2} d^{2} e -176 b^{3} d^{3}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{105 b^{4} \left (b x +a \right )}+\frac {2 \left (e^{4} a^{4}-4 b \,e^{3} d \,a^{3}+6 b^{2} e^{2} d^{2} a^{2}-4 a \,b^{3} d^{3} e +b^{4} d^{4}\right ) \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{4} \sqrt {\left (a e -b d \right ) b}\, \left (b x +a \right )}\) | \(237\) |
default | \(\frac {2 \left (b x +a \right ) \left (15 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {7}{2}} b^{3}-21 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} a \,b^{2} e +21 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {5}{2}} b^{3} d +35 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}-70 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e +35 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} e^{4}-420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b d \,e^{3}+630 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{2} d^{2} e^{2}-420 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{3} d^{3} e +105 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{4} d^{4}-105 \sqrt {e x +d}\, a^{3} e^{3} \sqrt {\left (a e -b d \right ) b}+315 \sqrt {e x +d}\, a^{2} d \,e^{2} b \sqrt {\left (a e -b d \right ) b}-315 \sqrt {e x +d}\, a \,d^{2} e \,b^{2} \sqrt {\left (a e -b d \right ) b}+105 \sqrt {e x +d}\, d^{3} b^{3} \sqrt {\left (a e -b d \right ) b}\right )}{105 \sqrt {\left (b x +a \right )^{2}}\, b^{4} \sqrt {\left (a e -b d \right ) b}}\) | \(462\) |
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Time = 0.38 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.61 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\left [-\frac {105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e + 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) - 2 \, {\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \, {\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + {\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, b^{4}}, -\frac {2 \, {\left (105 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (15 \, b^{3} e^{3} x^{3} + 176 \, b^{3} d^{3} - 406 \, a b^{2} d^{2} e + 350 \, a^{2} b d e^{2} - 105 \, a^{3} e^{3} + 3 \, {\left (22 \, b^{3} d e^{2} - 7 \, a b^{2} e^{3}\right )} x^{2} + {\left (122 \, b^{3} d^{2} e - 112 \, a b^{2} d e^{2} + 35 \, a^{2} b e^{3}\right )} x\right )} \sqrt {e x + d}\right )}}{105 \, b^{4}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {7}{2}}}{\sqrt {{\left (b x + a\right )}^{2}}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.30 \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\frac {2 \, {\left (b^{4} d^{4} \mathrm {sgn}\left (b x + a\right ) - 4 \, a b^{3} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 6 \, a^{2} b^{2} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 4 \, a^{3} b d e^{3} \mathrm {sgn}\left (b x + a\right ) + a^{4} e^{4} \mathrm {sgn}\left (b x + a\right )\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{\sqrt {-b^{2} d + a b e} b^{4}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{6} \mathrm {sgn}\left (b x + a\right ) + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{6} d \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) + 105 \, \sqrt {e x + d} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{5} e \mathrm {sgn}\left (b x + a\right ) - 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} d e \mathrm {sgn}\left (b x + a\right ) - 315 \, \sqrt {e x + d} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 315 \, \sqrt {e x + d} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 105 \, \sqrt {e x + d} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right )\right )}}{105 \, b^{7}} \]
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Timed out. \[ \int \frac {(d+e x)^{7/2}}{\sqrt {a^2+2 a b x+b^2 x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{7/2}}{\sqrt {{\left (a+b\,x\right )}^2}} \,d x \]
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