Integrand size = 30, antiderivative size = 294 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 \sqrt {b d-a e} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 52, 65, 214} \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {315 e^4 (a+b x) \sqrt {b d-a e} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^4 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (9 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (21 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (105 e^3 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^2} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^4 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{a b+b^2 x} \, dx}{128 b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^4 \left (b^2 d-a b e\right ) \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right ) \sqrt {d+e x}} \, dx}{128 b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (315 e^3 \left (b^2 d-a b e\right ) \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 )}{64 b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {315 e^4 (a+b x) \sqrt {d+e x}}{64 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {105 e^3 (d+e x)^{3/2}}{64 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {21 e^2 (d+e x)^{5/2}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {3 e (d+e x)^{7/2}}{8 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {315 e^4 \sqrt {b d-a e} (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.79 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {-\sqrt {b} \sqrt {d+e x} \left (-315 a^4 e^4+105 a^3 b e^3 (d-11 e x)+21 a^2 b^2 e^2 \left (2 d^2+19 d e x-73 e^2 x^2\right )+3 a b^3 e \left (8 d^3+52 d^2 e x+185 d e^2 x^2-279 e^3 x^3\right )+b^4 \left (16 d^4+88 d^3 e x+210 d^2 e^2 x^2+325 d e^3 x^3-128 e^4 x^4\right )\right )-315 e^4 \sqrt {-b d+a e} (a+b x)^4 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{64 b^{11/2} (a+b x)^3 \sqrt {(a+b x)^2}} \]
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Time = 2.44 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.79
method | result | size |
risch | \(\frac {2 e^{4} \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}-\frac {\left (2 a e -2 b d \right ) e^{4} \left (\frac {-\frac {325 \left (e x +d \right )^{\frac {7}{2}} b^{3}}{128}-\frac {765 \left (a e -b d \right ) b^{2} \left (e x +d \right )^{\frac {5}{2}}}{128}+\left (-\frac {643}{128} e^{2} a^{2} b +\frac {643}{64} a d e \,b^{2}-\frac {643}{128} b^{3} d^{2}\right ) \left (e x +d \right )^{\frac {3}{2}}+\left (-\frac {187}{128} a^{3} e^{3}+\frac {561}{128} a^{2} b d \,e^{2}-\frac {561}{128} a \,b^{2} d^{2} e +\frac {187}{128} b^{3} d^{3}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{4}}+\frac {315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{128 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}\) | \(233\) |
default | \(\frac {\left (-1530 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d e -1929 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} d \,e^{2}+1929 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} d^{2} e -643 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{3}+765 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{2} b^{2} e^{2}-325 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d +765 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{4} d^{2}-1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} b \,e^{5} x +315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} b d \,e^{4}-315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{4} e^{5} x^{4}+1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b^{2} d \,e^{4} x +643 \left (e x +d \right )^{\frac {3}{2}} \sqrt {\left (a e -b d \right ) b}\, a^{3} b \,e^{3}-315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{5} e^{5}+768 a^{2} b^{2} e^{4} x^{2} \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}+1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{4} d \,e^{4} x^{3}+1890 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{3} d \,e^{4} x^{2}+315 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{5} d \,e^{4} x^{4}+325 \left (e x +d \right )^{\frac {7}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{3} e +512 \sqrt {e x +d}\, \sqrt {\left (a e -b d \right ) b}\, a^{3} b \,e^{4} x -748 \sqrt {e x +d}\, e^{3} d \,a^{3} b \sqrt {\left (a e -b d \right ) b}+1122 \sqrt {e x +d}\, e^{2} d^{2} a^{2} b^{2} \sqrt {\left (a e -b d \right ) b}-748 \sqrt {e x +d}\, a \,d^{3} e \,b^{3} \sqrt {\left (a e -b d \right ) b}+512 a \,b^{3} e^{4} x^{3} \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}+315 \sqrt {e x +d}\, e^{4} a^{4} \sqrt {\left (a e -b d \right ) b}+187 \sqrt {e x +d}\, d^{4} b^{4} \sqrt {\left (a e -b d \right ) b}+128 b^{4} e^{4} x^{4} \sqrt {\left (a e -b d \right ) b}\, \sqrt {e x +d}-1260 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{3} e^{5} x^{3}-1890 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b^{2} e^{5} x^{2}\right ) \left (b x +a \right )}{64 \sqrt {\left (a e -b d \right ) b}\, b^{5} \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(892\) |
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Time = 0.32 (sec) , antiderivative size = 680, normalized size of antiderivative = 2.31 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [\frac {315 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{128 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}, -\frac {315 \, {\left (b^{4} e^{4} x^{4} + 4 \, a b^{3} e^{4} x^{3} + 6 \, a^{2} b^{2} e^{4} x^{2} + 4 \, a^{3} b e^{4} x + a^{4} e^{4}\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (128 \, b^{4} e^{4} x^{4} - 16 \, b^{4} d^{4} - 24 \, a b^{3} d^{3} e - 42 \, a^{2} b^{2} d^{2} e^{2} - 105 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} - {\left (325 \, b^{4} d e^{3} - 837 \, a b^{3} e^{4}\right )} x^{3} - 3 \, {\left (70 \, b^{4} d^{2} e^{2} + 185 \, a b^{3} d e^{3} - 511 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (88 \, b^{4} d^{3} e + 156 \, a b^{3} d^{2} e^{2} + 399 \, a^{2} b^{2} d e^{3} - 1155 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{64 \, {\left (b^{9} x^{4} + 4 \, a b^{8} x^{3} + 6 \, a^{2} b^{7} x^{2} + 4 \, a^{3} b^{6} x + a^{4} b^{5}\right )}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.29 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {2 \, \sqrt {e x + d} e^{4}}{b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {315 \, {\left (b d e^{4} - a e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {325 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{4} d e^{4} - 765 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{4} d^{2} e^{4} + 643 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{4} - 187 \, \sqrt {e x + d} b^{4} d^{4} e^{4} - 325 \, {\left (e x + d\right )}^{\frac {7}{2}} a b^{3} e^{5} + 1530 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{3} d e^{5} - 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{5} + 748 \, \sqrt {e x + d} a b^{3} d^{3} e^{5} - 765 \, {\left (e x + d\right )}^{\frac {5}{2}} a^{2} b^{2} e^{6} + 1929 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{6} - 1122 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{6} - 643 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{7} + 748 \, \sqrt {e x + d} a^{3} b d e^{7} - 187 \, \sqrt {e x + d} a^{4} e^{8}}{64 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4} b^{5} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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