Integrand size = 30, antiderivative size = 308 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {63 e^2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 e^2 (b d-a e)^{5/2} (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.12 (sec) , antiderivative size = 308, 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} \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=-\frac {63 e^2 (a+b x) (b d-a e)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) \sqrt {d+e x} (b d-a e)^2}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (a+b x) (d+e x)^{3/2} (b d-a e)}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 43
Rule 52
Rule 65
Rule 214
Rule 660
Rubi steps \begin{align*} \text {integral}& = \frac {\left (b^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{9/2}}{\left (a b+b^2 x\right )^3} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (9 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{7/2}}{\left (a b+b^2 x\right )^2} \, dx}{4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{5/2}}{a b+b^2 x} \, dx}{8 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \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}{8 b^4 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \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}{8 b^6 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {63 e^2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e^2 \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}{8 b^8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {63 e^2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (63 e \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 )}{4 b^8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = \frac {63 e^2 (b d-a e)^2 (a+b x) \sqrt {d+e x}}{4 b^5 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {21 e^2 (b d-a e) (a+b x) (d+e x)^{3/2}}{4 b^4 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {63 e^2 (a+b x) (d+e x)^{5/2}}{20 b^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {9 e (d+e x)^{7/2}}{4 b^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{9/2}}{2 b (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {63 e^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 )}{4 b^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.66 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.77 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {e^2 (a+b x)^3 \left (\frac {\sqrt {b} \sqrt {d+e x} \left (315 a^4 e^4+105 a^3 b e^3 (-7 d+5 e x)+21 a^2 b^2 e^2 \left (23 d^2-59 d e x+8 e^2 x^2\right )-3 a b^3 e \left (15 d^3-277 d^2 e x+136 d e^2 x^2+8 e^3 x^3\right )+b^4 \left (-10 d^4-85 d^3 e x+288 d^2 e^2 x^2+56 d e^3 x^3+8 e^4 x^4\right )\right )}{e^2 (a+b x)^2}-315 (-b d+a e)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )\right )}{20 b^{11/2} \left ((a+b x)^2\right )^{3/2}} \]
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Time = 2.31 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.73
method | result | size |
risch | \(\frac {2 e^{2} \left (x^{2} b^{2} e^{2}-5 x a b \,e^{2}+7 b^{2} d e x +30 a^{2} e^{2}-65 a b d e +36 b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{5 b^{5} \left (b x +a \right )}-\frac {\left (2 a^{3} e^{3}-6 a^{2} b d \,e^{2}+6 a \,b^{2} d^{2} e -2 b^{3} d^{3}\right ) e^{2} \left (\frac {-\frac {17 \left (e x +d \right )^{\frac {3}{2}} b}{8}+\left (-\frac {15 a e}{8}+\frac {15 b d}{8}\right ) \sqrt {e x +d}}{\left (b \left (e x +d \right )+a e -b d \right )^{2}}+\frac {63 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right )}{8 \sqrt {\left (a e -b d \right ) b}}\right ) \sqrt {\left (b x +a \right )^{2}}}{b^{5} \left (b x +a \right )}\) | \(226\) |
default | \(\text {Expression too large to display}\) | \(1115\) |
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Time = 0.29 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.37 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\left [\frac {315 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt {\frac {b d - a e}{b}} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {e x + d} b \sqrt {\frac {b d - a e}{b}}}{b x + a}\right ) + 2 \, {\left (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \, {\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \, {\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{40 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}, -\frac {315 \, {\left (a^{2} b^{2} d^{2} e^{2} - 2 \, a^{3} b d e^{3} + a^{4} e^{4} + {\left (b^{4} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + a^{2} b^{2} e^{4}\right )} x^{2} + 2 \, {\left (a b^{3} d^{2} e^{2} - 2 \, a^{2} b^{2} d e^{3} + a^{3} b e^{4}\right )} x\right )} \sqrt {-\frac {b d - a e}{b}} \arctan \left (-\frac {\sqrt {e x + d} b \sqrt {-\frac {b d - a e}{b}}}{b d - a e}\right ) - {\left (8 \, b^{4} e^{4} x^{4} - 10 \, b^{4} d^{4} - 45 \, a b^{3} d^{3} e + 483 \, a^{2} b^{2} d^{2} e^{2} - 735 \, a^{3} b d e^{3} + 315 \, a^{4} e^{4} + 8 \, {\left (7 \, b^{4} d e^{3} - 3 \, a b^{3} e^{4}\right )} x^{3} + 24 \, {\left (12 \, b^{4} d^{2} e^{2} - 17 \, a b^{3} d e^{3} + 7 \, a^{2} b^{2} e^{4}\right )} x^{2} - {\left (85 \, b^{4} d^{3} e - 831 \, a b^{3} d^{2} e^{2} + 1239 \, a^{2} b^{2} d e^{3} - 525 \, a^{3} b e^{4}\right )} x\right )} \sqrt {e x + d}}{20 \, {\left (b^{7} x^{2} + 2 \, a b^{6} x + a^{2} b^{5}\right )}}\right ] \]
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\[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {9}{2}}}{\left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {9}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.29 \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\frac {63 \, {\left (b^{3} d^{3} e^{2} - 3 \, a b^{2} d^{2} e^{3} + 3 \, a^{2} b d e^{4} - a^{3} e^{5}\right )} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{4 \, \sqrt {-b^{2} d + a b e} b^{5} \mathrm {sgn}\left (b x + a\right )} - \frac {17 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{4} d^{3} e^{2} - 15 \, \sqrt {e x + d} b^{4} d^{4} e^{2} - 51 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{3} d^{2} e^{3} + 60 \, \sqrt {e x + d} a b^{3} d^{3} e^{3} + 51 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{2} d e^{4} - 90 \, \sqrt {e x + d} a^{2} b^{2} d^{2} e^{4} - 17 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{3} b e^{5} + 60 \, \sqrt {e x + d} a^{3} b d e^{5} - 15 \, \sqrt {e x + d} a^{4} e^{6}}{4 \, {\left ({\left (e x + d\right )} b - b d + a e\right )}^{2} b^{5} \mathrm {sgn}\left (b x + a\right )} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {5}{2}} b^{12} e^{2} + 5 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{12} d e^{2} + 30 \, \sqrt {e x + d} b^{12} d^{2} e^{2} - 5 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{11} e^{3} - 60 \, \sqrt {e x + d} a b^{11} d e^{3} + 30 \, \sqrt {e x + d} a^{2} b^{10} e^{4}\right )}}{5 \, b^{15} \mathrm {sgn}\left (b x + a\right )} \]
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Timed out. \[ \int \frac {(d+e x)^{9/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{9/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}} \,d x \]
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