Integrand size = 30, antiderivative size = 260 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {5 e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^2 \sqrt {d+e x}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{7/2} (b d-a e)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {660, 43, 44, 65, 214} \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {5 e^4 (a+b x) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)^{3/2}}-\frac {5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^3 \sqrt {d+e x}}{64 b^3 \sqrt {a^2+2 a b x+b^2 x^2} (b d-a e)}-\frac {5 e^2 \sqrt {d+e x}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}} \]
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Rule 43
Rule 44
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)^{5/2}}{\left (a b+b^2 x\right )^5} \, dx}{\sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 b^2 e \left (a b+b^2 x\right )\right ) \int \frac {(d+e x)^{3/2}}{\left (a b+b^2 x\right )^4} \, dx}{8 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e^2 \left (a b+b^2 x\right )\right ) \int \frac {\sqrt {d+e x}}{\left (a b+b^2 x\right )^3} \, dx}{16 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {5 e^2 \sqrt {d+e x}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {\left (5 e^3 \left (a b+b^2 x\right )\right ) \int \frac {1}{\left (a b+b^2 x\right )^2 \sqrt {d+e x}} \, dx}{64 b^2 \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {5 e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^2 \sqrt {d+e x}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e^4 \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^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {5 e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^2 \sqrt {d+e x}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {\left (5 e^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 )}{64 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}} \\ & = -\frac {5 e^3 \sqrt {d+e x}}{64 b^3 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e^2 \sqrt {d+e x}}{32 b^3 (a+b x) \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {5 e (d+e x)^{3/2}}{24 b^2 (a+b x)^2 \sqrt {a^2+2 a b x+b^2 x^2}}-\frac {(d+e x)^{5/2}}{4 b (a+b x)^3 \sqrt {a^2+2 a b x+b^2 x^2}}+\frac {5 e^4 (a+b x) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {b d-a e}}\right )}{64 b^{7/2} (b d-a e)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.75 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\frac {e^4 (a+b x)^5 \left (\frac {\sqrt {b} \sqrt {d+e x} \left (-15 a^3 e^3-5 a^2 b e^2 (2 d+11 e x)-a b^2 e \left (8 d^2+36 d e x+73 e^2 x^2\right )+b^3 \left (48 d^3+136 d^2 e x+118 d e^2 x^2+15 e^3 x^3\right )\right )}{e^4 (-b d+a e) (a+b x)^4}+\frac {15 \arctan \left (\frac {\sqrt {b} \sqrt {d+e x}}{\sqrt {-b d+a e}}\right )}{(-b d+a e)^{3/2}}\right )}{192 b^{7/2} \left ((a+b x)^2\right )^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(476\) vs. \(2(177)=354\).
Time = 2.25 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.83
method | result | size |
default | \(\frac {\left (15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) b^{4} e^{4} x^{4}+60 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a \,b^{3} e^{4} x^{3}+15 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {7}{2}} b^{3}+90 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{2} b^{2} e^{4} x^{2}-73 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, a \,b^{2} e +73 \left (e x +d \right )^{\frac {5}{2}} \sqrt {\left (a e -b d \right ) b}\, b^{3} d +60 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{3} b \,e^{4} x -55 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a^{2} b \,e^{2}+110 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} a \,b^{2} d e -55 \sqrt {\left (a e -b d \right ) b}\, \left (e x +d \right )^{\frac {3}{2}} b^{3} d^{2}+15 \arctan \left (\frac {b \sqrt {e x +d}}{\sqrt {\left (a e -b d \right ) b}}\right ) a^{4} e^{4}-15 \sqrt {e x +d}\, a^{3} e^{3} \sqrt {\left (a e -b d \right ) b}+45 \sqrt {e x +d}\, a^{2} d \,e^{2} b \sqrt {\left (a e -b d \right ) b}-45 \sqrt {e x +d}\, a \,d^{2} e \,b^{2} \sqrt {\left (a e -b d \right ) b}+15 \sqrt {e x +d}\, d^{3} b^{3} \sqrt {\left (a e -b d \right ) b}\right ) \left (b x +a \right )}{192 \sqrt {\left (a e -b d \right ) b}\, b^{3} \left (a e -b d \right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}\) | \(477\) |
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Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (177) = 354\).
Time = 0.33 (sec) , antiderivative size = 894, normalized size of antiderivative = 3.44 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\left [-\frac {15 \, {\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 {b^{2} d - a b e} \log \left (\frac {b e x + 2 \, b d - a e - 2 \, \sqrt {b^{2} d - a b e} \sqrt {e x + d}}{b x + a}\right ) + 2 \, {\left (48 \, b^{5} d^{4} - 56 \, a b^{4} d^{3} e - 2 \, a^{2} b^{3} d^{2} e^{2} - 5 \, a^{3} b^{2} d e^{3} + 15 \, a^{4} b e^{4} + 15 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (118 \, b^{5} d^{2} e^{2} - 191 \, a b^{4} d e^{3} + 73 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (136 \, b^{5} d^{3} e - 172 \, a b^{4} d^{2} e^{2} - 19 \, a^{2} b^{3} d e^{3} + 55 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{384 \, {\left (a^{4} b^{6} d^{2} - 2 \, a^{5} b^{5} d e + a^{6} b^{4} e^{2} + {\left (b^{10} d^{2} - 2 \, a b^{9} d e + a^{2} b^{8} e^{2}\right )} x^{4} + 4 \, {\left (a b^{9} d^{2} - 2 \, a^{2} b^{8} d e + a^{3} b^{7} e^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{2} - 2 \, a^{3} b^{7} d e + a^{4} b^{6} e^{2}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{2} - 2 \, a^{4} b^{6} d e + a^{5} b^{5} e^{2}\right )} x\right )}}, -\frac {15 \, {\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 {-b^{2} d + a b e} \arctan \left (\frac {\sqrt {-b^{2} d + a b e} \sqrt {e x + d}}{b e x + b d}\right ) + {\left (48 \, b^{5} d^{4} - 56 \, a b^{4} d^{3} e - 2 \, a^{2} b^{3} d^{2} e^{2} - 5 \, a^{3} b^{2} d e^{3} + 15 \, a^{4} b e^{4} + 15 \, {\left (b^{5} d e^{3} - a b^{4} e^{4}\right )} x^{3} + {\left (118 \, b^{5} d^{2} e^{2} - 191 \, a b^{4} d e^{3} + 73 \, a^{2} b^{3} e^{4}\right )} x^{2} + {\left (136 \, b^{5} d^{3} e - 172 \, a b^{4} d^{2} e^{2} - 19 \, a^{2} b^{3} d e^{3} + 55 \, a^{3} b^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{192 \, {\left (a^{4} b^{6} d^{2} - 2 \, a^{5} b^{5} d e + a^{6} b^{4} e^{2} + {\left (b^{10} d^{2} - 2 \, a b^{9} d e + a^{2} b^{8} e^{2}\right )} x^{4} + 4 \, {\left (a b^{9} d^{2} - 2 \, a^{2} b^{8} d e + a^{3} b^{7} e^{2}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} d^{2} - 2 \, a^{3} b^{7} d e + a^{4} b^{6} e^{2}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} d^{2} - 2 \, a^{4} b^{6} d e + a^{5} b^{5} e^{2}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{\frac {5}{2}}} \,d x } \]
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Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.09 \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=-\frac {5 \, e^{4} \arctan \left (\frac {\sqrt {e x + d} b}{\sqrt {-b^{2} d + a b e}}\right )}{64 \, {\left (b^{4} d \mathrm {sgn}\left (b x + a\right ) - a b^{3} e \mathrm {sgn}\left (b x + a\right )\right )} \sqrt {-b^{2} d + a b e}} - \frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{3} e^{4} + 73 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{3} d e^{4} - 55 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{3} d^{2} e^{4} + 15 \, \sqrt {e x + d} b^{3} d^{3} e^{4} - 73 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{2} e^{5} + 110 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{2} d e^{5} - 45 \, \sqrt {e x + d} a b^{2} d^{2} e^{5} - 55 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b e^{6} + 45 \, \sqrt {e x + d} a^{2} b d e^{6} - 15 \, \sqrt {e x + d} a^{3} e^{7}}{192 \, {\left (b^{4} d \mathrm {sgn}\left (b x + a\right ) - a b^{3} e \mathrm {sgn}\left (b x + a\right )\right )} {\left ({\left (e x + d\right )} b - b d + a e\right )}^{4}} \]
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Timed out. \[ \int \frac {(d+e x)^{5/2}}{\left (a^2+2 a b x+b^2 x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{5/2}}{{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}} \,d x \]
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