Integrand size = 17, antiderivative size = 152 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}-\frac {10 b (b c-a d)^4}{d^6 \sqrt {c+d x}}-\frac {20 b^2 (b c-a d)^3 \sqrt {c+d x}}{d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{3/2}}{3 d^6}-\frac {2 b^4 (b c-a d) (c+d x)^{5/2}}{d^6}+\frac {2 b^5 (c+d x)^{7/2}}{7 d^6} \]
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Time = 0.04 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=-\frac {2 b^4 (c+d x)^{5/2} (b c-a d)}{d^6}+\frac {20 b^3 (c+d x)^{3/2} (b c-a d)^2}{3 d^6}-\frac {20 b^2 \sqrt {c+d x} (b c-a d)^3}{d^6}-\frac {10 b (b c-a d)^4}{d^6 \sqrt {c+d x}}+\frac {2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}+\frac {2 b^5 (c+d x)^{7/2}}{7 d^6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^5}{d^5 (c+d x)^{5/2}}+\frac {5 b (b c-a d)^4}{d^5 (c+d x)^{3/2}}-\frac {10 b^2 (b c-a d)^3}{d^5 \sqrt {c+d x}}+\frac {10 b^3 (b c-a d)^2 \sqrt {c+d x}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{3/2}}{d^5}+\frac {b^5 (c+d x)^{5/2}}{d^5}\right ) \, dx \\ & = \frac {2 (b c-a d)^5}{3 d^6 (c+d x)^{3/2}}-\frac {10 b (b c-a d)^4}{d^6 \sqrt {c+d x}}-\frac {20 b^2 (b c-a d)^3 \sqrt {c+d x}}{d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{3/2}}{3 d^6}-\frac {2 b^4 (b c-a d) (c+d x)^{5/2}}{d^6}+\frac {2 b^5 (c+d x)^{7/2}}{7 d^6} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=-\frac {2 \left (7 a^5 d^5+35 a^4 b d^4 (2 c+3 d x)-70 a^3 b^2 d^3 \left (8 c^2+12 c d x+3 d^2 x^2\right )+70 a^2 b^3 d^2 \left (16 c^3+24 c^2 d x+6 c d^2 x^2-d^3 x^3\right )-7 a b^4 d \left (128 c^4+192 c^3 d x+48 c^2 d^2 x^2-8 c d^3 x^3+3 d^4 x^4\right )+b^5 \left (256 c^5+384 c^4 d x+96 c^3 d^2 x^2-16 c^2 d^3 x^3+6 c d^4 x^4-3 d^5 x^5\right )\right )}{21 d^6 (c+d x)^{3/2}} \]
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Time = 0.33 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.28
method | result | size |
risch | \(\frac {2 b^{2} \left (3 d^{3} x^{3} b^{3}+21 x^{2} a \,b^{2} d^{3}-12 x^{2} b^{3} c \,d^{2}+70 x \,a^{2} b \,d^{3}-98 x a \,b^{2} c \,d^{2}+37 x \,b^{3} c^{2} d +210 a^{3} d^{3}-560 a^{2} b c \,d^{2}+511 a \,b^{2} c^{2} d -158 b^{3} c^{3}\right ) \sqrt {d x +c}}{21 d^{6}}-\frac {2 \left (15 b d x +a d +14 b c \right ) \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{3 d^{6} \left (d x +c \right )^{\frac {3}{2}}}\) | \(194\) |
pseudoelliptic | \(-\frac {2 \left (\left (-\frac {3}{7} b^{5} x^{5}-3 a \,b^{4} x^{4}-10 a^{2} b^{3} x^{3}-30 a^{3} b^{2} x^{2}+15 a^{4} b x +a^{5}\right ) d^{5}+10 \left (\frac {3}{35} b^{4} x^{4}+\frac {4}{5} a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}-12 a^{3} b x +a^{4}\right ) b c \,d^{4}-80 \left (\frac {1}{35} b^{3} x^{3}+\frac {3}{5} a \,b^{2} x^{2}-3 a^{2} b x +a^{3}\right ) b^{2} c^{2} d^{3}+160 b^{3} c^{3} \left (\frac {3}{35} b^{2} x^{2}-\frac {6}{5} a b x +a^{2}\right ) d^{2}-128 \left (-\frac {3 b x}{7}+a \right ) b^{4} c^{4} d +\frac {256 b^{5} c^{5}}{7}\right )}{3 \left (d x +c \right )^{\frac {3}{2}} d^{6}}\) | \(204\) |
gosper | \(-\frac {2 \left (-3 x^{5} b^{5} d^{5}-21 x^{4} a \,b^{4} d^{5}+6 x^{4} b^{5} c \,d^{4}-70 x^{3} a^{2} b^{3} d^{5}+56 x^{3} a \,b^{4} c \,d^{4}-16 x^{3} b^{5} c^{2} d^{3}-210 x^{2} a^{3} b^{2} d^{5}+420 x^{2} a^{2} b^{3} c \,d^{4}-336 x^{2} a \,b^{4} c^{2} d^{3}+96 x^{2} b^{5} c^{3} d^{2}+105 x \,a^{4} b \,d^{5}-840 x \,a^{3} b^{2} c \,d^{4}+1680 x \,a^{2} b^{3} c^{2} d^{3}-1344 x a \,b^{4} c^{3} d^{2}+384 x \,b^{5} c^{4} d +7 a^{5} d^{5}+70 a^{4} b c \,d^{4}-560 a^{3} b^{2} c^{2} d^{3}+1120 a^{2} b^{3} c^{3} d^{2}-896 a \,b^{4} c^{4} d +256 b^{5} c^{5}\right )}{21 \left (d x +c \right )^{\frac {3}{2}} d^{6}}\) | \(273\) |
trager | \(-\frac {2 \left (-3 x^{5} b^{5} d^{5}-21 x^{4} a \,b^{4} d^{5}+6 x^{4} b^{5} c \,d^{4}-70 x^{3} a^{2} b^{3} d^{5}+56 x^{3} a \,b^{4} c \,d^{4}-16 x^{3} b^{5} c^{2} d^{3}-210 x^{2} a^{3} b^{2} d^{5}+420 x^{2} a^{2} b^{3} c \,d^{4}-336 x^{2} a \,b^{4} c^{2} d^{3}+96 x^{2} b^{5} c^{3} d^{2}+105 x \,a^{4} b \,d^{5}-840 x \,a^{3} b^{2} c \,d^{4}+1680 x \,a^{2} b^{3} c^{2} d^{3}-1344 x a \,b^{4} c^{3} d^{2}+384 x \,b^{5} c^{4} d +7 a^{5} d^{5}+70 a^{4} b c \,d^{4}-560 a^{3} b^{2} c^{2} d^{3}+1120 a^{2} b^{3} c^{3} d^{2}-896 a \,b^{4} c^{4} d +256 b^{5} c^{5}\right )}{21 \left (d x +c \right )^{\frac {3}{2}} d^{6}}\) | \(273\) |
derivativedivides | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {7}{2}}}{7}+2 a \,b^{4} d \left (d x +c \right )^{\frac {5}{2}}-2 b^{5} c \left (d x +c \right )^{\frac {5}{2}}+\frac {20 a^{2} b^{3} d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {40 a \,b^{4} c d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {20 b^{5} c^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+20 a^{3} b^{2} d^{3} \sqrt {d x +c}-60 a^{2} b^{3} c \,d^{2} \sqrt {d x +c}+60 a \,b^{4} c^{2} d \sqrt {d x +c}-20 b^{5} c^{3} \sqrt {d x +c}-\frac {10 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{\sqrt {d x +c}}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{6}}\) | \(294\) |
default | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {7}{2}}}{7}+2 a \,b^{4} d \left (d x +c \right )^{\frac {5}{2}}-2 b^{5} c \left (d x +c \right )^{\frac {5}{2}}+\frac {20 a^{2} b^{3} d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}-\frac {40 a \,b^{4} c d \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {20 b^{5} c^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+20 a^{3} b^{2} d^{3} \sqrt {d x +c}-60 a^{2} b^{3} c \,d^{2} \sqrt {d x +c}+60 a \,b^{4} c^{2} d \sqrt {d x +c}-20 b^{5} c^{3} \sqrt {d x +c}-\frac {10 b \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}{\sqrt {d x +c}}-\frac {2 \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}}{d^{6}}\) | \(294\) |
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Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
Time = 0.23 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.86 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{5} d^{5} x^{5} - 256 \, b^{5} c^{5} + 896 \, a b^{4} c^{4} d - 1120 \, a^{2} b^{3} c^{3} d^{2} + 560 \, a^{3} b^{2} c^{2} d^{3} - 70 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5} - 3 \, {\left (2 \, b^{5} c d^{4} - 7 \, a b^{4} d^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} c^{2} d^{3} - 28 \, a b^{4} c d^{4} + 35 \, a^{2} b^{3} d^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} c^{3} d^{2} - 56 \, a b^{4} c^{2} d^{3} + 70 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} c^{4} d - 448 \, a b^{4} c^{3} d^{2} + 560 \, a^{2} b^{3} c^{2} d^{3} - 280 \, a^{3} b^{2} c d^{4} + 35 \, a^{4} b d^{5}\right )} x\right )} \sqrt {d x + c}}{21 \, {\left (d^{8} x^{2} + 2 \, c d^{7} x + c^{2} d^{6}\right )}} \]
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Time = 7.77 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{5}} - \frac {5 b \left (a d - b c\right )^{4}}{d^{5} \sqrt {c + d x}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{5 d^{5}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{3 d^{5}} + \frac {\sqrt {c + d x} \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{d^{5}} - \frac {\left (a d - b c\right )^{5}}{3 d^{5} \left (c + d x\right )^{\frac {3}{2}}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 b} & \text {otherwise} \end {cases}}{c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{5} - 21 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 70 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 210 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt {d x + c}}{d^{5}} + \frac {7 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5} - 15 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac {3}{2}} d^{5}}\right )}}{21 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (134) = 268\).
Time = 0.34 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.20 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (d x + c\right )} b^{5} c^{4} - b^{5} c^{5} - 60 \, {\left (d x + c\right )} a b^{4} c^{3} d + 5 \, a b^{4} c^{4} d + 90 \, {\left (d x + c\right )} a^{2} b^{3} c^{2} d^{2} - 10 \, a^{2} b^{3} c^{3} d^{2} - 60 \, {\left (d x + c\right )} a^{3} b^{2} c d^{3} + 10 \, a^{3} b^{2} c^{2} d^{3} + 15 \, {\left (d x + c\right )} a^{4} b d^{4} - 5 \, a^{4} b c d^{4} + a^{5} d^{5}\right )}}{3 \, {\left (d x + c\right )}^{\frac {3}{2}} d^{6}} + \frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{5} d^{36} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{5} c d^{36} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{5} c^{2} d^{36} - 210 \, \sqrt {d x + c} b^{5} c^{3} d^{36} + 21 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{4} d^{37} - 140 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{4} c d^{37} + 630 \, \sqrt {d x + c} a b^{4} c^{2} d^{37} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{3} d^{38} - 630 \, \sqrt {d x + c} a^{2} b^{3} c d^{38} + 210 \, \sqrt {d x + c} a^{3} b^{2} d^{39}\right )}}{21 \, d^{42}} \]
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Time = 0.16 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b x)^5}{(c+d x)^{5/2}} \, dx=\frac {2\,b^5\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}-\frac {\frac {2\,a^5\,d^5}{3}-\frac {2\,b^5\,c^5}{3}+\left (c+d\,x\right )\,\left (10\,a^4\,b\,d^4-40\,a^3\,b^2\,c\,d^3+60\,a^2\,b^3\,c^2\,d^2-40\,a\,b^4\,c^3\,d+10\,b^5\,c^4\right )-\frac {20\,a^2\,b^3\,c^3\,d^2}{3}+\frac {20\,a^3\,b^2\,c^2\,d^3}{3}+\frac {10\,a\,b^4\,c^4\,d}{3}-\frac {10\,a^4\,b\,c\,d^4}{3}}{d^6\,{\left (c+d\,x\right )}^{3/2}}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {c+d\,x}}{d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6} \]
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