Integrand size = 17, antiderivative size = 126 \[ \int \frac {\sqrt {b x+c x^2}}{x^7} \, dx=-\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}+\frac {16 c \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac {32 c^2 \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac {128 c^3 \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}-\frac {256 c^4 \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3} \]
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Time = 0.04 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {672, 664} \[ \int \frac {\sqrt {b x+c x^2}}{x^7} \, dx=-\frac {256 c^4 \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3}+\frac {128 c^3 \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}-\frac {32 c^2 \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac {16 c \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]
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Rule 664
Rule 672
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {(8 c) \int \frac {\sqrt {b x+c x^2}}{x^6} \, dx}{11 b} \\ & = -\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}+\frac {16 c \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac {\left (16 c^2\right ) \int \frac {\sqrt {b x+c x^2}}{x^5} \, dx}{33 b^2} \\ & = -\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}+\frac {16 c \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac {32 c^2 \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}-\frac {\left (64 c^3\right ) \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx}{231 b^3} \\ & = -\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}+\frac {16 c \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac {32 c^2 \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac {128 c^3 \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}+\frac {\left (128 c^4\right ) \int \frac {\sqrt {b x+c x^2}}{x^3} \, dx}{1155 b^4} \\ & = -\frac {2 \left (b x+c x^2\right )^{3/2}}{11 b x^7}+\frac {16 c \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac {32 c^2 \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac {128 c^3 \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}-\frac {256 c^4 \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.58 \[ \int \frac {\sqrt {b x+c x^2}}{x^7} \, dx=-\frac {2 \sqrt {x (b+c x)} \left (315 b^5+35 b^4 c x-40 b^3 c^2 x^2+48 b^2 c^3 x^3-64 b c^4 x^4+128 c^5 x^5\right )}{3465 b^5 x^6} \]
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Time = 2.17 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.52
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) \left (128 c^{4} x^{4}-192 b \,c^{3} x^{3}+240 b^{2} c^{2} x^{2}-280 b^{3} c x +315 b^{4}\right ) \sqrt {c \,x^{2}+b x}}{3465 b^{5} x^{6}}\) | \(66\) |
pseudoelliptic | \(\frac {2 \left (-128 c^{5} x^{5}+64 b \,x^{4} c^{4}-48 b^{2} c^{3} x^{3}+40 x^{2} b^{3} c^{2}-35 c x \,b^{4}-315 b^{5}\right ) \sqrt {x \left (c x +b \right )}}{3465 x^{6} b^{5}}\) | \(70\) |
trager | \(-\frac {2 \left (128 c^{5} x^{5}-64 b \,x^{4} c^{4}+48 b^{2} c^{3} x^{3}-40 x^{2} b^{3} c^{2}+35 c x \,b^{4}+315 b^{5}\right ) \sqrt {c \,x^{2}+b x}}{3465 b^{5} x^{6}}\) | \(72\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (128 c^{5} x^{5}-64 b \,x^{4} c^{4}+48 b^{2} c^{3} x^{3}-40 x^{2} b^{3} c^{2}+35 c x \,b^{4}+315 b^{5}\right )}{3465 x^{5} \sqrt {x \left (c x +b \right )}\, b^{5}}\) | \(75\) |
default | \(-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{11 b \,x^{7}}-\frac {8 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{9 b \,x^{6}}-\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{7 b \,x^{5}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 b^{2} x^{3}}\right )}{7 b}\right )}{3 b}\right )}{11 b}\) | \(119\) |
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Time = 0.26 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {b x+c x^2}}{x^7} \, dx=-\frac {2 \, {\left (128 \, c^{5} x^{5} - 64 \, b c^{4} x^{4} + 48 \, b^{2} c^{3} x^{3} - 40 \, b^{3} c^{2} x^{2} + 35 \, b^{4} c x + 315 \, b^{5}\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{5} x^{6}} \]
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\[ \int \frac {\sqrt {b x+c x^2}}{x^7} \, dx=\int \frac {\sqrt {x \left (b + c x\right )}}{x^{7}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {b x+c x^2}}{x^7} \, dx=-\frac {256 \, \sqrt {c x^{2} + b x} c^{5}}{3465 \, b^{5} x} + \frac {128 \, \sqrt {c x^{2} + b x} c^{4}}{3465 \, b^{4} x^{2}} - \frac {32 \, \sqrt {c x^{2} + b x} c^{3}}{1155 \, b^{3} x^{3}} + \frac {16 \, \sqrt {c x^{2} + b x} c^{2}}{693 \, b^{2} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} c}{99 \, b x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x}}{11 \, x^{6}} \]
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Time = 0.27 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.54 \[ \int \frac {\sqrt {b x+c x^2}}{x^7} \, dx=\frac {2 \, {\left (11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} c^{3} + 36960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} b c^{\frac {5}{2}} + 51480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} b^{2} c^{2} + 38115 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{3} c^{\frac {3}{2}} + 15785 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{4} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{5} \sqrt {c} + 315 \, b^{6}\right )}}{3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \]
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Time = 9.36 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {b x+c x^2}}{x^7} \, dx=\frac {16\,c^2\,\sqrt {c\,x^2+b\,x}}{693\,b^2\,x^4}-\frac {2\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {32\,c^3\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^3}+\frac {128\,c^4\,\sqrt {c\,x^2+b\,x}}{3465\,b^4\,x^2}-\frac {256\,c^5\,\sqrt {c\,x^2+b\,x}}{3465\,b^5\,x}-\frac {2\,c\,\sqrt {c\,x^2+b\,x}}{99\,b\,x^5} \]
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