Integrand size = 19, antiderivative size = 145 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=-\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{35 b^6 \left (b x+c x^2\right )^{3/2}}+\frac {1024 c^2 (2 c d-b e) (b+2 c x)}{35 b^8 \sqrt {b x+c x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {652, 628, 627} \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=\frac {1024 c^2 (b+2 c x) (2 c d-b e)}{35 b^8 \sqrt {b x+c x^2}}-\frac {128 c (b+2 c x) (2 c d-b e)}{35 b^6 \left (b x+c x^2\right )^{3/2}}+\frac {24 (b+2 c x) (2 c d-b e)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac {2 (x (2 c d-b e)+b d)}{7 b^2 \left (b x+c x^2\right )^{7/2}} \]
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Rule 627
Rule 628
Rule 652
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}-\frac {(12 (2 c d-b e)) \int \frac {1}{\left (b x+c x^2\right )^{7/2}} \, dx}{7 b^2} \\ & = -\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 b^4 \left (b x+c x^2\right )^{5/2}}+\frac {(192 c (2 c d-b e)) \int \frac {1}{\left (b x+c x^2\right )^{5/2}} \, dx}{35 b^4} \\ & = -\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{35 b^6 \left (b x+c x^2\right )^{3/2}}-\frac {\left (512 c^2 (2 c d-b e)\right ) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{35 b^6} \\ & = -\frac {2 (b d+(2 c d-b e) x)}{7 b^2 \left (b x+c x^2\right )^{7/2}}+\frac {24 (2 c d-b e) (b+2 c x)}{35 b^4 \left (b x+c x^2\right )^{5/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{35 b^6 \left (b x+c x^2\right )^{3/2}}+\frac {1024 c^2 (2 c d-b e) (b+2 c x)}{35 b^8 \sqrt {b x+c x^2}} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.03 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \sqrt {x (b+c x)} \left (-2048 c^7 d x^7-560 b^4 c^3 x^3 (d-4 e x)+1024 b c^6 x^6 (-7 d+e x)+4480 b^3 c^4 x^4 (-d+e x)+1792 b^2 c^5 x^5 (-5 d+2 e x)-14 b^6 c x (d+2 e x)+56 b^5 c^2 x^2 (d+5 e x)+b^7 (5 d+7 e x)\right )}{35 b^8 x^4 (b+c x)^4} \]
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Time = 0.19 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.24
method | result | size |
gosper | \(-\frac {2 x \left (c x +b \right ) \left (1024 b \,c^{6} e \,x^{7}-2048 c^{7} d \,x^{7}+3584 b^{2} c^{5} e \,x^{6}-7168 b \,c^{6} d \,x^{6}+4480 b^{3} c^{4} e \,x^{5}-8960 b^{2} c^{5} d \,x^{5}+2240 b^{4} c^{3} e \,x^{4}-4480 b^{3} c^{4} d \,x^{4}+280 b^{5} c^{2} e \,x^{3}-560 b^{4} c^{3} d \,x^{3}-28 b^{6} c e \,x^{2}+56 b^{5} c^{2} d \,x^{2}+7 b^{7} e x -14 b^{6} c d x +5 d \,b^{7}\right )}{35 b^{8} \left (c \,x^{2}+b x \right )^{\frac {9}{2}}}\) | \(180\) |
trager | \(-\frac {2 \left (1024 b \,c^{6} e \,x^{7}-2048 c^{7} d \,x^{7}+3584 b^{2} c^{5} e \,x^{6}-7168 b \,c^{6} d \,x^{6}+4480 b^{3} c^{4} e \,x^{5}-8960 b^{2} c^{5} d \,x^{5}+2240 b^{4} c^{3} e \,x^{4}-4480 b^{3} c^{4} d \,x^{4}+280 b^{5} c^{2} e \,x^{3}-560 b^{4} c^{3} d \,x^{3}-28 b^{6} c e \,x^{2}+56 b^{5} c^{2} d \,x^{2}+7 b^{7} e x -14 b^{6} c d x +5 d \,b^{7}\right ) \sqrt {c \,x^{2}+b x}}{35 b^{8} \left (c x +b \right )^{4} x^{4}}\) | \(184\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (462 b \,c^{2} e \,x^{3}-1024 c^{3} d \,x^{3}-56 b^{2} c e \,x^{2}+162 b \,c^{2} d \,x^{2}+7 b^{3} e x -34 b^{2} c d x +5 d \,b^{3}\right )}{35 b^{8} x^{3} \sqrt {x \left (c x +b \right )}}-\frac {2 c^{3} \left (562 x^{3} b \,c^{3} e -1024 x^{3} c^{4} d +1792 x^{2} b^{2} c^{2} e -3234 x^{2} b \,c^{3} d +1925 x \,b^{3} c e -3430 x \,b^{2} c^{2} d +700 b^{4} e -1225 b^{3} c d \right ) x}{35 \sqrt {x \left (c x +b \right )}\, \left (c^{3} x^{3}+3 c^{2} b \,x^{2}+3 c \,b^{2} x +b^{3}\right ) b^{8}}\) | \(205\) |
default | \(d \left (-\frac {2 \left (2 c x +b \right )}{7 b^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}-\frac {24 c \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{7 b^{2}}\right )+e \left (-\frac {1}{7 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{7 b^{2} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}-\frac {24 c \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{7 b^{2}}\right )}{2 c}\right )\) | \(237\) |
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Time = 0.27 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.54 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=-\frac {2 \, {\left (5 \, b^{7} d - 1024 \, {\left (2 \, c^{7} d - b c^{6} e\right )} x^{7} - 3584 \, {\left (2 \, b c^{6} d - b^{2} c^{5} e\right )} x^{6} - 4480 \, {\left (2 \, b^{2} c^{5} d - b^{3} c^{4} e\right )} x^{5} - 2240 \, {\left (2 \, b^{3} c^{4} d - b^{4} c^{3} e\right )} x^{4} - 280 \, {\left (2 \, b^{4} c^{3} d - b^{5} c^{2} e\right )} x^{3} + 28 \, {\left (2 \, b^{5} c^{2} d - b^{6} c e\right )} x^{2} - 7 \, {\left (2 \, b^{6} c d - b^{7} e\right )} x\right )} \sqrt {c x^{2} + b x}}{35 \, {\left (b^{8} c^{4} x^{8} + 4 \, b^{9} c^{3} x^{7} + 6 \, b^{10} c^{2} x^{6} + 4 \, b^{11} c x^{5} + b^{12} x^{4}\right )}} \]
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\[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=\int \frac {d + e x}{\left (x \left (b + c x\right )\right )^{\frac {9}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (129) = 258\).
Time = 0.19 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.01 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=-\frac {4 \, c d x}{7 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b^{2}} + \frac {96 \, c^{2} d x}{35 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{4}} - \frac {512 \, c^{3} d x}{35 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{6}} + \frac {4096 \, c^{4} d x}{35 \, \sqrt {c x^{2} + b x} b^{8}} + \frac {2 \, e x}{7 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b} - \frac {48 \, c e x}{35 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3}} + \frac {256 \, c^{2} e x}{35 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5}} - \frac {2048 \, c^{3} e x}{35 \, \sqrt {c x^{2} + b x} b^{7}} - \frac {2 \, d}{7 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}} b} + \frac {48 \, c d}{35 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{3}} - \frac {256 \, c^{2} d}{35 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{5}} + \frac {2048 \, c^{3} d}{35 \, \sqrt {c x^{2} + b x} b^{7}} - \frac {24 \, e}{35 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2}} + \frac {128 \, c e}{35 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4}} - \frac {1024 \, c^{2} e}{35 \, \sqrt {c x^{2} + b x} b^{6}} \]
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Time = 0.29 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.37 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (2 \, {\left (8 \, {\left (2 \, {\left (4 \, x {\left (\frac {2 \, {\left (2 \, c^{7} d - b c^{6} e\right )} x}{b^{8}} + \frac {7 \, {\left (2 \, b c^{6} d - b^{2} c^{5} e\right )}}{b^{8}}\right )} + \frac {35 \, {\left (2 \, b^{2} c^{5} d - b^{3} c^{4} e\right )}}{b^{8}}\right )} x + \frac {35 \, {\left (2 \, b^{3} c^{4} d - b^{4} c^{3} e\right )}}{b^{8}}\right )} x + \frac {35 \, {\left (2 \, b^{4} c^{3} d - b^{5} c^{2} e\right )}}{b^{8}}\right )} x - \frac {7 \, {\left (2 \, b^{5} c^{2} d - b^{6} c e\right )}}{b^{8}}\right )} x + \frac {7 \, {\left (2 \, b^{6} c d - b^{7} e\right )}}{b^{8}}\right )} x - \frac {5 \, d}{b}\right )}}{35 \, {\left (c x^{2} + b x\right )}^{\frac {7}{2}}} \]
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Time = 10.48 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{9/2}} \, dx=\frac {\frac {2048\,c^3\,d-1024\,b\,c^2\,e}{35\,b^7}+\frac {2\,c\,x\,\left (2048\,c^3\,d-1024\,b\,c^2\,e\right )}{35\,b^8}}{\sqrt {c\,x^2+b\,x}}-\frac {\frac {256\,c^2\,d-128\,b\,c\,e}{35\,b^5}+\frac {2\,c\,x\,\left (256\,c^2\,d-128\,b\,c\,e\right )}{35\,b^6}}{{\left (c\,x^2+b\,x\right )}^{3/2}}-\frac {\frac {2\,d}{7\,b}-x\,\left (\frac {2\,e}{7\,b}-\frac {4\,c\,d}{7\,b^2}\right )}{{\left (c\,x^2+b\,x\right )}^{7/2}}-\frac {\frac {24\,b\,e-48\,c\,d}{35\,b^3}+\frac {2\,c\,x\,\left (24\,b\,e-48\,c\,d\right )}{35\,b^4}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \]
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