Integrand size = 19, antiderivative size = 107 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 (b d+(2 c d-b e) x)}{5 b^2 \left (b x+c x^2\right )^{5/2}}+\frac {16 (2 c d-b e) (b+2 c x)}{15 b^4 \left (b x+c x^2\right )^{3/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{15 b^6 \sqrt {b x+c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {652, 628, 627} \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {128 c (b+2 c x) (2 c d-b e)}{15 b^6 \sqrt {b x+c x^2}}+\frac {16 (b+2 c x) (2 c d-b e)}{15 b^4 \left (b x+c x^2\right )^{3/2}}-\frac {2 (x (2 c d-b e)+b d)}{5 b^2 \left (b x+c x^2\right )^{5/2}} \]
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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)}{5 b^2 \left (b x+c x^2\right )^{5/2}}-\frac {(8 (2 c d-b e)) \int \frac {1}{\left (b x+c x^2\right )^{5/2}} \, dx}{5 b^2} \\ & = -\frac {2 (b d+(2 c d-b e) x)}{5 b^2 \left (b x+c x^2\right )^{5/2}}+\frac {16 (2 c d-b e) (b+2 c x)}{15 b^4 \left (b x+c x^2\right )^{3/2}}+\frac {(64 c (2 c d-b e)) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{15 b^4} \\ & = -\frac {2 (b d+(2 c d-b e) x)}{5 b^2 \left (b x+c x^2\right )^{5/2}}+\frac {16 (2 c d-b e) (b+2 c x)}{15 b^4 \left (b x+c x^2\right )^{3/2}}-\frac {128 c (2 c d-b e) (b+2 c x)}{15 b^6 \sqrt {b x+c x^2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.98 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \left (256 c^5 d x^5+80 b^3 c^2 x^2 (d-3 e x)+160 b^2 c^3 x^3 (3 d-2 e x)-128 b c^4 x^4 (-5 d+e x)-10 b^4 c x (d+4 e x)+b^5 (3 d+5 e x)\right )}{15 b^6 (x (b+c x))^{5/2}} \]
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Time = 0.19 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02
method | result | size |
pseudoelliptic | \(\frac {\left (-10 e x -6 d \right ) b^{5}+20 c x \left (4 e x +d \right ) b^{4}-160 c^{2} x^{2} \left (-3 e x +d \right ) b^{3}-960 c^{3} \left (-\frac {2 e x}{3}+d \right ) x^{3} b^{2}-1280 c^{4} x^{4} \left (-\frac {e x}{5}+d \right ) b -512 c^{5} d \,x^{5}}{15 \sqrt {x \left (c x +b \right )}\, x^{2} \left (c x +b \right )^{2} b^{6}}\) | \(109\) |
gosper | \(-\frac {2 x \left (c x +b \right ) \left (-128 b \,c^{4} e \,x^{5}+256 c^{5} d \,x^{5}-320 b^{2} c^{3} e \,x^{4}+640 b \,c^{4} d \,x^{4}-240 b^{3} c^{2} e \,x^{3}+480 b^{2} c^{3} d \,x^{3}-40 b^{4} c e \,x^{2}+80 b^{3} c^{2} d \,x^{2}+5 b^{5} e x -10 b^{4} c d x +3 d \,b^{5}\right )}{15 b^{6} \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}\) | \(132\) |
trager | \(-\frac {2 \left (-128 b \,c^{4} e \,x^{5}+256 c^{5} d \,x^{5}-320 b^{2} c^{3} e \,x^{4}+640 b \,c^{4} d \,x^{4}-240 b^{3} c^{2} e \,x^{3}+480 b^{2} c^{3} d \,x^{3}-40 b^{4} c e \,x^{2}+80 b^{3} c^{2} d \,x^{2}+5 b^{5} e x -10 b^{4} c d x +3 d \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{15 b^{6} \left (c x +b \right )^{3} x^{3}}\) | \(136\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (-55 b c e \,x^{2}+128 c^{2} d \,x^{2}+5 b^{2} e x -19 b c d x +3 b^{2} d \right )}{15 b^{6} x^{2} \sqrt {x \left (c x +b \right )}}+\frac {2 c^{2} \left (73 x^{2} b \,c^{2} e -128 x^{2} c^{3} d +160 x \,b^{2} c e -275 x b \,c^{2} d +90 b^{3} e -150 b^{2} c d \right ) x}{15 \sqrt {x \left (c x +b \right )}\, \left (c^{2} x^{2}+2 c b x +b^{2}\right ) b^{6}}\) | \(146\) |
default | \(d \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )+e \left (-\frac {1}{5 c \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {b \left (-\frac {2 \left (2 c x +b \right )}{5 b^{2} \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}-\frac {16 c \left (-\frac {2 \left (2 c x +b \right )}{3 b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 b^{4} \sqrt {c \,x^{2}+b x}}\right )}{5 b^{2}}\right )}{2 c}\right )\) | \(179\) |
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Time = 0.26 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.53 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left (3 \, b^{5} d + 128 \, {\left (2 \, c^{5} d - b c^{4} e\right )} x^{5} + 320 \, {\left (2 \, b c^{4} d - b^{2} c^{3} e\right )} x^{4} + 240 \, {\left (2 \, b^{2} c^{3} d - b^{3} c^{2} e\right )} x^{3} + 40 \, {\left (2 \, b^{3} c^{2} d - b^{4} c e\right )} x^{2} - 5 \, {\left (2 \, b^{4} c d - b^{5} e\right )} x\right )} \sqrt {c x^{2} + b x}}{15 \, {\left (b^{6} c^{3} x^{6} + 3 \, b^{7} c^{2} x^{5} + 3 \, b^{8} c x^{4} + b^{9} x^{3}\right )}} \]
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\[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=\int \frac {d + e x}{\left (x \left (b + c x\right )\right )^{\frac {7}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (95) = 190\).
Time = 0.18 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.96 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {4 \, c d x}{5 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b^{2}} + \frac {64 \, c^{2} d x}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{4}} - \frac {512 \, c^{3} d x}{15 \, \sqrt {c x^{2} + b x} b^{6}} + \frac {2 \, e x}{5 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b} - \frac {32 \, c e x}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}} + \frac {256 \, c^{2} e x}{15 \, \sqrt {c x^{2} + b x} b^{5}} - \frac {2 \, d}{5 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} b} + \frac {32 \, c d}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{3}} - \frac {256 \, c^{2} d}{15 \, \sqrt {c x^{2} + b x} b^{5}} - \frac {16 \, e}{15 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} b^{2}} + \frac {128 \, c e}{15 \, \sqrt {c x^{2} + b x} b^{4}} \]
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Time = 0.29 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.33 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2 \, {\left ({\left (8 \, {\left (2 \, {\left (4 \, x {\left (\frac {2 \, {\left (2 \, c^{5} d - b c^{4} e\right )} x}{b^{6}} + \frac {5 \, {\left (2 \, b c^{4} d - b^{2} c^{3} e\right )}}{b^{6}}\right )} + \frac {15 \, {\left (2 \, b^{2} c^{3} d - b^{3} c^{2} e\right )}}{b^{6}}\right )} x + \frac {5 \, {\left (2 \, b^{3} c^{2} d - b^{4} c e\right )}}{b^{6}}\right )} x - \frac {5 \, {\left (2 \, b^{4} c d - b^{5} e\right )}}{b^{6}}\right )} x + \frac {3 \, d}{b}\right )}}{15 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \]
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Time = 10.44 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.17 \[ \int \frac {d+e x}{\left (b x+c x^2\right )^{7/2}} \, dx=-\frac {2\,\left (5\,e\,b^5\,x+3\,d\,b^5-40\,e\,b^4\,c\,x^2-10\,d\,b^4\,c\,x-240\,e\,b^3\,c^2\,x^3+80\,d\,b^3\,c^2\,x^2-320\,e\,b^2\,c^3\,x^4+480\,d\,b^2\,c^3\,x^3-128\,e\,b\,c^4\,x^5+640\,d\,b\,c^4\,x^4+256\,d\,c^5\,x^5\right )}{15\,b^6\,{\left (c\,x^2+b\,x\right )}^{5/2}} \]
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