Integrand size = 37, antiderivative size = 113 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx=\frac {2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}-\frac {2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac {6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 c^3 d^3 \sqrt {d+e x}}{e^4} \]
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Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {640, 45} \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx=\frac {6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt {d+e x}}-\frac {2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac {2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}+\frac {2 c^3 d^3 \sqrt {d+e x}}{e^4} \]
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Rule 45
Rule 640
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a e+c d x)^3}{(d+e x)^{7/2}} \, dx \\ & = \int \left (\frac {\left (-c d^2+a e^2\right )^3}{e^3 (d+e x)^{7/2}}+\frac {3 c d \left (c d^2-a e^2\right )^2}{e^3 (d+e x)^{5/2}}-\frac {3 c^2 d^2 \left (c d^2-a e^2\right )}{e^3 (d+e x)^{3/2}}+\frac {c^3 d^3}{e^3 \sqrt {d+e x}}\right ) \, dx \\ & = \frac {2 \left (c d^2-a e^2\right )^3}{5 e^4 (d+e x)^{5/2}}-\frac {2 c d \left (c d^2-a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac {6 c^2 d^2 \left (c d^2-a e^2\right )}{e^4 \sqrt {d+e x}}+\frac {2 c^3 d^3 \sqrt {d+e x}}{e^4} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx=-\frac {2 \left (a^3 e^6+a^2 c d e^4 (2 d+5 e x)+a c^2 d^2 e^2 \left (8 d^2+20 d e x+15 e^2 x^2\right )-c^3 d^3 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )\right )}{5 e^4 (d+e x)^{5/2}} \]
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Time = 2.70 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\frac {2 \left (5 x^{3} d^{3} e^{3}+30 d^{4} e^{2} x^{2}+40 d^{5} e x +16 d^{6}\right ) c^{3}}{5}-\frac {16 \left (\frac {15}{8} x^{2} e^{2}+\frac {5}{2} d e x +d^{2}\right ) e^{2} d^{2} a \,c^{2}}{5}-\frac {4 e^{4} \left (\frac {5 e x}{2}+d \right ) d \,a^{2} c}{5}-\frac {2 e^{6} a^{3}}{5}}{\left (e x +d \right )^{\frac {5}{2}} e^{4}}\) | \(106\) |
risch | \(\frac {2 c^{3} d^{3} \sqrt {e x +d}}{e^{4}}-\frac {2 \left (15 x^{2} c^{2} d^{2} e^{2}+5 x a c d \,e^{3}+25 x \,c^{2} d^{3} e +a^{2} e^{4}+3 a c \,d^{2} e^{2}+11 c^{2} d^{4}\right ) \left (e^{2} a -c \,d^{2}\right )}{5 e^{4} \sqrt {e x +d}\, \left (x^{2} e^{2}+2 d e x +d^{2}\right )}\) | \(121\) |
gosper | \(-\frac {2 \left (-5 x^{3} c^{3} d^{3} e^{3}+15 x^{2} a \,c^{2} d^{2} e^{4}-30 x^{2} c^{3} d^{4} e^{2}+5 x \,a^{2} c d \,e^{5}+20 x a \,c^{2} d^{3} e^{3}-40 x \,c^{3} d^{5} e +e^{6} a^{3}+2 d^{2} e^{4} a^{2} c +8 d^{4} e^{2} c^{2} a -16 c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{4}}\) | \(130\) |
trager | \(-\frac {2 \left (-5 x^{3} c^{3} d^{3} e^{3}+15 x^{2} a \,c^{2} d^{2} e^{4}-30 x^{2} c^{3} d^{4} e^{2}+5 x \,a^{2} c d \,e^{5}+20 x a \,c^{2} d^{3} e^{3}-40 x \,c^{3} d^{5} e +e^{6} a^{3}+2 d^{2} e^{4} a^{2} c +8 d^{4} e^{2} c^{2} a -16 c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{4}}\) | \(130\) |
derivativedivides | \(\frac {2 c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(133\) |
default | \(\frac {2 c^{3} d^{3} \sqrt {e x +d}-\frac {2 \left (e^{6} a^{3}-3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a -c^{3} d^{6}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 c^{2} d^{2} \left (e^{2} a -c \,d^{2}\right )}{\sqrt {e x +d}}-\frac {2 c d \left (a^{2} e^{4}-2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\left (e x +d \right )^{\frac {3}{2}}}}{e^{4}}\) | \(133\) |
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Time = 0.29 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx=\frac {2 \, {\left (5 \, c^{3} d^{3} e^{3} x^{3} + 16 \, c^{3} d^{6} - 8 \, a c^{2} d^{4} e^{2} - 2 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 15 \, {\left (2 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 5 \, {\left (8 \, c^{3} d^{5} e - 4 \, a c^{2} d^{3} e^{3} - a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{5 \, {\left (e^{7} x^{3} + 3 \, d e^{6} x^{2} + 3 \, d^{2} e^{5} x + d^{3} e^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (105) = 210\).
Time = 1.60 (sec) , antiderivative size = 654, normalized size of antiderivative = 5.79 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx=\begin {cases} - \frac {2 a^{3} e^{6}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {4 a^{2} c d^{2} e^{4}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {10 a^{2} c d e^{5} x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {16 a c^{2} d^{4} e^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {40 a c^{2} d^{3} e^{3} x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} - \frac {30 a c^{2} d^{2} e^{4} x^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {32 c^{3} d^{6}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {80 c^{3} d^{5} e x}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {60 c^{3} d^{4} e^{2} x^{2}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} + \frac {10 c^{3} d^{3} e^{3} x^{3}}{5 d^{2} e^{4} \sqrt {d + e x} + 10 d e^{5} x \sqrt {d + e x} + 5 e^{6} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {c^{3} x^{4}}{4 \sqrt {d}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.24 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx=\frac {2 \, {\left (\frac {5 \, \sqrt {e x + d} c^{3} d^{3}}{e^{3}} + \frac {c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6} + 15 \, {\left (c^{3} d^{4} - a c^{2} d^{2} e^{2}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{3}}\right )}}{5 \, e} \]
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Time = 0.29 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx=\frac {2 \, \sqrt {e x + d} c^{3} d^{3}}{e^{4}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{2} c^{3} d^{4} - 5 \, {\left (e x + d\right )} c^{3} d^{5} + c^{3} d^{6} - 15 \, {\left (e x + d\right )}^{2} a c^{2} d^{2} e^{2} + 10 \, {\left (e x + d\right )} a c^{2} d^{3} e^{2} - 3 \, a c^{2} d^{4} e^{2} - 5 \, {\left (e x + d\right )} a^{2} c d e^{4} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}}{5 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{4}} \]
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Time = 10.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3}{(d+e x)^{13/2}} \, dx=-\frac {2\,\left (a^3\,e^6+2\,a^2\,c\,d^2\,e^4+5\,a^2\,c\,d\,e^5\,x+8\,a\,c^2\,d^4\,e^2+20\,a\,c^2\,d^3\,e^3\,x+15\,a\,c^2\,d^2\,e^4\,x^2-16\,c^3\,d^6-40\,c^3\,d^5\,e\,x-30\,c^3\,d^4\,e^2\,x^2-5\,c^3\,d^3\,e^3\,x^3\right )}{5\,e^4\,{\left (d+e\,x\right )}^{5/2}} \]
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