Integrand size = 26, antiderivative size = 121 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}-\frac {3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac {b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^4 d^7} \]
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Time = 0.03 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {697} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}-\frac {3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac {\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^4 d^7} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 (b d+2 c d x)^{13/2}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)^{9/2}}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^4 (b d+2 c d x)^{5/2}}+\frac {1}{64 c^3 d^6 \sqrt {b d+2 c d x}}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^3}{704 c^4 d (b d+2 c d x)^{11/2}}-\frac {3 \left (b^2-4 a c\right )^2}{448 c^4 d^3 (b d+2 c d x)^{7/2}}+\frac {b^2-4 a c}{64 c^4 d^5 (b d+2 c d x)^{3/2}}+\frac {\sqrt {b d+2 c d x}}{64 c^4 d^7} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {7 b^6-84 a b^4 c+336 a^2 b^2 c^2-448 a^3 c^3-33 b^4 (b+2 c x)^2+264 a b^2 c (b+2 c x)^2-528 a^2 c^2 (b+2 c x)^2+77 b^2 (b+2 c x)^4-308 a c (b+2 c x)^4+77 (b+2 c x)^6}{4928 c^4 d (d (b+2 c x))^{11/2}} \]
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Time = 2.38 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {\sqrt {2 c d x +b d}-\frac {d^{2} \left (4 a c -b^{2}\right )}{\left (2 c d x +b d \right )^{\frac {3}{2}}}-\frac {3 d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}-\frac {d^{6} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}{11 \left (2 c d x +b d \right )^{\frac {11}{2}}}}{64 c^{4} d^{7}}\) | \(130\) |
default | \(\frac {\sqrt {2 c d x +b d}-\frac {d^{2} \left (4 a c -b^{2}\right )}{\left (2 c d x +b d \right )^{\frac {3}{2}}}-\frac {3 d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}-\frac {d^{6} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}{11 \left (2 c d x +b d \right )^{\frac {11}{2}}}}{64 c^{4} d^{7}}\) | \(130\) |
gosper | \(-\frac {\left (2 c x +b \right ) \left (-77 c^{6} x^{6}-231 b \,c^{5} x^{5}+77 a \,c^{5} x^{4}-308 b^{2} c^{4} x^{4}+154 a b \,c^{4} x^{3}-231 x^{3} b^{3} c^{3}+33 a^{2} c^{4} x^{2}+99 a \,b^{2} c^{3} x^{2}-99 x^{2} b^{4} c^{2}+33 a^{2} b \,c^{3} x +22 x a \,b^{3} c^{2}-22 x \,b^{5} c +7 c^{3} a^{3}+3 a^{2} b^{2} c^{2}+2 a \,b^{4} c -2 b^{6}\right )}{77 c^{4} \left (2 c d x +b d \right )^{\frac {13}{2}}}\) | \(174\) |
pseudoelliptic | \(\frac {77 c^{6} x^{6}+231 b \,c^{5} x^{5}-77 a \,c^{5} x^{4}+308 b^{2} c^{4} x^{4}-154 a b \,c^{4} x^{3}+231 x^{3} b^{3} c^{3}-33 a^{2} c^{4} x^{2}-99 a \,b^{2} c^{3} x^{2}+99 x^{2} b^{4} c^{2}-33 a^{2} b \,c^{3} x -22 x a \,b^{3} c^{2}+22 x \,b^{5} c -7 c^{3} a^{3}-3 a^{2} b^{2} c^{2}-2 a \,b^{4} c +2 b^{6}}{77 d^{6} \left (2 c x +b \right )^{5} \sqrt {d \left (2 c x +b \right )}\, c^{4}}\) | \(178\) |
trager | \(-\frac {\left (-77 c^{6} x^{6}-231 b \,c^{5} x^{5}+77 a \,c^{5} x^{4}-308 b^{2} c^{4} x^{4}+154 a b \,c^{4} x^{3}-231 x^{3} b^{3} c^{3}+33 a^{2} c^{4} x^{2}+99 a \,b^{2} c^{3} x^{2}-99 x^{2} b^{4} c^{2}+33 a^{2} b \,c^{3} x +22 x a \,b^{3} c^{2}-22 x \,b^{5} c +7 c^{3} a^{3}+3 a^{2} b^{2} c^{2}+2 a \,b^{4} c -2 b^{6}\right ) \sqrt {2 c d x +b d}}{77 d^{7} \left (2 c x +b \right )^{6} c^{4}}\) | \(179\) |
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Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (105) = 210\).
Time = 0.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.08 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {{\left (77 \, c^{6} x^{6} + 231 \, b c^{5} x^{5} + 2 \, b^{6} - 2 \, a b^{4} c - 3 \, a^{2} b^{2} c^{2} - 7 \, a^{3} c^{3} + 77 \, {\left (4 \, b^{2} c^{4} - a c^{5}\right )} x^{4} + 77 \, {\left (3 \, b^{3} c^{3} - 2 \, a b c^{4}\right )} x^{3} + 33 \, {\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - a^{2} c^{4}\right )} x^{2} + 11 \, {\left (2 \, b^{5} c - 2 \, a b^{3} c^{2} - 3 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{77 \, {\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1975 vs. \(2 (116) = 232\).
Time = 1.67 (sec) , antiderivative size = 1975, normalized size of antiderivative = 16.32 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\text {Too large to display} \]
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Time = 0.20 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\frac {77 \, \sqrt {2 \, c d x + b d}}{c^{3} d^{6}} + \frac {77 \, {\left (2 \, c d x + b d\right )}^{4} {\left (b^{2} - 4 \, a c\right )} - 33 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{2} d^{2} + 7 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{4}}{{\left (2 \, c d x + b d\right )}^{\frac {11}{2}} c^{3} d^{4}}}{4928 \, c d} \]
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Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\sqrt {2 \, c d x + b d}}{64 \, c^{4} d^{7}} + \frac {7 \, b^{6} d^{4} - 84 \, a b^{4} c d^{4} + 336 \, a^{2} b^{2} c^{2} d^{4} - 448 \, a^{3} c^{3} d^{4} - 33 \, {\left (2 \, c d x + b d\right )}^{2} b^{4} d^{2} + 264 \, {\left (2 \, c d x + b d\right )}^{2} a b^{2} c d^{2} - 528 \, {\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2} d^{2} + 77 \, {\left (2 \, c d x + b d\right )}^{4} b^{2} - 308 \, {\left (2 \, c d x + b d\right )}^{4} a c}{4928 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} c^{4} d^{5}} \]
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Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{13/2}} \, dx=\frac {\sqrt {b\,d+2\,c\,d\,x}}{64\,c^4\,d^7}-\frac {{\left (b\,d+2\,c\,d\,x\right )}^2\,\left (\frac {48\,a^2\,c^2\,d^2}{7}-\frac {24\,a\,b^2\,c\,d^2}{7}+\frac {3\,b^4\,d^2}{7}\right )+{\left (b\,d+2\,c\,d\,x\right )}^4\,\left (4\,a\,c-b^2\right )-\frac {b^6\,d^4}{11}+\frac {64\,a^3\,c^3\,d^4}{11}-\frac {48\,a^2\,b^2\,c^2\,d^4}{11}+\frac {12\,a\,b^4\,c\,d^4}{11}}{64\,c^4\,d^5\,{\left (b\,d+2\,c\,d\,x\right )}^{11/2}} \]
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