Integrand size = 26, antiderivative size = 121 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx=\frac {\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}-\frac {\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}-\frac {3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{64 c^4 d^5}+\frac {(b d+2 c d x)^{5/2}}{320 c^4 d^7} \]
[Out]
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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx=-\frac {3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{64 c^4 d^5}-\frac {\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}+\frac {\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}+\frac {(b d+2 c d x)^{5/2}}{320 c^4 d^7} \]
[In]
[Out]
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)^{9/2}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (b d+2 c d x)^{5/2}}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^4 \sqrt {b d+2 c d x}}+\frac {(b d+2 c d x)^{3/2}}{64 c^3 d^6}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^3}{448 c^4 d (b d+2 c d x)^{7/2}}-\frac {\left (b^2-4 a c\right )^2}{64 c^4 d^3 (b d+2 c d x)^{3/2}}-\frac {3 \left (b^2-4 a c\right ) \sqrt {b d+2 c d x}}{64 c^4 d^5}+\frac {(b d+2 c d x)^{5/2}}{320 c^4 d^7} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx=\frac {5 b^6-60 a b^4 c+240 a^2 b^2 c^2-320 a^3 c^3-35 b^4 (b+2 c x)^2+280 a b^2 c (b+2 c x)^2-560 a^2 c^2 (b+2 c x)^2-105 b^2 (b+2 c x)^4+420 a c (b+2 c x)^4+7 (b+2 c x)^6}{2240 c^4 d (d (b+2 c x))^{7/2}} \]
[In]
[Out]
Time = 2.86 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.14
method | result | size |
pseudoelliptic | \(-\frac {-\frac {7 c^{6} x^{6}}{5}-21 x^{4} \left (\frac {b x}{5}+a \right ) c^{5}+7 a \,x^{2} \left (-6 b x +a \right ) c^{4}+\left (7 b^{3} x^{3}-35 a \,b^{2} x^{2}+7 a^{2} b x +a^{3}\right ) c^{3}+b^{2} \left (7 b^{2} x^{2}-14 a b x +a^{2}\right ) c^{2}-2 b^{4} \left (-\frac {7 b x}{5}+a \right ) c +\frac {2 b^{6}}{5}}{7 \sqrt {d \left (2 c x +b \right )}\, d^{4} \left (2 c x +b \right )^{3} c^{4}}\) | \(138\) |
derivativedivides | \(\frac {12 a c \,d^{2} \sqrt {2 c d x +b d}-3 b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}-\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{\left (2 c d x +b d \right )^{\frac {3}{2}}}-\frac {d^{6} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}}{64 c^{4} d^{7}}\) | \(143\) |
default | \(\frac {12 a c \,d^{2} \sqrt {2 c d x +b d}-3 b^{2} d^{2} \sqrt {2 c d x +b d}+\frac {\left (2 c d x +b d \right )^{\frac {5}{2}}}{5}-\frac {d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{\left (2 c d x +b d \right )^{\frac {3}{2}}}-\frac {d^{6} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}{7 \left (2 c d x +b d \right )^{\frac {7}{2}}}}{64 c^{4} d^{7}}\) | \(143\) |
gosper | \(-\frac {\left (2 c x +b \right ) \left (-7 c^{6} x^{6}-21 b \,c^{5} x^{5}-105 a \,c^{5} x^{4}-210 a b \,c^{4} x^{3}+35 x^{3} b^{3} c^{3}+35 a^{2} c^{4} x^{2}-175 a \,b^{2} c^{3} x^{2}+35 x^{2} b^{4} c^{2}+35 a^{2} b \,c^{3} x -70 x a \,b^{3} c^{2}+14 x \,b^{5} c +5 c^{3} a^{3}+5 a^{2} b^{2} c^{2}-10 a \,b^{4} c +2 b^{6}\right )}{35 c^{4} \left (2 c d x +b d \right )^{\frac {9}{2}}}\) | \(163\) |
trager | \(-\frac {\left (-7 c^{6} x^{6}-21 b \,c^{5} x^{5}-105 a \,c^{5} x^{4}-210 a b \,c^{4} x^{3}+35 x^{3} b^{3} c^{3}+35 a^{2} c^{4} x^{2}-175 a \,b^{2} c^{3} x^{2}+35 x^{2} b^{4} c^{2}+35 a^{2} b \,c^{3} x -70 x a \,b^{3} c^{2}+14 x \,b^{5} c +5 c^{3} a^{3}+5 a^{2} b^{2} c^{2}-10 a \,b^{4} c +2 b^{6}\right ) \sqrt {2 c d x +b d}}{35 d^{5} c^{4} \left (2 c x +b \right )^{4}}\) | \(168\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.74 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx=\frac {{\left (7 \, c^{6} x^{6} + 21 \, b c^{5} x^{5} + 105 \, a c^{5} x^{4} - 2 \, b^{6} + 10 \, a b^{4} c - 5 \, a^{2} b^{2} c^{2} - 5 \, a^{3} c^{3} - 35 \, {\left (b^{3} c^{3} - 6 \, a b c^{4}\right )} x^{3} - 35 \, {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + a^{2} c^{4}\right )} x^{2} - 7 \, {\left (2 \, b^{5} c - 10 \, a b^{3} c^{2} + 5 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{35 \, {\left (16 \, c^{8} d^{5} x^{4} + 32 \, b c^{7} d^{5} x^{3} + 24 \, b^{2} c^{6} d^{5} x^{2} + 8 \, b^{3} c^{5} d^{5} x + b^{4} c^{4} d^{5}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 1394 vs. \(2 (117) = 234\).
Time = 0.84 (sec) , antiderivative size = 1394, normalized size of antiderivative = 11.52 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx=\begin {cases} - \frac {5 a^{3} c^{3} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {5 a^{2} b^{2} c^{2} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {35 a^{2} b c^{3} x \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {35 a^{2} c^{4} x^{2} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {10 a b^{4} c \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {70 a b^{3} c^{2} x \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {175 a b^{2} c^{3} x^{2} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {210 a b c^{4} x^{3} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {105 a c^{5} x^{4} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {2 b^{6} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {14 b^{5} c x \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {35 b^{4} c^{2} x^{2} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} - \frac {35 b^{3} c^{3} x^{3} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {21 b c^{5} x^{5} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} + \frac {7 c^{6} x^{6} \sqrt {b d + 2 c d x}}{35 b^{4} c^{4} d^{5} + 280 b^{3} c^{5} d^{5} x + 840 b^{2} c^{6} d^{5} x^{2} + 1120 b c^{7} d^{5} x^{3} + 560 c^{8} d^{5} x^{4}} & \text {for}\: c \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}}{\left (b d\right )^{\frac {9}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx=-\frac {\frac {5 \, {\left (7 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{2} - {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{3} d^{2}} + \frac {7 \, {\left (15 \, \sqrt {2 \, c d x + b d} {\left (b^{2} - 4 \, a c\right )} d^{2} - {\left (2 \, c d x + b d\right )}^{\frac {5}{2}}\right )}}{c^{3} d^{6}}}{2240 \, c d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx=\frac {b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2} - 7 \, {\left (2 \, c d x + b d\right )}^{2} b^{4} + 56 \, {\left (2 \, c d x + b d\right )}^{2} a b^{2} c - 112 \, {\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2}}{448 \, {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{4} d^{3}} - \frac {15 \, \sqrt {2 \, c d x + b d} b^{2} c^{16} d^{30} - 60 \, \sqrt {2 \, c d x + b d} a c^{17} d^{30} - {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{16} d^{28}}{320 \, c^{20} d^{35}} \]
[In]
[Out]
Time = 9.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.31 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{9/2}} \, dx=-\frac {5\,a^3\,c^3+5\,a^2\,b^2\,c^2+35\,a^2\,b\,c^3\,x+35\,a^2\,c^4\,x^2-10\,a\,b^4\,c-70\,a\,b^3\,c^2\,x-175\,a\,b^2\,c^3\,x^2-210\,a\,b\,c^4\,x^3-105\,a\,c^5\,x^4+2\,b^6+14\,b^5\,c\,x+35\,b^4\,c^2\,x^2+35\,b^3\,c^3\,x^3-21\,b\,c^5\,x^5-7\,c^6\,x^6}{35\,c^4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}} \]
[In]
[Out]