Integrand size = 26, antiderivative size = 121 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx=\frac {\left (b^2-4 a c\right )^3}{320 c^4 d (b d+2 c d x)^{5/2}}-\frac {3 \left (b^2-4 a c\right )^2}{64 c^4 d^3 \sqrt {b d+2 c d x}}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{64 c^4 d^5}+\frac {(b d+2 c d x)^{7/2}}{448 c^4 d^7} \]
[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} \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx=-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{64 c^4 d^5}-\frac {3 \left (b^2-4 a c\right )^2}{64 c^4 d^3 \sqrt {b d+2 c d x}}+\frac {\left (b^2-4 a c\right )^3}{320 c^4 d (b d+2 c d x)^{5/2}}+\frac {(b d+2 c d x)^{7/2}}{448 c^4 d^7} \]
[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)^{7/2}}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^2 (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^3 d^4}+\frac {(b d+2 c d x)^{5/2}}{64 c^3 d^6}\right ) \, dx \\ & = \frac {\left (b^2-4 a c\right )^3}{320 c^4 d (b d+2 c d x)^{5/2}}-\frac {3 \left (b^2-4 a c\right )^2}{64 c^4 d^3 \sqrt {b d+2 c d x}}-\frac {\left (b^2-4 a c\right ) (b d+2 c d x)^{3/2}}{64 c^4 d^5}+\frac {(b d+2 c d x)^{7/2}}{448 c^4 d^7} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx=\frac {7 b^6-84 a b^4 c+336 a^2 b^2 c^2-448 a^3 c^3-105 b^4 (b+2 c x)^2+840 a b^2 c (b+2 c x)^2-1680 a^2 c^2 (b+2 c x)^2-35 b^2 (b+2 c x)^4+140 a c (b+2 c x)^4+5 (b+2 c x)^6}{2240 c^4 d (d (b+2 c x))^{5/2}} \]
[In]
[Out]
Time = 2.51 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.18
method | result | size |
derivativedivides | \(\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}-b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}-\frac {3 d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{\sqrt {2 c d x +b d}}-\frac {d^{6} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}{5 \left (2 c d x +b d \right )^{\frac {5}{2}}}}{64 c^{4} d^{7}}\) | \(143\) |
default | \(\frac {4 a c \,d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}-b^{2} d^{2} \left (2 c d x +b d \right )^{\frac {3}{2}}+\frac {\left (2 c d x +b d \right )^{\frac {7}{2}}}{7}-\frac {3 d^{4} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}{\sqrt {2 c d x +b d}}-\frac {d^{6} \left (64 c^{3} a^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}\right )}{5 \left (2 c d x +b d \right )^{\frac {5}{2}}}}{64 c^{4} d^{7}}\) | \(143\) |
pseudoelliptic | \(\frac {5 c^{6} x^{6}+\left (15 b \,x^{5}+35 a \,x^{4}\right ) c^{5}+\left (10 b^{2} x^{4}+70 a b \,x^{3}-105 a^{2} x^{2}\right ) c^{4}+\left (-5 b^{3} x^{3}+105 a \,b^{2} x^{2}-105 a^{2} b x -7 a^{3}\right ) c^{3}+\left (-15 b^{4} x^{2}+70 a \,b^{3} x -21 a^{2} b^{2}\right ) c^{2}+\left (-10 b^{5} x +14 a \,b^{4}\right ) c -2 b^{6}}{35 \sqrt {d \left (2 c x +b \right )}\, d^{3} \left (2 c x +b \right )^{2} c^{4}}\) | \(163\) |
gosper | \(-\frac {\left (2 c x +b \right ) \left (-5 c^{6} x^{6}-15 b \,c^{5} x^{5}-35 a \,c^{5} x^{4}-10 b^{2} c^{4} x^{4}-70 a b \,c^{4} x^{3}+5 x^{3} b^{3} c^{3}+105 a^{2} c^{4} x^{2}-105 a \,b^{2} c^{3} x^{2}+15 x^{2} b^{4} c^{2}+105 a^{2} b \,c^{3} x -70 x a \,b^{3} c^{2}+10 x \,b^{5} c +7 c^{3} a^{3}+21 a^{2} b^{2} c^{2}-14 a \,b^{4} c +2 b^{6}\right )}{35 c^{4} \left (2 c d x +b d \right )^{\frac {7}{2}}}\) | \(174\) |
trager | \(-\frac {\left (-5 c^{6} x^{6}-15 b \,c^{5} x^{5}-35 a \,c^{5} x^{4}-10 b^{2} c^{4} x^{4}-70 a b \,c^{4} x^{3}+5 x^{3} b^{3} c^{3}+105 a^{2} c^{4} x^{2}-105 a \,b^{2} c^{3} x^{2}+15 x^{2} b^{4} c^{2}+105 a^{2} b \,c^{3} x -70 x a \,b^{3} c^{2}+10 x \,b^{5} c +7 c^{3} a^{3}+21 a^{2} b^{2} c^{2}-14 a \,b^{4} c +2 b^{6}\right ) \sqrt {2 c d x +b d}}{35 d^{4} c^{4} \left (2 c x +b \right )^{3}}\) | \(179\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx=\frac {{\left (5 \, c^{6} x^{6} + 15 \, b c^{5} x^{5} - 2 \, b^{6} + 14 \, a b^{4} c - 21 \, a^{2} b^{2} c^{2} - 7 \, a^{3} c^{3} + 5 \, {\left (2 \, b^{2} c^{4} + 7 \, a c^{5}\right )} x^{4} - 5 \, {\left (b^{3} c^{3} - 14 \, a b c^{4}\right )} x^{3} - 15 \, {\left (b^{4} c^{2} - 7 \, a b^{2} c^{3} + 7 \, a^{2} c^{4}\right )} x^{2} - 5 \, {\left (2 \, b^{5} c - 14 \, a b^{3} c^{2} + 21 \, a^{2} b c^{3}\right )} x\right )} \sqrt {2 \, c d x + b d}}{35 \, {\left (8 \, c^{7} d^{4} x^{3} + 12 \, b c^{6} d^{4} x^{2} + 6 \, b^{2} c^{5} d^{4} x + b^{3} c^{4} d^{4}\right )}} \]
[In]
[Out]
Time = 2.11 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.79 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx=\begin {cases} \frac {- \frac {\left (4 a c - b^{2}\right )^{3}}{320 c^{3} \left (b d + 2 c d x\right )^{\frac {5}{2}}} - \frac {3 \left (4 a c - b^{2}\right )^{2}}{64 c^{3} d^{2} \sqrt {b d + 2 c d x}} + \frac {\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {3}{2}}}{192 c^{3} d^{4}} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}}}{448 c^{3} d^{6}}}{c d} & \text {for}\: c d \neq 0 \\\frac {a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}}{\left (b d\right )^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.17 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx=-\frac {\frac {7 \, {\left (15 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{2} - {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{3} d^{2}} + \frac {5 \, {\left (7 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - {\left (2 \, c d x + b d\right )}^{\frac {7}{2}}\right )}}{c^{3} d^{6}}}{2240 \, c d} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx=\frac {b^{6} d^{2} - 12 \, a b^{4} c d^{2} + 48 \, a^{2} b^{2} c^{2} d^{2} - 64 \, a^{3} c^{3} d^{2} - 15 \, {\left (2 \, c d x + b d\right )}^{2} b^{4} + 120 \, {\left (2 \, c d x + b d\right )}^{2} a b^{2} c - 240 \, {\left (2 \, c d x + b d\right )}^{2} a^{2} c^{2}}{320 \, {\left (2 \, c d x + b d\right )}^{\frac {5}{2}} c^{4} d^{3}} - \frac {7 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} b^{2} c^{24} d^{44} - 28 \, {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} a c^{25} d^{44} - {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} c^{24} d^{42}}{448 \, c^{28} d^{49}} \]
[In]
[Out]
Time = 9.22 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^{7/2}} \, dx=-\frac {7\,a^3\,c^3+21\,a^2\,b^2\,c^2+105\,a^2\,b\,c^3\,x+105\,a^2\,c^4\,x^2-14\,a\,b^4\,c-70\,a\,b^3\,c^2\,x-105\,a\,b^2\,c^3\,x^2-70\,a\,b\,c^4\,x^3-35\,a\,c^5\,x^4+2\,b^6+10\,b^5\,c\,x+15\,b^4\,c^2\,x^2+5\,b^3\,c^3\,x^3-10\,b^2\,c^4\,x^4-15\,b\,c^5\,x^5-5\,c^6\,x^6}{35\,c^4\,d\,{\left (b\,d+2\,c\,d\,x\right )}^{5/2}} \]
[In]
[Out]