Integrand size = 26, antiderivative size = 121 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx=-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac {(b d+2 c d x)^{15/2}}{960 c^4 d^7} \]
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Time = 0.04 (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} \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx=-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac {(b d+2 c d x)^{15/2}}{960 c^4 d^7} \]
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Rule 697
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (-b^2+4 a c\right )^3 \sqrt {b d+2 c d x}}{64 c^3}+\frac {3 \left (-b^2+4 a c\right )^2 (b d+2 c d x)^{5/2}}{64 c^3 d^2}+\frac {3 \left (-b^2+4 a c\right ) (b d+2 c d x)^{9/2}}{64 c^3 d^4}+\frac {(b d+2 c d x)^{13/2}}{64 c^3 d^6}\right ) \, dx \\ & = -\frac {\left (b^2-4 a c\right )^3 (b d+2 c d x)^{3/2}}{192 c^4 d}+\frac {3 \left (b^2-4 a c\right )^2 (b d+2 c d x)^{7/2}}{448 c^4 d^3}-\frac {3 \left (b^2-4 a c\right ) (b d+2 c d x)^{11/2}}{704 c^4 d^5}+\frac {(b d+2 c d x)^{15/2}}{960 c^4 d^7} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx=\frac {(d (b+2 c x))^{3/2} \left (-385 b^6+4620 a b^4 c-18480 a^2 b^2 c^2+24640 a^3 c^3+495 b^4 (b+2 c x)^2-3960 a b^2 c (b+2 c x)^2+7920 a^2 c^2 (b+2 c x)^2-315 b^2 (b+2 c x)^4+1260 a c (b+2 c x)^4+77 (b+2 c x)^6\right )}{73920 c^4 d} \]
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Time = 2.66 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.22
method | result | size |
derivativedivides | \(\frac {\frac {\left (2 c d x +b d \right )^{\frac {15}{2}}}{15}+\frac {\left (12 a c \,d^{2}-3 b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {11}{2}}}{11}+\frac {\left (\left (4 a c \,d^{2}-b^{2} d^{2}\right ) \left (8 a c \,d^{2}-2 b^{2} d^{2}\right )+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{2}\right ) \left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}}{64 d^{7} c^{4}}\) | \(148\) |
default | \(\frac {\frac {\left (2 c d x +b d \right )^{\frac {15}{2}}}{15}+\frac {\left (12 a c \,d^{2}-3 b^{2} d^{2}\right ) \left (2 c d x +b d \right )^{\frac {11}{2}}}{11}+\frac {\left (\left (4 a c \,d^{2}-b^{2} d^{2}\right ) \left (8 a c \,d^{2}-2 b^{2} d^{2}\right )+\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{2}\right ) \left (2 c d x +b d \right )^{\frac {7}{2}}}{7}+\frac {\left (4 a c \,d^{2}-b^{2} d^{2}\right )^{3} \left (2 c d x +b d \right )^{\frac {3}{2}}}{3}}{64 d^{7} c^{4}}\) | \(148\) |
pseudoelliptic | \(\frac {\left (2 c x +b \right ) \sqrt {d \left (2 c x +b \right )}\, \left (77 c^{6} x^{6}+231 b \,c^{5} x^{5}+315 a \,c^{5} x^{4}+210 b^{2} c^{4} x^{4}+630 a b \,c^{4} x^{3}+35 x^{3} b^{3} c^{3}+495 a^{2} c^{4} x^{2}+225 a \,b^{2} c^{3} x^{2}-15 x^{2} b^{4} c^{2}+495 a^{2} b \,c^{3} x -90 x a \,b^{3} c^{2}+6 x \,b^{5} c +385 c^{3} a^{3}-165 a^{2} b^{2} c^{2}+30 a \,b^{4} c -2 b^{6}\right )}{1155 c^{4}}\) | \(173\) |
gosper | \(\frac {\left (2 c x +b \right ) \left (77 c^{6} x^{6}+231 b \,c^{5} x^{5}+315 a \,c^{5} x^{4}+210 b^{2} c^{4} x^{4}+630 a b \,c^{4} x^{3}+35 x^{3} b^{3} c^{3}+495 a^{2} c^{4} x^{2}+225 a \,b^{2} c^{3} x^{2}-15 x^{2} b^{4} c^{2}+495 a^{2} b \,c^{3} x -90 x a \,b^{3} c^{2}+6 x \,b^{5} c +385 c^{3} a^{3}-165 a^{2} b^{2} c^{2}+30 a \,b^{4} c -2 b^{6}\right ) \sqrt {2 c d x +b d}}{1155 c^{4}}\) | \(174\) |
trager | \(\frac {\left (154 c^{7} x^{7}+539 b \,c^{6} x^{6}+630 a \,c^{6} x^{5}+651 b^{2} c^{5} x^{5}+1575 a b \,c^{5} x^{4}+280 b^{3} c^{4} x^{4}+990 a^{2} c^{5} x^{3}+1080 a \,b^{2} c^{4} x^{3}+5 b^{4} c^{3} x^{3}+1485 a^{2} b \,c^{4} x^{2}+45 a \,b^{3} c^{3} x^{2}-3 b^{5} c^{2} x^{2}+770 a^{3} c^{4} x +165 a^{2} b^{2} c^{3} x -30 c^{2} a \,b^{4} x +2 b^{6} c x +385 a^{3} c^{3} b -165 a^{2} c^{2} b^{3}+30 a \,b^{5} c -2 b^{7}\right ) \sqrt {2 c d x +b d}}{1155 c^{4}}\) | \(215\) |
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Time = 0.63 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.69 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (154 \, c^{7} x^{7} + 539 \, b c^{6} x^{6} - 2 \, b^{7} + 30 \, a b^{5} c - 165 \, a^{2} b^{3} c^{2} + 385 \, a^{3} b c^{3} + 21 \, {\left (31 \, b^{2} c^{5} + 30 \, a c^{6}\right )} x^{5} + 35 \, {\left (8 \, b^{3} c^{4} + 45 \, a b c^{5}\right )} x^{4} + 5 \, {\left (b^{4} c^{3} + 216 \, a b^{2} c^{4} + 198 \, a^{2} c^{5}\right )} x^{3} - 3 \, {\left (b^{5} c^{2} - 15 \, a b^{3} c^{3} - 495 \, a^{2} b c^{4}\right )} x^{2} + {\left (2 \, b^{6} c - 30 \, a b^{4} c^{2} + 165 \, a^{2} b^{2} c^{3} + 770 \, a^{3} c^{4}\right )} x\right )} \sqrt {2 \, c d x + b d}}{1155 \, c^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (119) = 238\).
Time = 1.07 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.05 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx=\begin {cases} \frac {\frac {\left (b d + 2 c d x\right )^{\frac {3}{2}} \cdot \left (64 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} + 12 a b^{4} c - b^{6}\right )}{192 c^{3}} + \frac {\left (b d + 2 c d x\right )^{\frac {7}{2}} \cdot \left (48 a^{2} c^{2} - 24 a b^{2} c + 3 b^{4}\right )}{448 c^{3} d^{2}} + \frac {\left (12 a c - 3 b^{2}\right ) \left (b d + 2 c d x\right )^{\frac {11}{2}}}{704 c^{3} d^{4}} + \frac {\left (b d + 2 c d x\right )^{\frac {15}{2}}}{960 c^{3} d^{6}}}{c d} & \text {for}\: c d \neq 0 \\\sqrt {b d} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + \frac {b c^{2} x^{6}}{2} + \frac {c^{3} x^{7}}{7} + \frac {x^{5} \cdot \left (3 a c^{2} + 3 b^{2} c\right )}{5} + \frac {x^{4} \cdot \left (6 a b c + b^{3}\right )}{4} + \frac {x^{3} \cdot \left (3 a^{2} c + 3 a b^{2}\right )}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.05 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx=-\frac {315 \, {\left (2 \, c d x + b d\right )}^{\frac {11}{2}} {\left (b^{2} - 4 \, a c\right )} d^{2} - 495 \, {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} {\left (2 \, c d x + b d\right )}^{\frac {7}{2}} d^{4} + 385 \, {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} {\left (2 \, c d x + b d\right )}^{\frac {3}{2}} d^{6} - 77 \, {\left (2 \, c d x + b d\right )}^{\frac {15}{2}}}{73920 \, c^{4} d^{7}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1165 vs. \(2 (105) = 210\).
Time = 0.28 (sec) , antiderivative size = 1165, normalized size of antiderivative = 9.63 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx=\text {Too large to display} \]
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Time = 9.18 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \sqrt {b d+2 c d x} \left (a+b x+c x^2\right )^3 \, dx=\frac {{\left (b\,d+2\,c\,d\,x\right )}^{15/2}}{960\,c^4\,d^7}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{11/2}\,\left (4\,a\,c-b^2\right )}{704\,c^4\,d^5}+\frac {{\left (b\,d+2\,c\,d\,x\right )}^{3/2}\,{\left (4\,a\,c-b^2\right )}^3}{192\,c^4\,d}+\frac {3\,{\left (b\,d+2\,c\,d\,x\right )}^{7/2}\,{\left (4\,a\,c-b^2\right )}^2}{448\,c^4\,d^3} \]
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