Integrand size = 26, antiderivative size = 59 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {4}{63} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^{7/2}+\frac {2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2} \]
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Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {706, 643} \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {4}{63} d^3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{7/2}+\frac {2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2} \]
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Rule 643
Rule 706
Rubi steps \begin{align*} \text {integral}& = \frac {2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2}+\frac {1}{9} \left (2 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^{5/2} \, dx \\ & = \frac {4}{63} \left (b^2-4 a c\right ) d^3 \left (a+b x+c x^2\right )^{7/2}+\frac {2}{9} d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^{7/2} \\ \end{align*}
Time = 2.05 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.75 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {2}{63} d^3 (a+x (b+c x))^{7/2} \left (-9 b^2-28 b c x+4 c \left (2 a-7 c x^2\right )\right ) \]
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Time = 2.60 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69
method | result | size |
gosper | \(-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (-28 c^{2} x^{2}-28 b c x +8 a c -9 b^{2}\right ) d^{3}}{63}\) | \(41\) |
pseudoelliptic | \(-\frac {2 \left (c \,x^{2}+b x +a \right )^{\frac {7}{2}} \left (-28 c^{2} x^{2}-28 b c x +8 a c -9 b^{2}\right ) d^{3}}{63}\) | \(41\) |
trager | \(d^{3} \left (\frac {8}{9} c^{5} x^{8}+\frac {32}{9} b \,c^{4} x^{7}+\frac {152}{63} a \,c^{4} x^{6}+\frac {118}{21} b^{2} c^{3} x^{6}+\frac {152}{21} a b \,c^{3} x^{5}+\frac {278}{63} c^{2} x^{5} b^{3}+\frac {40}{21} a^{2} c^{3} x^{4}+\frac {170}{21} a \,b^{2} c^{2} x^{4}+\frac {110}{63} b^{4} c \,x^{4}+\frac {80}{21} a^{2} b \,c^{2} x^{3}+\frac {260}{63} a \,b^{3} c \,x^{3}+\frac {2}{7} x^{3} b^{5}+\frac {8}{63} a^{3} c^{2} x^{2}+\frac {58}{21} c \,x^{2} a^{2} b^{2}+\frac {6}{7} b^{4} x^{2} a +\frac {8}{63} a^{3} b c x +\frac {6}{7} a^{2} b^{3} x -\frac {16}{63} a^{4} c +\frac {2}{7} a^{3} b^{2}\right ) \sqrt {c \,x^{2}+b x +a}\) | \(201\) |
risch | \(-\frac {2 d^{3} \left (-28 c^{5} x^{8}-112 b \,c^{4} x^{7}-76 a \,c^{4} x^{6}-177 b^{2} c^{3} x^{6}-228 a b \,c^{3} x^{5}-139 c^{2} x^{5} b^{3}-60 a^{2} c^{3} x^{4}-255 a \,b^{2} c^{2} x^{4}-55 b^{4} c \,x^{4}-120 a^{2} b \,c^{2} x^{3}-130 a \,b^{3} c \,x^{3}-9 x^{3} b^{5}-4 a^{3} c^{2} x^{2}-87 c \,x^{2} a^{2} b^{2}-27 b^{4} x^{2} a -4 a^{3} b c x -27 a^{2} b^{3} x +8 a^{4} c -9 a^{3} b^{2}\right ) \sqrt {c \,x^{2}+b x +a}}{63}\) | \(202\) |
default | \(\text {Expression too large to display}\) | \(1212\) |
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Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (51) = 102\).
Time = 0.37 (sec) , antiderivative size = 215, normalized size of antiderivative = 3.64 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{63} \, {\left (28 \, c^{5} d^{3} x^{8} + 112 \, b c^{4} d^{3} x^{7} + {\left (177 \, b^{2} c^{3} + 76 \, a c^{4}\right )} d^{3} x^{6} + {\left (139 \, b^{3} c^{2} + 228 \, a b c^{3}\right )} d^{3} x^{5} + 5 \, {\left (11 \, b^{4} c + 51 \, a b^{2} c^{2} + 12 \, a^{2} c^{3}\right )} d^{3} x^{4} + {\left (9 \, b^{5} + 130 \, a b^{3} c + 120 \, a^{2} b c^{2}\right )} d^{3} x^{3} + {\left (27 \, a b^{4} + 87 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d^{3} x^{2} + {\left (27 \, a^{2} b^{3} + 4 \, a^{3} b c\right )} d^{3} x + {\left (9 \, a^{3} b^{2} - 8 \, a^{4} c\right )} d^{3}\right )} \sqrt {c x^{2} + b x + a} \]
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Leaf count of result is larger than twice the leaf count of optimal. 559 vs. \(2 (58) = 116\).
Time = 0.32 (sec) , antiderivative size = 559, normalized size of antiderivative = 9.47 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=- \frac {16 a^{4} c d^{3} \sqrt {a + b x + c x^{2}}}{63} + \frac {2 a^{3} b^{2} d^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {8 a^{3} b c d^{3} x \sqrt {a + b x + c x^{2}}}{63} + \frac {8 a^{3} c^{2} d^{3} x^{2} \sqrt {a + b x + c x^{2}}}{63} + \frac {6 a^{2} b^{3} d^{3} x \sqrt {a + b x + c x^{2}}}{7} + \frac {58 a^{2} b^{2} c d^{3} x^{2} \sqrt {a + b x + c x^{2}}}{21} + \frac {80 a^{2} b c^{2} d^{3} x^{3} \sqrt {a + b x + c x^{2}}}{21} + \frac {40 a^{2} c^{3} d^{3} x^{4} \sqrt {a + b x + c x^{2}}}{21} + \frac {6 a b^{4} d^{3} x^{2} \sqrt {a + b x + c x^{2}}}{7} + \frac {260 a b^{3} c d^{3} x^{3} \sqrt {a + b x + c x^{2}}}{63} + \frac {170 a b^{2} c^{2} d^{3} x^{4} \sqrt {a + b x + c x^{2}}}{21} + \frac {152 a b c^{3} d^{3} x^{5} \sqrt {a + b x + c x^{2}}}{21} + \frac {152 a c^{4} d^{3} x^{6} \sqrt {a + b x + c x^{2}}}{63} + \frac {2 b^{5} d^{3} x^{3} \sqrt {a + b x + c x^{2}}}{7} + \frac {110 b^{4} c d^{3} x^{4} \sqrt {a + b x + c x^{2}}}{63} + \frac {278 b^{3} c^{2} d^{3} x^{5} \sqrt {a + b x + c x^{2}}}{63} + \frac {118 b^{2} c^{3} d^{3} x^{6} \sqrt {a + b x + c x^{2}}}{21} + \frac {32 b c^{4} d^{3} x^{7} \sqrt {a + b x + c x^{2}}}{9} + \frac {8 c^{5} d^{3} x^{8} \sqrt {a + b x + c x^{2}}}{9} \]
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Exception generated. \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} b^{2} d^{3} + \frac {8}{9} \, {\left (c x^{2} + b x + a\right )}^{\frac {9}{2}} c d^{3} - \frac {8}{7} \, {\left (c x^{2} + b x + a\right )}^{\frac {7}{2}} a c d^{3} \]
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Time = 10.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {8\,c\,d^3\,{\left (c\,x^2+b\,x+a\right )}^{9/2}}{9}+\frac {2\,b^2\,d^3\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7}-\frac {8\,a\,c\,d^3\,{\left (c\,x^2+b\,x+a\right )}^{7/2}}{7} \]
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