Integrand size = 17, antiderivative size = 71 \[ \int (a+b x)^2 \sqrt {c+d x} \, dx=\frac {2 (b c-a d)^2 (c+d x)^{3/2}}{3 d^3}-\frac {4 b (b c-a d) (c+d x)^{5/2}}{5 d^3}+\frac {2 b^2 (c+d x)^{7/2}}{7 d^3} \]
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Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {45} \[ \int (a+b x)^2 \sqrt {c+d x} \, dx=-\frac {4 b (c+d x)^{5/2} (b c-a d)}{5 d^3}+\frac {2 (c+d x)^{3/2} (b c-a d)^2}{3 d^3}+\frac {2 b^2 (c+d x)^{7/2}}{7 d^3} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^2 \sqrt {c+d x}}{d^2}-\frac {2 b (b c-a d) (c+d x)^{3/2}}{d^2}+\frac {b^2 (c+d x)^{5/2}}{d^2}\right ) \, dx \\ & = \frac {2 (b c-a d)^2 (c+d x)^{3/2}}{3 d^3}-\frac {4 b (b c-a d) (c+d x)^{5/2}}{5 d^3}+\frac {2 b^2 (c+d x)^{7/2}}{7 d^3} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86 \[ \int (a+b x)^2 \sqrt {c+d x} \, dx=\frac {2 (c+d x)^{3/2} \left (35 a^2 d^2+14 a b d (-2 c+3 d x)+b^2 \left (8 c^2-12 c d x+15 d^2 x^2\right )\right )}{105 d^3} \]
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Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {3}{7} b^{2} x^{2}+\frac {6}{5} a b x +a^{2}\right ) d^{2}-\frac {4 \left (\frac {3 b x}{7}+a \right ) b c d}{5}+\frac {8 b^{2} c^{2}}{35}\right ) \left (d x +c \right )^{\frac {3}{2}}}{3 d^{3}}\) | \(54\) |
derivativedivides | \(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{3}}\) | \(56\) |
default | \(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {4 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{3}}\) | \(56\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (15 d^{2} x^{2} b^{2}+42 x a b \,d^{2}-12 x \,b^{2} c d +35 a^{2} d^{2}-28 a b c d +8 b^{2} c^{2}\right )}{105 d^{3}}\) | \(63\) |
trager | \(\frac {2 \left (15 b^{2} d^{3} x^{3}+42 a b \,d^{3} x^{2}+3 b^{2} c \,d^{2} x^{2}+35 a^{2} d^{3} x +14 a b c \,d^{2} x -4 b^{2} c^{2} d x +35 a^{2} c \,d^{2}-28 a b \,c^{2} d +8 b^{2} c^{3}\right ) \sqrt {d x +c}}{105 d^{3}}\) | \(100\) |
risch | \(\frac {2 \left (15 b^{2} d^{3} x^{3}+42 a b \,d^{3} x^{2}+3 b^{2} c \,d^{2} x^{2}+35 a^{2} d^{3} x +14 a b c \,d^{2} x -4 b^{2} c^{2} d x +35 a^{2} c \,d^{2}-28 a b \,c^{2} d +8 b^{2} c^{3}\right ) \sqrt {d x +c}}{105 d^{3}}\) | \(100\) |
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Time = 0.23 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.39 \[ \int (a+b x)^2 \sqrt {c+d x} \, dx=\frac {2 \, {\left (15 \, b^{2} d^{3} x^{3} + 8 \, b^{2} c^{3} - 28 \, a b c^{2} d + 35 \, a^{2} c d^{2} + 3 \, {\left (b^{2} c d^{2} + 14 \, a b d^{3}\right )} x^{2} - {\left (4 \, b^{2} c^{2} d - 14 \, a b c d^{2} - 35 \, a^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{105 \, d^{3}} \]
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Time = 0.73 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.48 \[ \int (a+b x)^2 \sqrt {c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{2} \left (c + d x\right )^{\frac {7}{2}}}{7 d^{2}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (2 a b d - 2 b^{2} c\right )}{5 d^{2}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 d^{2}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\begin {cases} a^{2} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{3}}{3 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int (a+b x)^2 \sqrt {c+d x} \, dx=\frac {2 \, {\left (15 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{2} - 42 \, {\left (b^{2} c - a b d\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 35 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{105 \, d^{3}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (59) = 118\).
Time = 0.29 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.82 \[ \int (a+b x)^2 \sqrt {c+d x} \, dx=\frac {2 \, {\left (105 \, \sqrt {d x + c} a^{2} c + 35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2} + \frac {70 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a b c}{d} + \frac {7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} b^{2} c}{d^{2}} + \frac {14 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a b}{d} + \frac {3 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b^{2}}{d^{2}}\right )}}{105 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.96 \[ \int (a+b x)^2 \sqrt {c+d x} \, dx=\frac {2\,{\left (c+d\,x\right )}^{3/2}\,\left (15\,b^2\,{\left (c+d\,x\right )}^2+35\,a^2\,d^2+35\,b^2\,c^2-42\,b^2\,c\,\left (c+d\,x\right )+42\,a\,b\,d\,\left (c+d\,x\right )-70\,a\,b\,c\,d\right )}{105\,d^3} \]
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