Integrand size = 17, antiderivative size = 129 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 (b c-a d)^4 (c+d x)^{5/2}}{5 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{7/2}}{7 d^5}+\frac {4 b^2 (b c-a d)^2 (c+d x)^{9/2}}{3 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{11/2}}{11 d^5}+\frac {2 b^4 (c+d x)^{13/2}}{13 d^5} \]
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Time = 0.03 (sec) , antiderivative size = 129, 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} \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=-\frac {8 b^3 (c+d x)^{11/2} (b c-a d)}{11 d^5}+\frac {4 b^2 (c+d x)^{9/2} (b c-a d)^2}{3 d^5}-\frac {8 b (c+d x)^{7/2} (b c-a d)^3}{7 d^5}+\frac {2 (c+d x)^{5/2} (b c-a d)^4}{5 d^5}+\frac {2 b^4 (c+d x)^{13/2}}{13 d^5} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^4 (c+d x)^{3/2}}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^{5/2}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{7/2}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{9/2}}{d^4}+\frac {b^4 (c+d x)^{11/2}}{d^4}\right ) \, dx \\ & = \frac {2 (b c-a d)^4 (c+d x)^{5/2}}{5 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{7/2}}{7 d^5}+\frac {4 b^2 (b c-a d)^2 (c+d x)^{9/2}}{3 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{11/2}}{11 d^5}+\frac {2 b^4 (c+d x)^{13/2}}{13 d^5} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 (c+d x)^{5/2} \left (3003 a^4 d^4+1716 a^3 b d^3 (-2 c+5 d x)+286 a^2 b^2 d^2 \left (8 c^2-20 c d x+35 d^2 x^2\right )+52 a b^3 d \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )+b^4 \left (128 c^4-320 c^3 d x+560 c^2 d^2 x^2-840 c d^3 x^3+1155 d^4 x^4\right )\right )}{15015 d^5} \]
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Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{5}}\) | \(100\) |
default | \(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {4 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{5}}\) | \(100\) |
pseudoelliptic | \(\frac {2 \left (\left (\frac {5}{13} b^{4} x^{4}+\frac {20}{11} a \,b^{3} x^{3}+\frac {10}{3} a^{2} b^{2} x^{2}+\frac {20}{7} a^{3} b x +a^{4}\right ) d^{4}-\frac {8 \left (\frac {35}{143} b^{3} x^{3}+\frac {35}{33} a \,b^{2} x^{2}+\frac {5}{3} a^{2} b x +a^{3}\right ) b c \,d^{3}}{7}+\frac {16 b^{2} \left (\frac {35}{143} b^{2} x^{2}+\frac {10}{11} a b x +a^{2}\right ) c^{2} d^{2}}{21}-\frac {64 b^{3} \left (\frac {5 b x}{13}+a \right ) c^{3} d}{231}+\frac {128 b^{4} c^{4}}{3003}\right ) \left (d x +c \right )^{\frac {5}{2}}}{5 d^{5}}\) | \(143\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (1155 d^{4} x^{4} b^{4}+5460 a \,b^{3} d^{4} x^{3}-840 b^{4} c \,d^{3} x^{3}+10010 a^{2} b^{2} d^{4} x^{2}-3640 a \,b^{3} c \,d^{3} x^{2}+560 b^{4} c^{2} d^{2} x^{2}+8580 a^{3} b \,d^{4} x -5720 a^{2} b^{2} c \,d^{3} x +2080 a \,b^{3} c^{2} d^{2} x -320 b^{4} c^{3} d x +3003 a^{4} d^{4}-3432 a^{3} b c \,d^{3}+2288 a^{2} b^{2} c^{2} d^{2}-832 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{15015 d^{5}}\) | \(186\) |
trager | \(\frac {2 \left (1155 b^{4} d^{6} x^{6}+5460 a \,b^{3} d^{6} x^{5}+1470 b^{4} c \,d^{5} x^{5}+10010 a^{2} b^{2} d^{6} x^{4}+7280 a \,b^{3} c \,d^{5} x^{4}+35 b^{4} c^{2} d^{4} x^{4}+8580 a^{3} b \,d^{6} x^{3}+14300 a^{2} b^{2} c \,d^{5} x^{3}+260 a \,b^{3} c^{2} d^{4} x^{3}-40 b^{4} c^{3} d^{3} x^{3}+3003 a^{4} d^{6} x^{2}+13728 a^{3} b c \,d^{5} x^{2}+858 a^{2} b^{2} c^{2} d^{4} x^{2}-312 a \,b^{3} c^{3} d^{3} x^{2}+48 b^{4} c^{4} d^{2} x^{2}+6006 a^{4} c \,d^{5} x +1716 a^{3} b \,c^{2} d^{4} x -1144 a^{2} b^{2} c^{3} d^{3} x +416 a \,b^{3} c^{4} d^{2} x -64 b^{4} c^{5} d x +3003 a^{4} c^{2} d^{4}-3432 a^{3} b \,c^{3} d^{3}+2288 a^{2} b^{2} c^{4} d^{2}-832 a \,b^{3} c^{5} d +128 b^{4} c^{6}\right ) \sqrt {d x +c}}{15015 d^{5}}\) | \(332\) |
risch | \(\frac {2 \left (1155 b^{4} d^{6} x^{6}+5460 a \,b^{3} d^{6} x^{5}+1470 b^{4} c \,d^{5} x^{5}+10010 a^{2} b^{2} d^{6} x^{4}+7280 a \,b^{3} c \,d^{5} x^{4}+35 b^{4} c^{2} d^{4} x^{4}+8580 a^{3} b \,d^{6} x^{3}+14300 a^{2} b^{2} c \,d^{5} x^{3}+260 a \,b^{3} c^{2} d^{4} x^{3}-40 b^{4} c^{3} d^{3} x^{3}+3003 a^{4} d^{6} x^{2}+13728 a^{3} b c \,d^{5} x^{2}+858 a^{2} b^{2} c^{2} d^{4} x^{2}-312 a \,b^{3} c^{3} d^{3} x^{2}+48 b^{4} c^{4} d^{2} x^{2}+6006 a^{4} c \,d^{5} x +1716 a^{3} b \,c^{2} d^{4} x -1144 a^{2} b^{2} c^{3} d^{3} x +416 a \,b^{3} c^{4} d^{2} x -64 b^{4} c^{5} d x +3003 a^{4} c^{2} d^{4}-3432 a^{3} b \,c^{3} d^{3}+2288 a^{2} b^{2} c^{4} d^{2}-832 a \,b^{3} c^{5} d +128 b^{4} c^{6}\right ) \sqrt {d x +c}}{15015 d^{5}}\) | \(332\) |
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Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (109) = 218\).
Time = 0.22 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.41 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 \, {\left (1155 \, b^{4} d^{6} x^{6} + 128 \, b^{4} c^{6} - 832 \, a b^{3} c^{5} d + 2288 \, a^{2} b^{2} c^{4} d^{2} - 3432 \, a^{3} b c^{3} d^{3} + 3003 \, a^{4} c^{2} d^{4} + 210 \, {\left (7 \, b^{4} c d^{5} + 26 \, a b^{3} d^{6}\right )} x^{5} + 35 \, {\left (b^{4} c^{2} d^{4} + 208 \, a b^{3} c d^{5} + 286 \, a^{2} b^{2} d^{6}\right )} x^{4} - 20 \, {\left (2 \, b^{4} c^{3} d^{3} - 13 \, a b^{3} c^{2} d^{4} - 715 \, a^{2} b^{2} c d^{5} - 429 \, a^{3} b d^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{4} c^{4} d^{2} - 104 \, a b^{3} c^{3} d^{3} + 286 \, a^{2} b^{2} c^{2} d^{4} + 4576 \, a^{3} b c d^{5} + 1001 \, a^{4} d^{6}\right )} x^{2} - 2 \, {\left (32 \, b^{4} c^{5} d - 208 \, a b^{3} c^{4} d^{2} + 572 \, a^{2} b^{2} c^{3} d^{3} - 858 \, a^{3} b c^{2} d^{4} - 3003 \, a^{4} c d^{5}\right )} x\right )} \sqrt {d x + c}}{15015 \, d^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (119) = 238\).
Time = 0.93 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.88 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (c + d x\right )^{\frac {13}{2}}}{13 d^{4}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \cdot \left (4 a b^{3} d - 4 b^{4} c\right )}{11 d^{4}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (6 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 6 b^{4} c^{2}\right )}{9 d^{4}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (4 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 4 b^{4} c^{3}\right )}{7 d^{4}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{5 d^{4}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\begin {cases} a^{4} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{5}}{5 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.40 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2 \, {\left (1155 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{4} - 5460 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 10010 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 8580 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}} + 3003 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{15015 \, d^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (109) = 218\).
Time = 0.33 (sec) , antiderivative size = 807, normalized size of antiderivative = 6.26 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\text {Too large to display} \]
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Time = 0.26 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int (a+b x)^4 (c+d x)^{3/2} \, dx=\frac {2\,b^4\,{\left (c+d\,x\right )}^{13/2}}{13\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5}+\frac {4\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{3\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5} \]
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