Integrand size = 17, antiderivative size = 158 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=-\frac {2 (b c-a d)^5 (c+d x)^{5/2}}{5 d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{7/2}}{7 d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{9/2}}{9 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{11/2}}{11 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{13/2}}{13 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6} \]
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Time = 0.04 (sec) , antiderivative size = 158, 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} \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=-\frac {10 b^4 (c+d x)^{13/2} (b c-a d)}{13 d^6}+\frac {20 b^3 (c+d x)^{11/2} (b c-a d)^2}{11 d^6}-\frac {20 b^2 (c+d x)^{9/2} (b c-a d)^3}{9 d^6}+\frac {10 b (c+d x)^{7/2} (b c-a d)^4}{7 d^6}-\frac {2 (c+d x)^{5/2} (b c-a d)^5}{5 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^5 (c+d x)^{3/2}}{d^5}+\frac {5 b (b c-a d)^4 (c+d x)^{5/2}}{d^5}-\frac {10 b^2 (b c-a d)^3 (c+d x)^{7/2}}{d^5}+\frac {10 b^3 (b c-a d)^2 (c+d x)^{9/2}}{d^5}-\frac {5 b^4 (b c-a d) (c+d x)^{11/2}}{d^5}+\frac {b^5 (c+d x)^{13/2}}{d^5}\right ) \, dx \\ & = -\frac {2 (b c-a d)^5 (c+d x)^{5/2}}{5 d^6}+\frac {10 b (b c-a d)^4 (c+d x)^{7/2}}{7 d^6}-\frac {20 b^2 (b c-a d)^3 (c+d x)^{9/2}}{9 d^6}+\frac {20 b^3 (b c-a d)^2 (c+d x)^{11/2}}{11 d^6}-\frac {10 b^4 (b c-a d) (c+d x)^{13/2}}{13 d^6}+\frac {2 b^5 (c+d x)^{15/2}}{15 d^6} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2 (c+d x)^{5/2} \left (9009 a^5 d^5+6435 a^4 b d^4 (-2 c+5 d x)+1430 a^3 b^2 d^3 \left (8 c^2-20 c d x+35 d^2 x^2\right )+390 a^2 b^3 d^2 \left (-16 c^3+40 c^2 d x-70 c d^2 x^2+105 d^3 x^3\right )+15 a b^4 d \left (128 c^4-320 c^3 d x+560 c^2 d^2 x^2-840 c d^3 x^3+1155 d^4 x^4\right )+b^5 \left (-256 c^5+640 c^4 d x-1120 c^3 d^2 x^2+1680 c^2 d^3 x^3-2310 c d^4 x^4+3003 d^5 x^5\right )\right )}{45045 d^6} \]
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Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {15}{2}}}{15}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(122\) |
default | \(\frac {\frac {2 b^{5} \left (d x +c \right )^{\frac {15}{2}}}{15}+\frac {10 \left (a d -b c \right ) b^{4} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {20 \left (a d -b c \right )^{2} b^{3} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {20 \left (a d -b c \right )^{3} b^{2} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {10 \left (a d -b c \right )^{4} b \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (a d -b c \right )^{5} \left (d x +c \right )^{\frac {5}{2}}}{5}}{d^{6}}\) | \(122\) |
pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{3} b^{5} x^{5}+\frac {25}{13} a \,b^{4} x^{4}+\frac {50}{11} a^{2} b^{3} x^{3}+\frac {50}{9} a^{3} b^{2} x^{2}+\frac {25}{7} a^{4} b x +a^{5}\right ) d^{5}-\frac {10 \left (\frac {7}{39} b^{4} x^{4}+\frac {140}{143} a \,b^{3} x^{3}+\frac {70}{33} a^{2} b^{2} x^{2}+\frac {20}{9} a^{3} b x +a^{4}\right ) b c \,d^{4}}{7}+\frac {80 \left (\frac {21}{143} b^{3} x^{3}+\frac {105}{143} a \,b^{2} x^{2}+\frac {15}{11} a^{2} b x +a^{3}\right ) b^{2} c^{2} d^{3}}{63}-\frac {160 b^{3} \left (\frac {7}{39} b^{2} x^{2}+\frac {10}{13} a b x +a^{2}\right ) c^{3} d^{2}}{231}+\frac {640 \left (\frac {b x}{3}+a \right ) b^{4} c^{4} d}{3003}-\frac {256 b^{5} c^{5}}{9009}\right ) \left (d x +c \right )^{\frac {5}{2}}}{5 d^{6}}\) | \(204\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {5}{2}} \left (3003 x^{5} b^{5} d^{5}+17325 x^{4} a \,b^{4} d^{5}-2310 x^{4} b^{5} c \,d^{4}+40950 x^{3} a^{2} b^{3} d^{5}-12600 x^{3} a \,b^{4} c \,d^{4}+1680 x^{3} b^{5} c^{2} d^{3}+50050 x^{2} a^{3} b^{2} d^{5}-27300 x^{2} a^{2} b^{3} c \,d^{4}+8400 x^{2} a \,b^{4} c^{2} d^{3}-1120 x^{2} b^{5} c^{3} d^{2}+32175 x \,a^{4} b \,d^{5}-28600 x \,a^{3} b^{2} c \,d^{4}+15600 x \,a^{2} b^{3} c^{2} d^{3}-4800 x a \,b^{4} c^{3} d^{2}+640 x \,b^{5} c^{4} d +9009 a^{5} d^{5}-12870 a^{4} b c \,d^{4}+11440 a^{3} b^{2} c^{2} d^{3}-6240 a^{2} b^{3} c^{3} d^{2}+1920 a \,b^{4} c^{4} d -256 b^{5} c^{5}\right )}{45045 d^{6}}\) | \(273\) |
trager | \(\frac {2 \left (3003 b^{5} d^{7} x^{7}+17325 a \,b^{4} d^{7} x^{6}+3696 b^{5} c \,d^{6} x^{6}+40950 a^{2} b^{3} d^{7} x^{5}+22050 a \,b^{4} c \,d^{6} x^{5}+63 b^{5} c^{2} d^{5} x^{5}+50050 a^{3} b^{2} d^{7} x^{4}+54600 a^{2} b^{3} c \,d^{6} x^{4}+525 a \,b^{4} c^{2} d^{5} x^{4}-70 b^{5} c^{3} d^{4} x^{4}+32175 a^{4} b \,d^{7} x^{3}+71500 a^{3} b^{2} c \,d^{6} x^{3}+1950 a^{2} b^{3} c^{2} d^{5} x^{3}-600 a \,b^{4} c^{3} d^{4} x^{3}+80 b^{5} c^{4} d^{3} x^{3}+9009 a^{5} d^{7} x^{2}+51480 a^{4} b c \,d^{6} x^{2}+4290 a^{3} b^{2} c^{2} d^{5} x^{2}-2340 a^{2} b^{3} c^{3} d^{4} x^{2}+720 a \,b^{4} c^{4} d^{3} x^{2}-96 b^{5} c^{5} d^{2} x^{2}+18018 a^{5} c \,d^{6} x +6435 a^{4} b \,c^{2} d^{5} x -5720 a^{3} b^{2} c^{3} d^{4} x +3120 a^{2} b^{3} c^{4} d^{3} x -960 a \,b^{4} c^{5} d^{2} x +128 b^{5} c^{6} d x +9009 a^{5} c^{2} d^{5}-12870 a^{4} b \,c^{3} d^{4}+11440 a^{3} b^{2} c^{4} d^{3}-6240 a^{2} b^{3} c^{5} d^{2}+1920 a \,b^{4} c^{6} d -256 b^{5} c^{7}\right ) \sqrt {d x +c}}{45045 d^{6}}\) | \(453\) |
risch | \(\frac {2 \left (3003 b^{5} d^{7} x^{7}+17325 a \,b^{4} d^{7} x^{6}+3696 b^{5} c \,d^{6} x^{6}+40950 a^{2} b^{3} d^{7} x^{5}+22050 a \,b^{4} c \,d^{6} x^{5}+63 b^{5} c^{2} d^{5} x^{5}+50050 a^{3} b^{2} d^{7} x^{4}+54600 a^{2} b^{3} c \,d^{6} x^{4}+525 a \,b^{4} c^{2} d^{5} x^{4}-70 b^{5} c^{3} d^{4} x^{4}+32175 a^{4} b \,d^{7} x^{3}+71500 a^{3} b^{2} c \,d^{6} x^{3}+1950 a^{2} b^{3} c^{2} d^{5} x^{3}-600 a \,b^{4} c^{3} d^{4} x^{3}+80 b^{5} c^{4} d^{3} x^{3}+9009 a^{5} d^{7} x^{2}+51480 a^{4} b c \,d^{6} x^{2}+4290 a^{3} b^{2} c^{2} d^{5} x^{2}-2340 a^{2} b^{3} c^{3} d^{4} x^{2}+720 a \,b^{4} c^{4} d^{3} x^{2}-96 b^{5} c^{5} d^{2} x^{2}+18018 a^{5} c \,d^{6} x +6435 a^{4} b \,c^{2} d^{5} x -5720 a^{3} b^{2} c^{3} d^{4} x +3120 a^{2} b^{3} c^{4} d^{3} x -960 a \,b^{4} c^{5} d^{2} x +128 b^{5} c^{6} d x +9009 a^{5} c^{2} d^{5}-12870 a^{4} b \,c^{3} d^{4}+11440 a^{3} b^{2} c^{4} d^{3}-6240 a^{2} b^{3} c^{5} d^{2}+1920 a \,b^{4} c^{6} d -256 b^{5} c^{7}\right ) \sqrt {d x +c}}{45045 d^{6}}\) | \(453\) |
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Leaf count of result is larger than twice the leaf count of optimal. 418 vs. \(2 (134) = 268\).
Time = 0.22 (sec) , antiderivative size = 418, normalized size of antiderivative = 2.65 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2 \, {\left (3003 \, b^{5} d^{7} x^{7} - 256 \, b^{5} c^{7} + 1920 \, a b^{4} c^{6} d - 6240 \, a^{2} b^{3} c^{5} d^{2} + 11440 \, a^{3} b^{2} c^{4} d^{3} - 12870 \, a^{4} b c^{3} d^{4} + 9009 \, a^{5} c^{2} d^{5} + 231 \, {\left (16 \, b^{5} c d^{6} + 75 \, a b^{4} d^{7}\right )} x^{6} + 63 \, {\left (b^{5} c^{2} d^{5} + 350 \, a b^{4} c d^{6} + 650 \, a^{2} b^{3} d^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} c^{3} d^{4} - 15 \, a b^{4} c^{2} d^{5} - 1560 \, a^{2} b^{3} c d^{6} - 1430 \, a^{3} b^{2} d^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} c^{4} d^{3} - 120 \, a b^{4} c^{3} d^{4} + 390 \, a^{2} b^{3} c^{2} d^{5} + 14300 \, a^{3} b^{2} c d^{6} + 6435 \, a^{4} b d^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} c^{5} d^{2} - 240 \, a b^{4} c^{4} d^{3} + 780 \, a^{2} b^{3} c^{3} d^{4} - 1430 \, a^{3} b^{2} c^{2} d^{5} - 17160 \, a^{4} b c d^{6} - 3003 \, a^{5} d^{7}\right )} x^{2} + {\left (128 \, b^{5} c^{6} d - 960 \, a b^{4} c^{5} d^{2} + 3120 \, a^{2} b^{3} c^{4} d^{3} - 5720 \, a^{3} b^{2} c^{3} d^{4} + 6435 \, a^{4} b c^{2} d^{5} + 18018 \, a^{5} c d^{6}\right )} x\right )} \sqrt {d x + c}}{45045 \, d^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (146) = 292\).
Time = 0.99 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.12 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (c + d x\right )^{\frac {15}{2}}}{15 d^{5}} + \frac {\left (c + d x\right )^{\frac {13}{2}} \cdot \left (5 a b^{4} d - 5 b^{5} c\right )}{13 d^{5}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \cdot \left (10 a^{2} b^{3} d^{2} - 20 a b^{4} c d + 10 b^{5} c^{2}\right )}{11 d^{5}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (10 a^{3} b^{2} d^{3} - 30 a^{2} b^{3} c d^{2} + 30 a b^{4} c^{2} d - 10 b^{5} c^{3}\right )}{9 d^{5}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (5 a^{4} b d^{4} - 20 a^{3} b^{2} c d^{3} + 30 a^{2} b^{3} c^{2} d^{2} - 20 a b^{4} c^{3} d + 5 b^{5} c^{4}\right )}{7 d^{5}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (a^{5} d^{5} - 5 a^{4} b c d^{4} + 10 a^{3} b^{2} c^{2} d^{3} - 10 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d - b^{5} c^{5}\right )}{5 d^{5}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{6}}{6 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.64 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2 \, {\left (3003 \, {\left (d x + c\right )}^{\frac {15}{2}} b^{5} - 17325 \, {\left (b^{5} c - a b^{4} d\right )} {\left (d x + c\right )}^{\frac {13}{2}} + 40950 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d + a^{2} b^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} - 50050 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 32175 \, {\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 9009 \, {\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} {\left (d x + c\right )}^{\frac {5}{2}}\right )}}{45045 \, d^{6}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1084 vs. \(2 (134) = 268\).
Time = 0.32 (sec) , antiderivative size = 1084, normalized size of antiderivative = 6.86 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\text {Too large to display} \]
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Time = 0.27 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int (a+b x)^5 (c+d x)^{3/2} \, dx=\frac {2\,b^5\,{\left (c+d\,x\right )}^{15/2}}{15\,d^6}-\frac {\left (10\,b^5\,c-10\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}+\frac {2\,{\left (a\,d-b\,c\right )}^5\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}+\frac {20\,b^2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {20\,b^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {10\,b\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6} \]
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