Integrand size = 17, antiderivative size = 129 \[ \int (a+b x)^4 \sqrt {c+d x} \, dx=\frac {2 (b c-a d)^4 (c+d x)^{3/2}}{3 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{5/2}}{5 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{9/2}}{9 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 d^5} \]
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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} \[ \int (a+b x)^4 \sqrt {c+d x} \, dx=-\frac {8 b^3 (c+d x)^{9/2} (b c-a d)}{9 d^5}+\frac {12 b^2 (c+d x)^{7/2} (b c-a d)^2}{7 d^5}-\frac {8 b (c+d x)^{5/2} (b c-a d)^3}{5 d^5}+\frac {2 (c+d x)^{3/2} (b c-a d)^4}{3 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 d^5} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^4 \sqrt {c+d x}}{d^4}-\frac {4 b (b c-a d)^3 (c+d x)^{3/2}}{d^4}+\frac {6 b^2 (b c-a d)^2 (c+d x)^{5/2}}{d^4}-\frac {4 b^3 (b c-a d) (c+d x)^{7/2}}{d^4}+\frac {b^4 (c+d x)^{9/2}}{d^4}\right ) \, dx \\ & = \frac {2 (b c-a d)^4 (c+d x)^{3/2}}{3 d^5}-\frac {8 b (b c-a d)^3 (c+d x)^{5/2}}{5 d^5}+\frac {12 b^2 (b c-a d)^2 (c+d x)^{7/2}}{7 d^5}-\frac {8 b^3 (b c-a d) (c+d x)^{9/2}}{9 d^5}+\frac {2 b^4 (c+d x)^{11/2}}{11 d^5} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.19 \[ \int (a+b x)^4 \sqrt {c+d x} \, dx=\frac {2 (c+d x)^{3/2} \left (1155 a^4 d^4+924 a^3 b d^3 (-2 c+3 d x)+198 a^2 b^2 d^2 \left (8 c^2-12 c d x+15 d^2 x^2\right )+44 a b^3 d \left (-16 c^3+24 c^2 d x-30 c d^2 x^2+35 d^3 x^3\right )+b^4 \left (128 c^4-192 c^3 d x+240 c^2 d^2 x^2-280 c d^3 x^3+315 d^4 x^4\right )\right )}{3465 d^5} \]
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Time = 0.27 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {12 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{5}}\) | \(100\) |
default | \(\frac {\frac {2 b^{4} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {8 \left (a d -b c \right ) b^{3} \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {12 \left (a d -b c \right )^{2} b^{2} \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {8 \left (a d -b c \right )^{3} b \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (a d -b c \right )^{4} \left (d x +c \right )^{\frac {3}{2}}}{3}}{d^{5}}\) | \(100\) |
pseudoelliptic | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (\left (\frac {3}{11} b^{4} x^{4}+\frac {4}{3} a \,b^{3} x^{3}+\frac {18}{7} a^{2} b^{2} x^{2}+\frac {12}{5} a^{3} b x +a^{4}\right ) d^{4}-\frac {8 b c \left (\frac {5}{33} b^{3} x^{3}+\frac {5}{7} a \,b^{2} x^{2}+\frac {9}{7} a^{2} b x +a^{3}\right ) d^{3}}{5}+\frac {48 b^{2} c^{2} \left (\frac {5}{33} b^{2} x^{2}+\frac {2}{3} a b x +a^{2}\right ) d^{2}}{35}-\frac {64 b^{3} \left (\frac {3 b x}{11}+a \right ) c^{3} d}{105}+\frac {128 b^{4} c^{4}}{1155}\right )}{3 d^{5}}\) | \(143\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (315 d^{4} x^{4} b^{4}+1540 a \,b^{3} d^{4} x^{3}-280 b^{4} c \,d^{3} x^{3}+2970 a^{2} b^{2} d^{4} x^{2}-1320 a \,b^{3} c \,d^{3} x^{2}+240 b^{4} c^{2} d^{2} x^{2}+2772 a^{3} b \,d^{4} x -2376 a^{2} b^{2} c \,d^{3} x +1056 a \,b^{3} c^{2} d^{2} x -192 b^{4} c^{3} d x +1155 a^{4} d^{4}-1848 a^{3} b c \,d^{3}+1584 a^{2} b^{2} c^{2} d^{2}-704 a \,b^{3} c^{3} d +128 b^{4} c^{4}\right )}{3465 d^{5}}\) | \(186\) |
trager | \(\frac {2 \left (315 b^{4} d^{5} x^{5}+1540 a \,b^{3} d^{5} x^{4}+35 b^{4} c \,d^{4} x^{4}+2970 a^{2} b^{2} d^{5} x^{3}+220 a \,b^{3} c \,d^{4} x^{3}-40 b^{4} c^{2} d^{3} x^{3}+2772 a^{3} b \,d^{5} x^{2}+594 a^{2} b^{2} c \,d^{4} x^{2}-264 a \,c^{2} b^{3} d^{3} x^{2}+48 b^{4} c^{3} d^{2} x^{2}+1155 a^{4} d^{5} x +924 a^{3} b c \,d^{4} x -792 a^{2} b^{2} c^{2} d^{3} x +352 a \,b^{3} c^{3} d^{2} x -64 b^{4} c^{4} d x +1155 a^{4} c \,d^{4}-1848 a^{3} b \,c^{2} d^{3}+1584 a^{2} b^{2} c^{3} d^{2}-704 a \,b^{3} c^{4} d +128 b^{4} c^{5}\right ) \sqrt {d x +c}}{3465 d^{5}}\) | \(257\) |
risch | \(\frac {2 \left (315 b^{4} d^{5} x^{5}+1540 a \,b^{3} d^{5} x^{4}+35 b^{4} c \,d^{4} x^{4}+2970 a^{2} b^{2} d^{5} x^{3}+220 a \,b^{3} c \,d^{4} x^{3}-40 b^{4} c^{2} d^{3} x^{3}+2772 a^{3} b \,d^{5} x^{2}+594 a^{2} b^{2} c \,d^{4} x^{2}-264 a \,c^{2} b^{3} d^{3} x^{2}+48 b^{4} c^{3} d^{2} x^{2}+1155 a^{4} d^{5} x +924 a^{3} b c \,d^{4} x -792 a^{2} b^{2} c^{2} d^{3} x +352 a \,b^{3} c^{3} d^{2} x -64 b^{4} c^{4} d x +1155 a^{4} c \,d^{4}-1848 a^{3} b \,c^{2} d^{3}+1584 a^{2} b^{2} c^{3} d^{2}-704 a \,b^{3} c^{4} d +128 b^{4} c^{5}\right ) \sqrt {d x +c}}{3465 d^{5}}\) | \(257\) |
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (109) = 218\).
Time = 0.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.90 \[ \int (a+b x)^4 \sqrt {c+d x} \, dx=\frac {2 \, {\left (315 \, b^{4} d^{5} x^{5} + 128 \, b^{4} c^{5} - 704 \, a b^{3} c^{4} d + 1584 \, a^{2} b^{2} c^{3} d^{2} - 1848 \, a^{3} b c^{2} d^{3} + 1155 \, a^{4} c d^{4} + 35 \, {\left (b^{4} c d^{4} + 44 \, a b^{3} d^{5}\right )} x^{4} - 10 \, {\left (4 \, b^{4} c^{2} d^{3} - 22 \, a b^{3} c d^{4} - 297 \, a^{2} b^{2} d^{5}\right )} x^{3} + 6 \, {\left (8 \, b^{4} c^{3} d^{2} - 44 \, a b^{3} c^{2} d^{3} + 99 \, a^{2} b^{2} c d^{4} + 462 \, a^{3} b d^{5}\right )} x^{2} - {\left (64 \, b^{4} c^{4} d - 352 \, a b^{3} c^{3} d^{2} + 792 \, a^{2} b^{2} c^{2} d^{3} - 924 \, a^{3} b c d^{4} - 1155 \, a^{4} d^{5}\right )} x\right )} \sqrt {d x + c}}{3465 \, d^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (119) = 238\).
Time = 0.90 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.88 \[ \int (a+b x)^4 \sqrt {c+d x} \, dx=\begin {cases} \frac {2 \left (\frac {b^{4} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{4}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (4 a b^{3} d - 4 b^{4} c\right )}{9 d^{4}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (6 a^{2} b^{2} d^{2} - 12 a b^{3} c d + 6 b^{4} c^{2}\right )}{7 d^{4}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \cdot \left (4 a^{3} b d^{3} - 12 a^{2} b^{2} c d^{2} + 12 a b^{3} c^{2} d - 4 b^{4} c^{3}\right )}{5 d^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{4} d^{4} - 4 a^{3} b c d^{3} + 6 a^{2} b^{2} c^{2} d^{2} - 4 a b^{3} c^{3} d + b^{4} c^{4}\right )}{3 d^{4}}\right )}{d} & \text {for}\: d \neq 0 \\\sqrt {c} \left (\begin {cases} a^{4} x & \text {for}\: b = 0 \\\frac {\left (a + b x\right )^{5}}{5 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.20 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.40 \[ \int (a+b x)^4 \sqrt {c+d x} \, dx=\frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{4} - 1540 \, {\left (b^{4} c - a b^{3} d\right )} {\left (d x + c\right )}^{\frac {9}{2}} + 2970 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 2772 \, {\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 1155 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}}\right )}}{3465 \, d^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (109) = 218\).
Time = 0.32 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.64 \[ \int (a+b x)^4 \sqrt {c+d x} \, dx=\frac {2 \, {\left (3465 \, \sqrt {d x + c} a^{4} c + 1155 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{4} + \frac {4620 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{3} b c}{d} + \frac {1386 \, {\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^{2} b^{2} c}{d^{2}} + \frac {924 \, {\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^{3} b}{d} + \frac {396 \, {\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 )} a b^{3} c}{d^{3}} + \frac {594 \, {\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 )} a^{2} b^{2}}{d^{2}} + \frac {11 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b^{4} c}{d^{4}} + \frac {44 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} a b^{3}}{d^{3}} + \frac {5 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{4}}{d^{4}}\right )}}{3465 \, d} \]
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Time = 0.27 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.87 \[ \int (a+b x)^4 \sqrt {c+d x} \, dx=\frac {2\,b^4\,{\left (c+d\,x\right )}^{11/2}}{11\,d^5}-\frac {\left (8\,b^4\,c-8\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^5}+\frac {2\,{\left (a\,d-b\,c\right )}^4\,{\left (c+d\,x\right )}^{3/2}}{3\,d^5}+\frac {12\,b^2\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{7/2}}{7\,d^5}+\frac {8\,b\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{5/2}}{5\,d^5} \]
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