Integrand size = 17, antiderivative size = 100 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=-\frac {2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^4}+\frac {2 b (b c-a d)^2 (c+d x)^{9/2}}{3 d^4}-\frac {6 b^2 (b c-a d) (c+d x)^{11/2}}{11 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4} \]
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Time = 0.02 (sec) , antiderivative size = 100, 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} \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=-\frac {6 b^2 (c+d x)^{11/2} (b c-a d)}{11 d^4}+\frac {2 b (c+d x)^{9/2} (b c-a d)^2}{3 d^4}-\frac {2 (c+d x)^{7/2} (b c-a d)^3}{7 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4} \]
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Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c+a d)^3 (c+d x)^{5/2}}{d^3}+\frac {3 b (b c-a d)^2 (c+d x)^{7/2}}{d^3}-\frac {3 b^2 (b c-a d) (c+d x)^{9/2}}{d^3}+\frac {b^3 (c+d x)^{11/2}}{d^3}\right ) \, dx \\ & = -\frac {2 (b c-a d)^3 (c+d x)^{7/2}}{7 d^4}+\frac {2 b (b c-a d)^2 (c+d x)^{9/2}}{3 d^4}-\frac {6 b^2 (b c-a d) (c+d x)^{11/2}}{11 d^4}+\frac {2 b^3 (c+d x)^{13/2}}{13 d^4} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2 (c+d x)^{7/2} \left (429 a^3 d^3+143 a^2 b d^2 (-2 c+7 d x)+13 a b^2 d \left (8 c^2-28 c d x+63 d^2 x^2\right )+b^3 \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )\right )}{3003 d^4} \]
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Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a d -b c \right ) b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a d -b c \right )^{2} b \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{4}}\) | \(78\) |
default | \(\frac {\frac {2 b^{3} \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {6 \left (a d -b c \right ) b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a d -b c \right )^{2} b \left (d x +c \right )^{\frac {9}{2}}}{3}+\frac {2 \left (a d -b c \right )^{3} \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{4}}\) | \(78\) |
pseudoelliptic | \(\frac {2 \left (\left (\frac {7}{13} d^{3} x^{3}-\frac {42}{143} c \,d^{2} x^{2}+\frac {56}{429} c^{2} d x -\frac {16}{429} c^{3}\right ) b^{3}+\frac {8 \left (\frac {63}{8} d^{2} x^{2}-\frac {7}{2} c d x +c^{2}\right ) d a \,b^{2}}{33}-\frac {2 \left (-\frac {7 d x}{2}+c \right ) d^{2} a^{2} b}{3}+a^{3} d^{3}\right ) \left (d x +c \right )^{\frac {7}{2}}}{7 d^{4}}\) | \(94\) |
gosper | \(\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (231 d^{3} x^{3} b^{3}+819 x^{2} a \,b^{2} d^{3}-126 x^{2} b^{3} c \,d^{2}+1001 x \,a^{2} b \,d^{3}-364 x a \,b^{2} c \,d^{2}+56 x \,b^{3} c^{2} d +429 a^{3} d^{3}-286 a^{2} b c \,d^{2}+104 a \,b^{2} c^{2} d -16 b^{3} c^{3}\right )}{3003 d^{4}}\) | \(116\) |
trager | \(\frac {2 \left (231 b^{3} d^{6} x^{6}+819 a \,b^{2} d^{6} x^{5}+567 b^{3} c \,d^{5} x^{5}+1001 a^{2} b \,d^{6} x^{4}+2093 a \,b^{2} c \,d^{5} x^{4}+371 b^{3} c^{2} d^{4} x^{4}+429 a^{3} d^{6} x^{3}+2717 a^{2} b c \,d^{5} x^{3}+1469 a \,b^{2} c^{2} d^{4} x^{3}+5 b^{3} c^{3} d^{3} x^{3}+1287 a^{3} c \,d^{5} x^{2}+2145 a^{2} b \,c^{2} d^{4} x^{2}+39 a \,b^{2} c^{3} d^{3} x^{2}-6 b^{3} c^{4} d^{2} x^{2}+1287 a^{3} c^{2} d^{4} x +143 a^{2} b \,c^{3} d^{3} x -52 a \,b^{2} c^{4} d^{2} x +8 b^{3} c^{5} d x +429 a^{3} c^{3} d^{3}-286 a^{2} b \,c^{4} d^{2}+104 a \,b^{2} c^{5} d -16 b^{3} c^{6}\right ) \sqrt {d x +c}}{3003 d^{4}}\) | \(286\) |
risch | \(\frac {2 \left (231 b^{3} d^{6} x^{6}+819 a \,b^{2} d^{6} x^{5}+567 b^{3} c \,d^{5} x^{5}+1001 a^{2} b \,d^{6} x^{4}+2093 a \,b^{2} c \,d^{5} x^{4}+371 b^{3} c^{2} d^{4} x^{4}+429 a^{3} d^{6} x^{3}+2717 a^{2} b c \,d^{5} x^{3}+1469 a \,b^{2} c^{2} d^{4} x^{3}+5 b^{3} c^{3} d^{3} x^{3}+1287 a^{3} c \,d^{5} x^{2}+2145 a^{2} b \,c^{2} d^{4} x^{2}+39 a \,b^{2} c^{3} d^{3} x^{2}-6 b^{3} c^{4} d^{2} x^{2}+1287 a^{3} c^{2} d^{4} x +143 a^{2} b \,c^{3} d^{3} x -52 a \,b^{2} c^{4} d^{2} x +8 b^{3} c^{5} d x +429 a^{3} c^{3} d^{3}-286 a^{2} b \,c^{4} d^{2}+104 a \,b^{2} c^{5} d -16 b^{3} c^{6}\right ) \sqrt {d x +c}}{3003 d^{4}}\) | \(286\) |
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Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (84) = 168\).
Time = 0.22 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.68 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2 \, {\left (231 \, b^{3} d^{6} x^{6} - 16 \, b^{3} c^{6} + 104 \, a b^{2} c^{5} d - 286 \, a^{2} b c^{4} d^{2} + 429 \, a^{3} c^{3} d^{3} + 63 \, {\left (9 \, b^{3} c d^{5} + 13 \, a b^{2} d^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} c^{2} d^{4} + 299 \, a b^{2} c d^{5} + 143 \, a^{2} b d^{6}\right )} x^{4} + {\left (5 \, b^{3} c^{3} d^{3} + 1469 \, a b^{2} c^{2} d^{4} + 2717 \, a^{2} b c d^{5} + 429 \, a^{3} d^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} c^{4} d^{2} - 13 \, a b^{2} c^{3} d^{3} - 715 \, a^{2} b c^{2} d^{4} - 429 \, a^{3} c d^{5}\right )} x^{2} + {\left (8 \, b^{3} c^{5} d - 52 \, a b^{2} c^{4} d^{2} + 143 \, a^{2} b c^{3} d^{3} + 1287 \, a^{3} c^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{3003 \, d^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (92) = 184\).
Time = 0.46 (sec) , antiderivative size = 549, normalized size of antiderivative = 5.49 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\begin {cases} \frac {2 a^{3} c^{3} \sqrt {c + d x}}{7 d} + \frac {6 a^{3} c^{2} x \sqrt {c + d x}}{7} + \frac {6 a^{3} c d x^{2} \sqrt {c + d x}}{7} + \frac {2 a^{3} d^{2} x^{3} \sqrt {c + d x}}{7} - \frac {4 a^{2} b c^{4} \sqrt {c + d x}}{21 d^{2}} + \frac {2 a^{2} b c^{3} x \sqrt {c + d x}}{21 d} + \frac {10 a^{2} b c^{2} x^{2} \sqrt {c + d x}}{7} + \frac {38 a^{2} b c d x^{3} \sqrt {c + d x}}{21} + \frac {2 a^{2} b d^{2} x^{4} \sqrt {c + d x}}{3} + \frac {16 a b^{2} c^{5} \sqrt {c + d x}}{231 d^{3}} - \frac {8 a b^{2} c^{4} x \sqrt {c + d x}}{231 d^{2}} + \frac {2 a b^{2} c^{3} x^{2} \sqrt {c + d x}}{77 d} + \frac {226 a b^{2} c^{2} x^{3} \sqrt {c + d x}}{231} + \frac {46 a b^{2} c d x^{4} \sqrt {c + d x}}{33} + \frac {6 a b^{2} d^{2} x^{5} \sqrt {c + d x}}{11} - \frac {32 b^{3} c^{6} \sqrt {c + d x}}{3003 d^{4}} + \frac {16 b^{3} c^{5} x \sqrt {c + d x}}{3003 d^{3}} - \frac {4 b^{3} c^{4} x^{2} \sqrt {c + d x}}{1001 d^{2}} + \frac {10 b^{3} c^{3} x^{3} \sqrt {c + d x}}{3003 d} + \frac {106 b^{3} c^{2} x^{4} \sqrt {c + d x}}{429} + \frac {54 b^{3} c d x^{5} \sqrt {c + d x}}{143} + \frac {2 b^{3} d^{2} x^{6} \sqrt {c + d x}}{13} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
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none
Time = 0.29 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.18 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2 \, {\left (231 \, {\left (d x + c\right )}^{\frac {13}{2}} b^{3} - 819 \, {\left (b^{3} c - a b^{2} d\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 1001 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 429 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{3003 \, d^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (84) = 168\).
Time = 0.32 (sec) , antiderivative size = 857, normalized size of antiderivative = 8.57 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\text {Too large to display} \]
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Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int (a+b x)^3 (c+d x)^{5/2} \, dx=\frac {2\,b^3\,{\left (c+d\,x\right )}^{13/2}}{13\,d^4}-\frac {\left (6\,b^3\,c-6\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^4}+\frac {2\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^{7/2}}{7\,d^4}+\frac {2\,b\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^{9/2}}{3\,d^4} \]
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