Integrand size = 13, antiderivative size = 30 \[ \int \left (c (a+b x)^3\right )^{5/2} \, dx=\frac {2 c^2 (a+b x)^7 \sqrt {c (a+b x)^3}}{17 b} \]
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Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {253, 15, 30} \[ \int \left (c (a+b x)^3\right )^{5/2} \, dx=\frac {2 c^2 (a+b x)^7 \sqrt {c (a+b x)^3}}{17 b} \]
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Rule 15
Rule 30
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (c x^3\right )^{5/2} \, dx,x,a+b x\right )}{b} \\ & = \frac {\left (c^2 \sqrt {c (a+b x)^3}\right ) \text {Subst}\left (\int x^{15/2} \, dx,x,a+b x\right )}{b (a+b x)^{3/2}} \\ & = \frac {2 c^2 (a+b x)^7 \sqrt {c (a+b x)^3}}{17 b} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \left (c (a+b x)^3\right )^{5/2} \, dx=\frac {2 (a+b x) \left (c (a+b x)^3\right )^{5/2}}{17 b} \]
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Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(\frac {2 \left (b x +a \right ) \left (c \left (b x +a \right )^{3}\right )^{\frac {5}{2}}}{17 b}\) | \(22\) |
default | \(\frac {2 \left (c \left (b x +a \right )^{3}\right )^{\frac {5}{2}} \left (b c x +a c \right )^{\frac {17}{2}}}{17 \left (b x +a \right )^{5} \left (c \left (b x +a \right )\right )^{\frac {5}{2}} c^{6} b}\) | \(46\) |
risch | \(\frac {2 c^{2} \sqrt {c \left (b x +a \right )^{3}}\, \left (b^{8} x^{8}+8 a \,x^{7} b^{7}+28 a^{2} x^{6} b^{6}+56 a^{3} x^{5} b^{5}+70 a^{4} x^{4} b^{4}+56 a^{5} b^{3} x^{3}+28 a^{6} x^{2} b^{2}+8 a^{7} x b +a^{8}\right )}{17 \left (b x +a \right ) b}\) | \(109\) |
trager | \(\frac {2 c^{2} \left (b^{7} x^{7}+7 a \,b^{6} x^{6}+21 a^{2} b^{5} x^{5}+35 a^{3} b^{4} x^{4}+35 a^{4} b^{3} x^{3}+21 a^{5} b^{2} x^{2}+7 a^{6} b x +a^{7}\right ) \sqrt {b^{3} c \,x^{3}+3 a \,b^{2} c \,x^{2}+3 a^{2} b c x +c \,a^{3}}}{17 b}\) | \(114\) |
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Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 4.50 \[ \int \left (c (a+b x)^3\right )^{5/2} \, dx=\frac {2 \, {\left (b^{7} c^{2} x^{7} + 7 \, a b^{6} c^{2} x^{6} + 21 \, a^{2} b^{5} c^{2} x^{5} + 35 \, a^{3} b^{4} c^{2} x^{4} + 35 \, a^{4} b^{3} c^{2} x^{3} + 21 \, a^{5} b^{2} c^{2} x^{2} + 7 \, a^{6} b c^{2} x + a^{7} c^{2}\right )} \sqrt {b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{17 \, b} \]
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\[ \int \left (c (a+b x)^3\right )^{5/2} \, dx=\int \left (c \left (a + b x\right )^{3}\right )^{\frac {5}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (26) = 52\).
Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.60 \[ \int \left (c (a+b x)^3\right )^{5/2} \, dx=\frac {2 \, {\left (b^{7} c^{\frac {5}{2}} x^{7} + 7 \, a b^{6} c^{\frac {5}{2}} x^{6} + 21 \, a^{2} b^{5} c^{\frac {5}{2}} x^{5} + 35 \, a^{3} b^{4} c^{\frac {5}{2}} x^{4} + 35 \, a^{4} b^{3} c^{\frac {5}{2}} x^{3} + 21 \, a^{5} b^{2} c^{\frac {5}{2}} x^{2} + 7 \, a^{6} b c^{\frac {5}{2}} x + a^{7} c^{\frac {5}{2}}\right )} {\left (b x + a\right )}^{\frac {3}{2}}}{17 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 851 vs. \(2 (26) = 52\).
Time = 0.30 (sec) , antiderivative size = 851, normalized size of antiderivative = 28.37 \[ \int \left (c (a+b x)^3\right )^{5/2} \, dx=\text {Too large to display} \]
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Time = 6.94 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.67 \[ \int \left (c (a+b x)^3\right )^{5/2} \, dx=\sqrt {c\,{\left (a+b\,x\right )}^3}\,\left (\frac {14\,a^6\,c^2\,x}{17}+\frac {2\,a^7\,c^2}{17\,b}+\frac {2\,b^6\,c^2\,x^7}{17}+\frac {42\,a^5\,b\,c^2\,x^2}{17}+\frac {14\,a\,b^5\,c^2\,x^6}{17}+\frac {70\,a^4\,b^2\,c^2\,x^3}{17}+\frac {70\,a^3\,b^3\,c^2\,x^4}{17}+\frac {42\,a^2\,b^4\,c^2\,x^5}{17}\right ) \]
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