Integrand size = 13, antiderivative size = 30 \[ \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx=-\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}} \]
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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} \[ \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx=-\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}} \]
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Rule 15
Rule 30
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (c x^3\right )^{5/2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x)^{3/2} \text {Subst}\left (\int \frac {1}{x^{15/2}} \, dx,x,a+b x\right )}{b c^2 \sqrt {c (a+b x)^3}} \\ & = -\frac {2}{13 b c^2 (a+b x)^5 \sqrt {c (a+b x)^3}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx=-\frac {2 (a+b x)}{13 b \left (c (a+b x)^3\right )^{5/2}} \]
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Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right )}{13 b \left (c \left (b x +a \right )^{3}\right )^{\frac {5}{2}}}\) | \(22\) |
default | \(-\frac {2 \left (b x +a \right )^{5} \left (c \left (b x +a \right )\right )^{\frac {5}{2}} c^{4}}{13 \left (c \left (b x +a \right )^{3}\right )^{\frac {5}{2}} b \left (b c x +a c \right )^{\frac {13}{2}}}\) | \(46\) |
trager | \(-\frac {2 \sqrt {b^{3} c \,x^{3}+3 a \,b^{2} c \,x^{2}+3 a^{2} b c x +c \,a^{3}}}{13 c^{3} \left (b x +a \right )^{8} b}\) | \(50\) |
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Leaf count of result is larger than twice the leaf count of optimal. 151 vs. \(2 (26) = 52\).
Time = 0.26 (sec) , antiderivative size = 151, normalized size of antiderivative = 5.03 \[ \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx=-\frac {2 \, \sqrt {b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{13 \, {\left (b^{9} c^{3} x^{8} + 8 \, a b^{8} c^{3} x^{7} + 28 \, a^{2} b^{7} c^{3} x^{6} + 56 \, a^{3} b^{6} c^{3} x^{5} + 70 \, a^{4} b^{5} c^{3} x^{4} + 56 \, a^{5} b^{4} c^{3} x^{3} + 28 \, a^{6} b^{3} c^{3} x^{2} + 8 \, a^{7} b^{2} c^{3} x + a^{8} b c^{3}\right )}} \]
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Time = 4.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx=\begin {cases} - \frac {2 \left (\frac {a}{b} + x\right )}{13 \left (c \left (a + b x\right )^{3}\right )^{\frac {5}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\left (a^{3} c\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (26) = 52\).
Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.83 \[ \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx=-\frac {2 \, \sqrt {c}}{13 \, {\left (b^{6} c^{3} x^{5} + 5 \, a b^{5} c^{3} x^{4} + 10 \, a^{2} b^{4} c^{3} x^{3} + 10 \, a^{3} b^{3} c^{3} x^{2} + 5 \, a^{4} b^{2} c^{3} x + a^{5} b c^{3}\right )} {\left (b x + a\right )}^{\frac {3}{2}}} \]
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none
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx=-\frac {2 \, c^{4}}{13 \, {\left (b c x + a c\right )}^{\frac {13}{2}} b \mathrm {sgn}\left (b x + a\right )} \]
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Time = 6.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (c (a+b x)^3\right )^{5/2}} \, dx=-\frac {2\,\sqrt {c\,{\left (a+b\,x\right )}^3}}{13\,b\,c^3\,{\left (a+b\,x\right )}^8} \]
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