Integrand size = 13, antiderivative size = 30 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 c^2 \sqrt {\frac {c}{(a+b x)^3}}}{13 b (a+b x)^5} \]
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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} \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 c^2 \sqrt {\frac {c}{(a+b x)^3}}}{13 b (a+b x)^5} \]
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Rule 15
Rule 30
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {c}{x^3}\right )^{5/2} \, dx,x,a+b x\right )}{b} \\ & = \frac {\left (c^2 \sqrt {\frac {c}{(a+b x)^3}} (a+b x)^{3/2}\right ) \text {Subst}\left (\int \frac {1}{x^{15/2}} \, dx,x,a+b x\right )}{b} \\ & = -\frac {2 c^2 \sqrt {\frac {c}{(a+b x)^3}}}{13 b (a+b x)^5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 \left (\frac {c}{(a+b x)^3}\right )^{5/2} (a+b x)}{13 b} \]
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Time = 0.13 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
gosper | \(-\frac {2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{3}}\right )^{\frac {5}{2}}}{13 b}\) | \(22\) |
default | \(-\frac {2 \left (b x +a \right ) \left (\frac {c}{\left (b x +a \right )^{3}}\right )^{\frac {5}{2}}}{13 b}\) | \(22\) |
trager | \(-\frac {2 c^{2} \sqrt {\frac {c}{b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}}}}{13 b \left (b x +a \right )^{5}}\) | \(49\) |
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Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.10 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 \, c^{2} \sqrt {\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{13 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1243 vs. \(2 (27) = 54\).
Time = 27.34 (sec) , antiderivative size = 1243, normalized size of antiderivative = 41.43 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=\begin {cases} - \frac {638 a^{6} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {5}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {3828 a^{5} b x \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {5}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {9570 a^{4} b^{2} x^{2} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {5}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {12760 a^{3} b^{3} x^{3} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {5}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {1170 a^{3} c \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {3}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {9570 a^{2} b^{4} x^{4} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {5}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {3510 a^{2} b c x \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {3}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {3828 a b^{5} x^{5} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {5}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {3510 a b^{2} c x^{2} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {3}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {638 b^{6} x^{6} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {5}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} - \frac {1170 b^{3} c x^{3} \left (\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}\right )^{\frac {3}{2}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} + \frac {1794 c^{2} \sqrt {\frac {c}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}}}{91 a^{5} b + 455 a^{4} b^{2} x + 910 a^{3} b^{3} x^{2} + 910 a^{2} b^{4} x^{3} + 455 a b^{5} x^{4} + 91 b^{6} x^{5}} & \text {for}\: b \neq 0 \\x \left (\frac {c}{a^{3}}\right )^{\frac {5}{2}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 \, {\left (b c^{\frac {5}{2}} x + a c^{\frac {5}{2}}\right )}}{13 \, {\left (b x + a\right )}^{\frac {15}{2}} b} \]
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Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.73 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2 \, c^{9} \mathrm {sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \mathrm {sgn}\left (b x + a\right )}{13 \, {\left (b c x + a c\right )}^{\frac {13}{2}} b} \]
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Time = 6.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.83 \[ \int \left (\frac {c}{(a+b x)^3}\right )^{5/2} \, dx=-\frac {2\,b^2\,c^2\,\sqrt {\frac {c}{{\left (a+b\,x\right )}^3}}\,\left (x^3+\frac {a^3}{b^3}+\frac {3\,a\,x^2}{b}+\frac {3\,a^2\,x}{b^2}\right )}{13\,{\left (a+b\,x\right )}^8} \]
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