Integrand size = 13, antiderivative size = 30 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {(a+b x)^5}{6 b c^2 \sqrt {\frac {c}{(a+b x)^2}}} \]
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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} \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {(a+b x)^5}{6 b c^2 \sqrt {\frac {c}{(a+b x)^2}}} \]
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Rule 15
Rule 30
Rule 253
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (\frac {c}{x^2}\right )^{5/2}} \, dx,x,a+b x\right )}{b} \\ & = \frac {\text {Subst}\left (\int x^5 \, dx,x,a+b x\right )}{b c^2 \sqrt {\frac {c}{(a+b x)^2}} (a+b x)} \\ & = \frac {(a+b x)^5}{6 b c^2 \sqrt {\frac {c}{(a+b x)^2}}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {a+b x}{6 b \left (\frac {c}{(a+b x)^2}\right )^{5/2}} \]
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Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {b x +a}{6 \left (\frac {c}{\left (b x +a \right )^{2}}\right )^{\frac {5}{2}} b}\) | \(22\) |
risch | \(\frac {\left (b x +a \right )^{5}}{6 b \,c^{2} \sqrt {\frac {c}{\left (b x +a \right )^{2}}}}\) | \(27\) |
gosper | \(\frac {x \left (b^{5} x^{5}+6 a \,b^{4} x^{4}+15 a^{2} b^{3} x^{3}+20 a^{3} b^{2} x^{2}+15 a^{4} b x +6 a^{5}\right )}{6 \left (b x +a \right )^{5} \left (\frac {c}{\left (b x +a \right )^{2}}\right )^{\frac {5}{2}}}\) | \(73\) |
trager | \(\frac {\left (b x +a \right ) \left (b^{5} x^{5}+6 a \,b^{4} x^{4}+15 a^{2} b^{3} x^{3}+20 a^{3} b^{2} x^{2}+15 a^{4} b x +6 a^{5}\right ) x \sqrt {\frac {c}{b^{2} x^{2}+2 a b x +a^{2}}}}{6 c^{3}}\) | \(85\) |
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Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (26) = 52\).
Time = 0.24 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.07 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {{\left (b^{6} x^{7} + 7 \, a b^{5} x^{6} + 21 \, a^{2} b^{4} x^{5} + 35 \, a^{3} b^{3} x^{4} + 35 \, a^{4} b^{2} x^{3} + 21 \, a^{5} b x^{2} + 6 \, a^{6} x\right )} \sqrt {\frac {c}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{6 \, c^{3}} \]
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Time = 1.92 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\begin {cases} \frac {\frac {a}{b} + x}{6 \left (\frac {c}{\left (a + b x\right )^{2}}\right )^{\frac {5}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\frac {c}{a^{2}}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (26) = 52\).
Time = 0.20 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.97 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {b^{5} x^{6} + 6 \, a b^{4} x^{5} + 15 \, a^{2} b^{3} x^{4} + 20 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x^{2} + 6 \, a^{5} x}{6 \, c^{\frac {5}{2}}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (26) = 52\).
Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.83 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\frac {b^{5} \sqrt {c} x^{6} + 6 \, a b^{4} \sqrt {c} x^{5} + 15 \, a^{2} b^{3} \sqrt {c} x^{4} + 20 \, a^{3} b^{2} \sqrt {c} x^{3} + 15 \, a^{4} b \sqrt {c} x^{2} + 6 \, a^{5} \sqrt {c} x}{6 \, c^{3} \mathrm {sgn}\left (b x + a\right )} \]
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Time = 5.96 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.27 \[ \int \frac {1}{\left (\frac {c}{(a+b x)^2}\right )^{5/2}} \, dx=\sqrt {\frac {c}{{\left (a+b\,x\right )}^2}}\,\left (\frac {a^6\,x}{c^3}+\frac {b^6\,x^7}{6\,c^3}+\frac {7\,a^5\,b\,x^2}{2\,c^3}+\frac {7\,a\,b^5\,x^6}{6\,c^3}+\frac {35\,a^4\,b^2\,x^3}{6\,c^3}+\frac {35\,a^3\,b^3\,x^4}{6\,c^3}+\frac {7\,a^2\,b^4\,x^5}{2\,c^3}\right ) \]
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