Integrand size = 21, antiderivative size = 90 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^5} \, dx=-\frac {b \sqrt {1+(c+d x)^2}}{12 d e^5 (c+d x)^3}+\frac {b \sqrt {1+(c+d x)^2}}{6 d e^5 (c+d x)}-\frac {a+b \text {arcsinh}(c+d x)}{4 d e^5 (c+d x)^4} \]
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Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5859, 12, 5776, 277, 270} \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^5} \, dx=-\frac {a+b \text {arcsinh}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \sqrt {(c+d x)^2+1}}{6 d e^5 (c+d x)}-\frac {b \sqrt {(c+d x)^2+1}}{12 d e^5 (c+d x)^3} \]
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Rule 12
Rule 270
Rule 277
Rule 5776
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{e^5 x^5} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \text {arcsinh}(x)}{x^5} \, dx,x,c+d x\right )}{d e^5} \\ & = -\frac {a+b \text {arcsinh}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \text {Subst}\left (\int \frac {1}{x^4 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d e^5} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{12 d e^5 (c+d x)^3}-\frac {a+b \text {arcsinh}(c+d x)}{4 d e^5 (c+d x)^4}-\frac {b \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{6 d e^5} \\ & = -\frac {b \sqrt {1+(c+d x)^2}}{12 d e^5 (c+d x)^3}+\frac {b \sqrt {1+(c+d x)^2}}{6 d e^5 (c+d x)}-\frac {a+b \text {arcsinh}(c+d x)}{4 d e^5 (c+d x)^4} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.68 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^5} \, dx=-\frac {b (c+d x) \left (1-2 (c+d x)^2\right ) \sqrt {1+(c+d x)^2}+3 (a+b \text {arcsinh}(c+d x))}{12 d e^5 (c+d x)^4} \]
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Time = 0.36 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(\frac {-\frac {a}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{12 \left (d x +c \right )^{3}}+\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 d x +6 c}\right )}{e^{5}}}{d}\) | \(80\) |
default | \(\frac {-\frac {a}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{12 \left (d x +c \right )^{3}}+\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 d x +6 c}\right )}{e^{5}}}{d}\) | \(80\) |
parts | \(-\frac {a}{4 e^{5} \left (d x +c \right )^{4} d}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{12 \left (d x +c \right )^{3}}+\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 d x +6 c}\right )}{e^{5} d}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (80) = 160\).
Time = 0.28 (sec) , antiderivative size = 210, normalized size of antiderivative = 2.33 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^5} \, dx=\frac {3 \, a d^{4} x^{4} + 12 \, a c d^{3} x^{3} + 18 \, a c^{2} d^{2} x^{2} + 12 \, a c^{3} d x - 3 \, b c^{4} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left (2 \, b c^{4} d^{3} x^{3} + 6 \, b c^{5} d^{2} x^{2} + 2 \, b c^{7} - b c^{5} + {\left (6 \, b c^{6} - b c^{4}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{12 \, {\left (c^{4} d^{5} e^{5} x^{4} + 4 \, c^{5} d^{4} e^{5} x^{3} + 6 \, c^{6} d^{3} e^{5} x^{2} + 4 \, c^{7} d^{2} e^{5} x + c^{8} d e^{5}\right )}} \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^5} \, dx=\frac {\int \frac {a}{c^{5} + 5 c^{4} d x + 10 c^{3} d^{2} x^{2} + 10 c^{2} d^{3} x^{3} + 5 c d^{4} x^{4} + d^{5} x^{5}}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c^{5} + 5 c^{4} d x + 10 c^{3} d^{2} x^{2} + 10 c^{2} d^{3} x^{3} + 5 c d^{4} x^{4} + d^{5} x^{5}}\, dx}{e^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 258 vs. \(2 (80) = 160\).
Time = 0.21 (sec) , antiderivative size = 258, normalized size of antiderivative = 2.87 \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^5} \, dx=\frac {1}{12} \, b {\left (\frac {{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 2 \, c^{4} + {\left (12 \, c^{2} d^{2} + d^{2}\right )} x^{2} + c^{2} + 2 \, {\left (4 \, c^{3} d + c d\right )} x - 1\right )} d}{{\left (d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}} - \frac {3 \, \operatorname {arsinh}\left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} - \frac {a}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \]
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\[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^5} \, dx=\int { \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{5}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arcsinh}(c+d x)}{(c e+d e x)^5} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^5} \,d x \]
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