Integrand size = 28, antiderivative size = 120 \[ \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {b g x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}+\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c^2 \sqrt {d+c^2 d x^2}}+\frac {f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+c^2 d x^2}} \]
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Time = 0.15 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5845, 5838, 5783, 5798, 8} \[ \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {f \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {c^2 d x^2+d}}+\frac {g \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{c^2 \sqrt {c^2 d x^2+d}}-\frac {b g x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}} \]
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Rule 8
Rule 5783
Rule 5798
Rule 5838
Rule 5845
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+c^2 x^2} \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}} \\ & = \frac {\sqrt {1+c^2 x^2} \int \left (\frac {f (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {g x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+c^2 d x^2}} \\ & = \frac {\left (f \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}+\frac {\left (g \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}} \\ & = \frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c^2 \sqrt {d+c^2 d x^2}}+\frac {f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+c^2 d x^2}}-\frac {\left (b g \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{c \sqrt {d+c^2 d x^2}} \\ & = -\frac {b g x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}+\frac {g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c^2 \sqrt {d+c^2 d x^2}}+\frac {f \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.32 \[ \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {2 \sqrt {d} g \left (a+a c^2 x^2-b c x \sqrt {1+c^2 x^2}\right )+2 b \sqrt {d} g \left (1+c^2 x^2\right ) \text {arcsinh}(c x)+b c \sqrt {d} f \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2+2 a c f \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{2 c^2 \sqrt {d} \sqrt {d+c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(226\) vs. \(2(108)=216\).
Time = 0.83 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.89
method | result | size |
default | \(\frac {a f \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+\frac {a g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}\right )\) | \(227\) |
parts | \(\frac {a f \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+\frac {a g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}\right )\) | \(227\) |
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\[ \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \]
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\[ \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
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none
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.72 \[ \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {b f \operatorname {arsinh}\left (c x\right )^{2}}{2 \, c \sqrt {d}} - \frac {b g x}{c \sqrt {d}} + \frac {a f \operatorname {arsinh}\left (c x\right )}{c \sqrt {d}} + \frac {\sqrt {c^{2} d x^{2} + d} b g \operatorname {arsinh}\left (c x\right )}{c^{2} d} + \frac {\sqrt {c^{2} d x^{2} + d} a g}{c^{2} d} \]
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\[ \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \]
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Timed out. \[ \int \frac {(f+g x) (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]
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