Integrand size = 23, antiderivative size = 142 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=-\frac {4 b \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 d e}+\frac {2 b \sqrt {e} (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{9 d \sqrt {1+(c+d x)^2}} \]
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Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5859, 5776, 327, 335, 226} \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 d e}+\frac {2 b \sqrt {e} (c+d x+1) \sqrt {\frac {(c+d x)^2+1}{(c+d x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{9 d \sqrt {(c+d x)^2+1}}-\frac {4 b \sqrt {(c+d x)^2+1} \sqrt {e (c+d x)}}{9 d} \]
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Rule 226
Rule 327
Rule 335
Rule 5776
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {e x} (a+b \text {arcsinh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e} \\ & = -\frac {4 b \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 d e}+\frac {(2 b e) \text {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x^2}} \, dx,x,c+d x\right )}{9 d} \\ & = -\frac {4 b \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 d e}+\frac {(4 b) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{e^2}}} \, dx,x,\sqrt {e (c+d x)}\right )}{9 d} \\ & = -\frac {4 b \sqrt {e (c+d x)} \sqrt {1+(c+d x)^2}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x))}{3 d e}+\frac {2 b \sqrt {e} (1+c+d x) \sqrt {\frac {1+(c+d x)^2}{(1+c+d x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),\frac {1}{2}\right )}{9 d \sqrt {1+(c+d x)^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.03 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.61 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\frac {2 \sqrt {e (c+d x)} \left (3 a c+3 a d x-2 b \sqrt {1+(c+d x)^2}+3 b c \text {arcsinh}(c+d x)+3 b d x \text {arcsinh}(c+d x)+2 b \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-(c+d x)^2\right )\right )}{9 d} \]
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Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.26
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}-\frac {e^{2} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{3 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e}\) | \(179\) |
default | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}-\frac {e^{2} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{3 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e}\) | \(179\) |
parts | \(\frac {2 a \left (d e x +c e \right )^{\frac {3}{2}}}{3 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arcsinh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\frac {e^{2} \sqrt {d e x +c e}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}{3}-\frac {e^{2} \sqrt {1-\frac {i \left (d e x +c e \right )}{e}}\, \sqrt {1+\frac {i \left (d e x +c e \right )}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {\frac {i}{e}}, i\right )}{3 \sqrt {\frac {i}{e}}\, \sqrt {\frac {\left (d e x +c e \right )^{2}}{e^{2}}+1}}\right )}{3 e}\right )}{d e}\) | \(184\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=-\frac {2 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} \sqrt {d e x + c e} b d^{2} - 3 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, \sqrt {d^{3} e} b {\rm weierstrassPInverse}\left (-\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) - 3 \, {\left (a d^{3} x + a c d^{2}\right )} \sqrt {d e x + c e}\right )}}{9 \, d^{3}} \]
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\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )\, dx \]
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Exception generated. \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\text {Exception raised: ValueError} \]
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\[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int \sqrt {c e+d e x} (a+b \text {arcsinh}(c+d x)) \, dx=\int \sqrt {c\,e+d\,e\,x}\,\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right ) \,d x \]
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