Integrand size = 25, antiderivative size = 132 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^2}{d e}-\frac {8 b (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-(c+d x)^2\right )}{3 d e^2}+\frac {16 b^2 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};-(c+d x)^2\right )}{15 d e^3} \]
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Time = 0.15 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5859, 5776, 5817} \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {c e+d e x}} \, dx=\frac {16 b^2 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};-(c+d x)^2\right )}{15 d e^3}-\frac {8 b (e (c+d x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-(c+d x)^2\right ) (a+b \text {arcsinh}(c+d x))}{3 d e^2}+\frac {2 \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^2}{d e} \]
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Rule 5776
Rule 5817
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arcsinh}(x))^2}{\sqrt {e x}} \, dx,x,c+d x\right )}{d} \\ & = \frac {2 \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^2}{d e}-\frac {(4 b) \text {Subst}\left (\int \frac {\sqrt {e x} (a+b \text {arcsinh}(x))}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e} \\ & = \frac {2 \sqrt {e (c+d x)} (a+b \text {arcsinh}(c+d x))^2}{d e}-\frac {8 b (e (c+d x))^{3/2} (a+b \text {arcsinh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-(c+d x)^2\right )}{3 d e^2}+\frac {16 b^2 (e (c+d x))^{5/2} \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};-(c+d x)^2\right )}{15 d e^3} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} \left (15 (a+b \text {arcsinh}(c+d x))^2-20 b (c+d x) (a+b \text {arcsinh}(c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-(c+d x)^2\right )+8 b^2 (c+d x)^2 \, _3F_2\left (1,\frac {5}{4},\frac {5}{4};\frac {7}{4},\frac {9}{4};-(c+d x)^2\right )\right )}{15 d e} \]
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\[\int \frac {\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}{\sqrt {d e x +c e}}d x\]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {c e+d e x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{\sqrt {d e x + c e}} \,d x } \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {c e+d e x}} \, dx=\int \frac {\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{2}}{\sqrt {e \left (c + d x\right )}}\, dx \]
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Exception generated. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {c e+d e x}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {c e+d e x}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2}}{\sqrt {d e x + c e}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arcsinh}(c+d x))^2}{\sqrt {c e+d e x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2}{\sqrt {c\,e+d\,e\,x}} \,d x \]
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