Integrand size = 26, antiderivative size = 267 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {16 b^2 d \sqrt {d+c^2 d x^2}}{75 c^2}+\frac {8 b^2 d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{225 c^2}+\frac {2 b^2 d \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{125 c^2}-\frac {2 b d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c \sqrt {1+c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d} \]
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Time = 0.15 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5798, 200, 5784, 12, 1261, 712} \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {2 b d x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c \sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {4 b c d x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{15 \sqrt {c^2 x^2+1}}-\frac {2 b c^3 d x^5 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{25 \sqrt {c^2 x^2+1}}+\frac {2 b^2 d \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}}{125 c^2}+\frac {16 b^2 d \sqrt {c^2 d x^2+d}}{75 c^2}+\frac {8 b^2 d \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{225 c^2} \]
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Rule 12
Rule 200
Rule 712
Rule 1261
Rule 5784
Rule 5798
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {\left (2 b d \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx}{5 c \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c \sqrt {1+c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {\left (2 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {1+c^2 x^2}} \, dx}{5 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c \sqrt {1+c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {\left (2 b^2 d \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx}{75 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c \sqrt {1+c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {15+10 c^2 x+3 c^4 x^2}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{75 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c \sqrt {1+c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}+\frac {\left (b^2 d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {1+c^2 x}}+4 \sqrt {1+c^2 x}+3 \left (1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \sqrt {1+c^2 x^2}} \\ & = \frac {16 b^2 d \sqrt {d+c^2 d x^2}}{75 c^2}+\frac {8 b^2 d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{225 c^2}+\frac {2 b^2 d \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{125 c^2}-\frac {2 b d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c \sqrt {1+c^2 x^2}}-\frac {4 b c d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d} \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.74 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \sqrt {d+c^2 d x^2} \left (225 a^2 \left (1+c^2 x^2\right )^3-30 a b c x \sqrt {1+c^2 x^2} \left (15+10 c^2 x^2+3 c^4 x^4\right )+2 b^2 \left (149+187 c^2 x^2+47 c^4 x^4+9 c^6 x^6\right )+30 b \left (15 a \left (1+c^2 x^2\right )^3-b c x \sqrt {1+c^2 x^2} \left (15+10 c^2 x^2+3 c^4 x^4\right )\right ) \text {arcsinh}(c x)+225 b^2 \left (1+c^2 x^2\right )^3 \text {arcsinh}(c x)^2\right )}{1125 c^2 \left (1+c^2 x^2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1148\) vs. \(2(233)=466\).
Time = 0.37 (sec) , antiderivative size = 1149, normalized size of antiderivative = 4.30
method | result | size |
default | \(\text {Expression too large to display}\) | \(1149\) |
parts | \(\text {Expression too large to display}\) | \(1149\) |
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Time = 0.27 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.24 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {225 \, {\left (b^{2} c^{6} d x^{6} + 3 \, b^{2} c^{4} d x^{4} + 3 \, b^{2} c^{2} d x^{2} + b^{2} d\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (15 \, a b c^{6} d x^{6} + 45 \, a b c^{4} d x^{4} + 45 \, a b c^{2} d x^{2} + 15 \, a b d - {\left (3 \, b^{2} c^{5} d x^{5} + 10 \, b^{2} c^{3} d x^{3} + 15 \, b^{2} c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (9 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} d x^{6} + {\left (675 \, a^{2} + 94 \, b^{2}\right )} c^{4} d x^{4} + {\left (675 \, a^{2} + 374 \, b^{2}\right )} c^{2} d x^{2} + {\left (225 \, a^{2} + 298 \, b^{2}\right )} d - 30 \, {\left (3 \, a b c^{5} d x^{5} + 10 \, a b c^{3} d x^{3} + 15 \, a b c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{1125 \, {\left (c^{4} x^{2} + c^{2}\right )}} \]
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\[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.86 \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \operatorname {arsinh}\left (c x\right )^{2}}{5 \, c^{2} d} + \frac {2}{1125} \, b^{2} {\left (\frac {9 \, \sqrt {c^{2} x^{2} + 1} c^{2} d^{\frac {5}{2}} x^{4} + 38 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {5}{2}} x^{2} + \frac {149 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {5}{2}}}{c^{2}}}{d} - \frac {15 \, {\left (3 \, c^{4} d^{\frac {5}{2}} x^{5} + 10 \, c^{2} d^{\frac {5}{2}} x^{3} + 15 \, d^{\frac {5}{2}} x\right )} \operatorname {arsinh}\left (c x\right )}{c d}\right )} + \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \operatorname {arsinh}\left (c x\right )}{5 \, c^{2} d} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a^{2}}{5 \, c^{2} d} - \frac {2 \, {\left (3 \, c^{4} d^{\frac {5}{2}} x^{5} + 10 \, c^{2} d^{\frac {5}{2}} x^{3} + 15 \, d^{\frac {5}{2}} x\right )} a b}{75 \, c d} \]
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Exception generated. \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
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