Integrand size = 26, antiderivative size = 180 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {4 b^2 \sqrt {d+c^2 d x^2}}{9 c^2}+\frac {2 b^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{27 c^2}-\frac {2 b x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d} \]
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Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5798, 5784, 455, 45} \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {2 b x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c \sqrt {c^2 x^2+1}}+\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b c x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{9 \sqrt {c^2 x^2+1}}+\frac {2 b^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{27 c^2}+\frac {4 b^2 \sqrt {c^2 d x^2+d}}{9 c^2} \]
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Rule 45
Rule 455
Rule 5784
Rule 5798
Rubi steps \begin{align*} \text {integral}& = \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {\left (2 b \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x)) \, dx}{3 c \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {\left (2 b^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (1+\frac {c^2 x^2}{3}\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {\left (b^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {c^2 x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {\left (b^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1+c^2 x}}+\frac {1}{3} \sqrt {1+c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}} \\ & = \frac {4 b^2 \sqrt {d+c^2 d x^2}}{9 c^2}+\frac {2 b^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{27 c^2}-\frac {2 b x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d} \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.92 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d+c^2 d x^2} \left (-6 a b c x \sqrt {1+c^2 x^2} \left (3+c^2 x^2\right )+9 \left (a+a c^2 x^2\right )^2+2 b^2 \left (7+8 c^2 x^2+c^4 x^4\right )+6 b \left (3 a \left (1+c^2 x^2\right )^2-b c x \sqrt {1+c^2 x^2} \left (3+c^2 x^2\right )\right ) \text {arcsinh}(c x)+9 \left (b+b c^2 x^2\right )^2 \text {arcsinh}(c x)^2\right )}{27 c^2 \left (1+c^2 x^2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(656\) vs. \(2(156)=312\).
Time = 0.26 (sec) , antiderivative size = 657, normalized size of antiderivative = 3.65
method | result | size |
default | \(\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (9 \operatorname {arcsinh}\left (c x \right )^{2}-6 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{216 c^{2} \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 ) \left (\operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{8 c^{2} \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 ) \left (\operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (9 \operatorname {arcsinh}\left (c x \right )^{2}+6 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{216 c^{2} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{72 c^{2} \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 ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{8 c^{2} \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 ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\) | \(657\) |
parts | \(\frac {a^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (9 \operatorname {arcsinh}\left (c x \right )^{2}-6 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{216 c^{2} \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 ) \left (\operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{8 c^{2} \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 ) \left (\operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (9 \operatorname {arcsinh}\left (c x \right )^{2}+6 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{216 c^{2} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{72 c^{2} \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 ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{8 c^{2} \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 ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{8 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{72 c^{2} \left (c^{2} x^{2}+1\right )}\right )\) | \(657\) |
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Time = 0.27 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.38 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {9 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b c^{4} x^{4} + 6 \, a b c^{2} x^{2} + 3 \, a b - {\left (b^{2} c^{3} x^{3} + 3 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} + 2 \, {\left (9 \, a^{2} + 8 \, b^{2}\right )} c^{2} x^{2} + 9 \, a^{2} + 14 \, b^{2} - 6 \, {\left (a b c^{3} x^{3} + 3 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{27 \, {\left (c^{4} x^{2} + c^{2}\right )}} \]
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\[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.02 \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} + 1} d^{\frac {3}{2}} x^{2} + \frac {7 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {3}{2}}}{c^{2}}}{d} - \frac {3 \, {\left (c^{2} d^{\frac {3}{2}} x^{3} + 3 \, d^{\frac {3}{2}} x\right )} \operatorname {arsinh}\left (c x\right )}{c d}\right )} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} b^{2} \operatorname {arsinh}\left (c x\right )^{2}}{3 \, c^{2} d} + \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a b \operatorname {arsinh}\left (c x\right )}{3 \, c^{2} d} - \frac {2 \, {\left (c^{2} d^{\frac {3}{2}} x^{3} + 3 \, d^{\frac {3}{2}} x\right )} a b}{9 \, c d} + \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} a^{2}}{3 \, c^{2} d} \]
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Exception generated. \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d} \,d x \]
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