Integrand size = 28, antiderivative size = 291 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {b^2 x \sqrt {d+c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d+c^2 d x^2}-\frac {b^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 c^3 \sqrt {1+c^2 x^2}}-\frac {b x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {1+c^2 x^2}}-\frac {b c x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{24 b c^3 \sqrt {1+c^2 x^2}} \]
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Time = 0.28 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5806, 5812, 5783, 5776, 327, 221} \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {b x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{8 c \sqrt {c^2 x^2+1}}+\frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{8 c^2}-\frac {b c x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{8 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^3}{24 b c^3 \sqrt {c^2 x^2+1}}-\frac {b^2 \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{64 c^3 \sqrt {c^2 x^2+1}}+\frac {b^2 x \sqrt {c^2 d x^2+d}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {c^2 d x^2+d} \]
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Rule 221
Rule 327
Rule 5776
Rule 5783
Rule 5806
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {\sqrt {d+c^2 d x^2} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int x^3 (a+b \text {arcsinh}(c x)) \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\sqrt {d+c^2 d x^2} \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{8 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b \sqrt {d+c^2 d x^2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{4 c \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}} \\ & = \frac {1}{32} b^2 x^3 \sqrt {d+c^2 d x^2}-\frac {b x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {1+c^2 x^2}}-\frac {b c x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{24 b c^3 \sqrt {1+c^2 x^2}}-\frac {\left (3 b^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{32 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}} \\ & = \frac {b^2 x \sqrt {d+c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d+c^2 d x^2}-\frac {b x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {1+c^2 x^2}}-\frac {b c x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{24 b c^3 \sqrt {1+c^2 x^2}}+\frac {\left (3 b^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{64 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2 \sqrt {1+c^2 x^2}} \\ & = \frac {b^2 x \sqrt {d+c^2 d x^2}}{64 c^2}+\frac {1}{32} b^2 x^3 \sqrt {d+c^2 d x^2}-\frac {b^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{64 c^3 \sqrt {1+c^2 x^2}}-\frac {b x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {1+c^2 x^2}}-\frac {b c x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 \sqrt {1+c^2 x^2}}+\frac {x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^2}+\frac {1}{4} x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{24 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 1.30 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.71 \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {-96 a^2 c x \left (1+2 c^2 x^2\right ) \sqrt {d+c^2 d x^2}+96 a^2 \sqrt {d} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {12 a b \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{\sqrt {1+c^2 x^2}}+\frac {b^2 \sqrt {d+c^2 d x^2} \left (32 \text {arcsinh}(c x)^3+12 \text {arcsinh}(c x) \cosh (4 \text {arcsinh}(c x))-3 \left (1+8 \text {arcsinh}(c x)^2\right ) \sinh (4 \text {arcsinh}(c x))\right )}{\sqrt {1+c^2 x^2}}}{768 c^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(617\) vs. \(2(251)=502\).
Time = 0.27 (sec) , antiderivative size = 618, normalized size of antiderivative = 2.12
method | result | size |
default | \(\frac {a^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{2}}-\frac {a^{2} d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{3}}{24 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (c x \right )^{2}-4 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{512 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (c x \right )^{2}+4 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{512 c^{3} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\) | \(618\) |
parts | \(\frac {a^{2} x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{2}}-\frac {a^{2} d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{2} \sqrt {c^{2} d}}+b^{2} \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{3}}{24 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (c x \right )^{2}-4 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{512 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (c x \right )^{2}+4 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{512 c^{3} \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\) | \(618\) |
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\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^{2} \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
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Exception generated. \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { \sqrt {c^{2} d x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {d\,c^2\,x^2+d} \,d x \]
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