Integrand size = 26, antiderivative size = 175 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {b x^3 \sqrt {d+c^2 d x^2}}{45 c \sqrt {1+c^2 x^2}}-\frac {b c x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2} \]
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Time = 0.09 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45, 5804, 12} \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}-\frac {b c x^5 \sqrt {c^2 d x^2+d}}{25 \sqrt {c^2 x^2+1}}-\frac {b x^3 \sqrt {c^2 d x^2+d}}{45 c \sqrt {c^2 x^2+1}}+\frac {2 b x \sqrt {c^2 d x^2+d}}{15 c^3 \sqrt {c^2 x^2+1}} \]
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Rule 12
Rule 45
Rule 272
Rule 5804
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2}-\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int \frac {-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2}-\frac {\left (b \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{15 c^3 \sqrt {1+c^2 x^2}} \\ & = \frac {2 b x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {b x^3 \sqrt {d+c^2 d x^2}}{45 c \sqrt {1+c^2 x^2}}-\frac {b c x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d^2} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {d+c^2 d x^2} \left (15 a \left (1+c^2 x^2\right )^2 \left (-2+3 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (30-5 c^2 x^2-9 c^4 x^4\right )+15 b \left (1+c^2 x^2\right )^2 \left (-2+3 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{225 c^4 \left (1+c^2 x^2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(577\) vs. \(2(149)=298\).
Time = 0.27 (sec) , antiderivative size = 578, normalized size of antiderivative = 3.30
method | result | size |
default | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{5 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{15 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (c x \right )\right )}{800 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{288 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{16 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{288 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \,\operatorname {arcsinh}\left (c x \right )\right )}{800 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(578\) |
parts | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{5 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{15 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}+16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}+5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+5 \,\operatorname {arcsinh}\left (c x \right )\right )}{800 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{288 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{16 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{288 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (16 c^{6} x^{6}-16 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+28 c^{4} x^{4}-20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+13 c^{2} x^{2}-5 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+5 \,\operatorname {arcsinh}\left (c x \right )\right )}{800 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(578\) |
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Time = 0.27 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.90 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {15 \, {\left (3 \, b c^{6} x^{6} + 4 \, b c^{4} x^{4} - b c^{2} x^{2} - 2 \, b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (45 \, a c^{6} x^{6} + 60 \, a c^{4} x^{4} - 15 \, a c^{2} x^{2} - {\left (9 \, b c^{5} x^{5} + 5 \, b c^{3} x^{3} - 30 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 30 \, a\right )} \sqrt {c^{2} d x^{2} + d}}{225 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]
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\[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^{3} \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \]
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Time = 0.21 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.77 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{15} \, b {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, a {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} - \frac {{\left (9 \, c^{4} \sqrt {d} x^{5} + 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} b}{225 \, c^{3}} \]
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Exception generated. \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \]
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