Integrand size = 26, antiderivative size = 217 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b d x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^3 \sqrt {d+c^2 d x^2}}{105 c \sqrt {1+c^2 x^2}}-\frac {8 b c d x^5 \sqrt {d+c^2 d x^2}}{175 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2} \]
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Time = 0.11 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {272, 45, 5804, 12, 380} \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\left (c^2 d x^2+d\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}-\frac {8 b c d x^5 \sqrt {c^2 d x^2+d}}{175 \sqrt {c^2 x^2+1}}-\frac {b d x^3 \sqrt {c^2 d x^2+d}}{105 c \sqrt {c^2 x^2+1}}+\frac {2 b d x \sqrt {c^2 d x^2+d}}{35 c^3 \sqrt {c^2 x^2+1}}-\frac {b c^3 d x^7 \sqrt {c^2 d x^2+d}}{49 \sqrt {c^2 x^2+1}} \]
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Rule 12
Rule 45
Rule 272
Rule 380
Rule 5804
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}-\frac {\left (b c \sqrt {d+c^2 d x^2}\right ) \int \frac {d \left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right )}{35 c^4} \, dx}{\sqrt {1+c^2 x^2}} \\ & = -\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}-\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right ) \, dx}{35 c^3 \sqrt {1+c^2 x^2}} \\ & = -\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2}-\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \int \left (-2+c^2 x^2+8 c^4 x^4+5 c^6 x^6\right ) \, dx}{35 c^3 \sqrt {1+c^2 x^2}} \\ & = \frac {2 b d x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^3 \sqrt {d+c^2 d x^2}}{105 c \sqrt {1+c^2 x^2}}-\frac {8 b c d x^5 \sqrt {d+c^2 d x^2}}{175 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^7 \sqrt {d+c^2 d x^2}}{49 \sqrt {1+c^2 x^2}}-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^4 d}+\frac {\left (d+c^2 d x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 d^2} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.60 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {d \sqrt {d+c^2 d x^2} \left (105 a \left (1+c^2 x^2\right )^3 \left (-2+5 c^2 x^2\right )-b c x \sqrt {1+c^2 x^2} \left (-210+35 c^2 x^2+168 c^4 x^4+75 c^6 x^6\right )+105 b \left (1+c^2 x^2\right )^3 \left (-2+5 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{3675 c^4 \left (1+c^2 x^2\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(871\) vs. \(2(185)=370\).
Time = 0.22 (sec) , antiderivative size = 872, normalized size of antiderivative = 4.02
method | result | size |
default | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}+56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}+7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{6272 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 ) d}{3200 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 ) d}{384 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \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 ) d}{128 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \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 ) d}{128 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 ) d}{384 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 ) d}{3200 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}-56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}-7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(872\) |
parts | \(a \left (\frac {x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{7 c^{2} d}-\frac {2 \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{35 d \,c^{4}}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}+64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}+112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}+56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}+7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{6272 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 ) d}{3200 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 ) d}{384 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \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 ) d}{128 c^{4} \left (c^{2} x^{2}+1\right )}-\frac {3 \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 ) d}{128 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 ) d}{384 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 ) d}{3200 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (64 c^{8} x^{8}-64 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+144 c^{6} x^{6}-112 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+104 c^{4} x^{4}-56 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+25 c^{2} x^{2}-7 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+7 \,\operatorname {arcsinh}\left (c x \right )\right ) d}{6272 c^{4} \left (c^{2} x^{2}+1\right )}\right )\) | \(872\) |
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Time = 0.27 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.92 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {105 \, {\left (5 \, b c^{8} d x^{8} + 13 \, b c^{6} d x^{6} + 9 \, b c^{4} d x^{4} - b c^{2} d x^{2} - 2 \, b d\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (525 \, a c^{8} d x^{8} + 1365 \, a c^{6} d x^{6} + 945 \, a c^{4} d x^{4} - 105 \, a c^{2} d x^{2} - 210 \, a d - {\left (75 \, b c^{7} d x^{7} + 168 \, b c^{5} d x^{5} + 35 \, b c^{3} d x^{3} - 210 \, b c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3675 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]
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\[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )\, dx \]
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Time = 0.20 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.67 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a - \frac {{\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} + 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} - 210 \, d^{\frac {3}{2}} x\right )} b}{3675 \, c^{3}} \]
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Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]
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