Integrand size = 24, antiderivative size = 180 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {73 b d^2 x \sqrt {1+c^2 x^2}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}-\frac {73 b d^2 \text {arcsinh}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x)) \]
[Out]
Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 45, 5803, 12, 1281, 470, 327, 221} \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))-\frac {73 b d^2 \text {arcsinh}(c x)}{3072 c^4}-\frac {43 b c d^2 x^5 \sqrt {c^2 x^2+1}}{1152}-\frac {73 b d^2 x^3 \sqrt {c^2 x^2+1}}{4608 c}+\frac {73 b d^2 x \sqrt {c^2 x^2+1}}{3072 c^3}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {c^2 x^2+1} \]
[In]
[Out]
Rule 12
Rule 45
Rule 221
Rule 272
Rule 327
Rule 470
Rule 1281
Rule 5803
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))-(b c) \int \frac {d^2 x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {1+c^2 x^2}} \, dx \\ & = \frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))-\frac {1}{24} \left (b c d^2\right ) \int \frac {x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))-\frac {\left (b d^2\right ) \int \frac {x^4 \left (48 c^2+43 c^4 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx}{192 c} \\ & = -\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))-\frac {\left (73 b c d^2\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{1152} \\ & = -\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))+\frac {\left (73 b d^2\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{1536 c} \\ & = \frac {73 b d^2 x \sqrt {1+c^2 x^2}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x))-\frac {\left (73 b d^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3072 c^3} \\ & = \frac {73 b d^2 x \sqrt {1+c^2 x^2}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}-\frac {73 b d^2 \text {arcsinh}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 d^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{8} c^4 d^2 x^8 (a+b \text {arcsinh}(c x)) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.64 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^2 \left (384 a c^4 x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )-b c x \sqrt {1+c^2 x^2} \left (-219+146 c^2 x^2+344 c^4 x^4+144 c^6 x^6\right )+3 b \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right ) \text {arcsinh}(c x)\right )}{9216 c^4} \]
[In]
[Out]
Time = 0.24 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.84
method | result | size |
parts | \(d^{2} a \left (\frac {1}{8} c^{4} x^{8}+\frac {1}{3} c^{2} x^{6}+\frac {1}{4} x^{4}\right )+\frac {d^{2} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{1152}-\frac {73 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {c^{2} x^{2}+1}}{3072}-\frac {73 \,\operatorname {arcsinh}\left (c x \right )}{3072}\right )}{c^{4}}\) | \(152\) |
derivativedivides | \(\frac {d^{2} a \left (\frac {1}{8} c^{8} x^{8}+\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{1152}-\frac {73 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {c^{2} x^{2}+1}}{3072}-\frac {73 \,\operatorname {arcsinh}\left (c x \right )}{3072}\right )}{c^{4}}\) | \(156\) |
default | \(\frac {d^{2} a \left (\frac {1}{8} c^{8} x^{8}+\frac {1}{3} c^{6} x^{6}+\frac {1}{4} c^{4} x^{4}\right )+d^{2} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{3}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {43 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{1152}-\frac {73 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4608}+\frac {73 c x \sqrt {c^{2} x^{2}+1}}{3072}-\frac {73 \,\operatorname {arcsinh}\left (c x \right )}{3072}\right )}{c^{4}}\) | \(156\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.89 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1152 \, a c^{8} d^{2} x^{8} + 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} d^{2} x^{8} + 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (144 \, b c^{7} d^{2} x^{7} + 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} - 219 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{9216 \, c^{4}} \]
[In]
[Out]
Time = 1.00 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.21 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{4} d^{2} x^{8}}{8} + \frac {a c^{2} d^{2} x^{6}}{3} + \frac {a d^{2} x^{4}}{4} + \frac {b c^{4} d^{2} x^{8} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {b c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1}}{64} + \frac {b c^{2} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {43 b c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{1152} + \frac {b d^{2} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {73 b d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4608 c} + \frac {73 b d^{2} x \sqrt {c^{2} x^{2} + 1}}{3072 c^{3}} - \frac {73 b d^{2} \operatorname {asinh}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.62 \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{8} \, a c^{4} d^{2} x^{8} + \frac {1}{3} \, a c^{2} d^{2} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} \]
[In]
[Out]
Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]
[In]
[Out]