Integrand size = 24, antiderivative size = 97 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {b x^3}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {b \text {arcsinh}(c x)}{4 c^4 d^3}+\frac {x^4 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5800, 294, 221} \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {x^4 (a+b \text {arcsinh}(c x))}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {b \text {arcsinh}(c x)}{4 c^4 d^3}+\frac {b x^3}{12 c d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {c^2 x^2+1}} \]
[In]
[Out]
Rule 221
Rule 294
Rule 5800
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {(b c) \int \frac {x^4}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3} \\ & = \frac {b x^3}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {x^4 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b \int \frac {x^2}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{4 c d^3} \\ & = \frac {b x^3}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {1+c^2 x^2}}+\frac {x^4 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 c^3 d^3} \\ & = \frac {b x^3}{12 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x}{4 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {b \text {arcsinh}(c x)}{4 c^4 d^3}+\frac {x^4 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {-3 a \left (1+2 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (3+4 c^2 x^2\right )-3 \left (b+2 b c^2 x^2\right ) \text {arcsinh}(c x)}{12 c^4 d^3 \left (1+c^2 x^2\right )^2} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{2 \left (c^{2} x^{2}+1\right )}\right )}{d^{3}}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}}\) | \(108\) |
default | \(\frac {\frac {a \left (\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{2 \left (c^{2} x^{2}+1\right )}\right )}{d^{3}}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}}\) | \(108\) |
parts | \(\frac {a \left (-\frac {1}{2 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {1}{4 c^{4} \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b \left (\frac {\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3} c^{4}}\) | \(113\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {3 \, a c^{4} x^{4} - 3 \, {\left (2 \, b c^{2} x^{2} + b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (4 \, b c^{3} x^{3} + 3 \, b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
[In]
[Out]
\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a x^{3}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{3} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]
[In]
[Out]
\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} + d\right )}^{3}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^3} \, dx=\int \frac {x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
[In]
[Out]