Integrand size = 16, antiderivative size = 81 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \left (3 c^2 d-e\right ) \sqrt {1+c^2 x^2}}{3 c^3}-\frac {b e \left (1+c^2 x^2\right )^{3/2}}{9 c^3}+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x)) \]
[Out]
Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5792, 455, 45} \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))-\frac {b \sqrt {c^2 x^2+1} \left (3 c^2 d-e\right )}{3 c^3}-\frac {b e \left (c^2 x^2+1\right )^{3/2}}{9 c^3} \]
[In]
[Out]
Rule 45
Rule 455
Rule 5792
Rubi steps \begin{align*} \text {integral}& = d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))-(b c) \int \frac {x \left (d+\frac {e x^2}{3}\right )}{\sqrt {1+c^2 x^2}} \, dx \\ & = d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))-\frac {1}{2} (b c) \text {Subst}\left (\int \frac {d+\frac {e x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right ) \\ & = d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x))-\frac {1}{2} (b c) \text {Subst}\left (\int \left (\frac {3 c^2 d-e}{3 c^2 \sqrt {1+c^2 x}}+\frac {e \sqrt {1+c^2 x}}{3 c^2}\right ) \, dx,x,x^2\right ) \\ & = -\frac {b \left (3 c^2 d-e\right ) \sqrt {1+c^2 x^2}}{3 c^3}-\frac {b e \left (1+c^2 x^2\right )^{3/2}}{9 c^3}+d x (a+b \text {arcsinh}(c x))+\frac {1}{3} e x^3 (a+b \text {arcsinh}(c x)) \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{9} \left (3 a x \left (3 d+e x^2\right )-\frac {b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )}{c^3}+3 b x \left (3 d+e x^2\right ) \text {arcsinh}(c x)\right ) \]
[In]
[Out]
Time = 0.02 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.20
method | result | size |
parts | \(a \left (\frac {1}{3} x^{3} e +d x \right )+\frac {b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) x^{3} e}{3}+\operatorname {arcsinh}\left (c x \right ) d c x -\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d \,c^{2} \sqrt {c^{2} x^{2}+1}}{3 c^{2}}\right )}{c}\) | \(97\) |
derivativedivides | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(109\) |
default | \(\frac {\frac {a \left (d \,c^{3} x +\frac {1}{3} e \,c^{3} x^{3}\right )}{c^{2}}+\frac {b \left (\operatorname {arcsinh}\left (c x \right ) d \,c^{3} x +\frac {\operatorname {arcsinh}\left (c x \right ) e \,c^{3} x^{3}}{3}-\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\) | \(109\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.16 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {3 \, a c^{3} e x^{3} + 9 \, a c^{3} d x + 3 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c^{2} e x^{2} + 9 \, b c^{2} d - 2 \, b e\right )} \sqrt {c^{2} x^{2} + 1}}{9 \, c^{3}} \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.35 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} a d x + \frac {a e x^{3}}{3} + b d x \operatorname {asinh}{\left (c x \right )} + \frac {b e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {b e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {2 b e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d x + \frac {e x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12 \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{3} \, a e x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} b d}{c} \]
[In]
[Out]
Exception generated. \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int \left (d+e x^2\right ) (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
[In]
[Out]