Integrand size = 26, antiderivative size = 277 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=-\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d}+\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {2 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^5 d} \]
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Time = 0.36 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5812, 5789, 4265, 2611, 2320, 6724, 5798, 8, 30} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{c^5 d}-\frac {2 i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^5 d}+\frac {2 i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^5 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {22 b \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c^3 d}+\frac {2 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d} \]
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Rule 8
Rule 30
Rule 2320
Rule 2611
Rule 4265
Rule 5789
Rule 5798
Rule 5812
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx}{c^2}-\frac {(2 b) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{3 c d} \\ & = -\frac {2 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx}{c^4}+\frac {(4 b) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{9 c^3 d}+\frac {(2 b) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{c^3 d}+\frac {\left (2 b^2\right ) \int x^2 \, dx}{9 c^2 d} \\ & = \frac {2 b^2 x^3}{27 c^2 d}+\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {\text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d}-\frac {\left (4 b^2\right ) \int 1 \, dx}{9 c^4 d}-\frac {\left (2 b^2\right ) \int 1 \, dx}{c^4 d} \\ & = -\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d}+\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {(2 i b) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d}+\frac {(2 i b) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d} \\ & = -\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d}+\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d}-\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d} \\ & = -\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d}+\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {\left (2 i b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{c^5 d} \\ & = -\frac {22 b^2 x}{9 c^4 d}+\frac {2 b^2 x^3}{27 c^2 d}+\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^5 d}-\frac {2 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3 d}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d}+\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {2 (a+b \text {arcsinh}(c x))^2 \arctan \left (e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {2 i b (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^5 d}+\frac {2 i b^2 \operatorname {PolyLog}\left (3,-i e^{\text {arcsinh}(c x)}\right )}{c^5 d}-\frac {2 i b^2 \operatorname {PolyLog}\left (3,i e^{\text {arcsinh}(c x)}\right )}{c^5 d} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.32 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {-3 a^2 c x+a^2 c^3 x^3+3 a^2 \arctan (c x)-\frac {2}{3} a b \left (-11 \sqrt {1+c^2 x^2}+c^2 x^2 \sqrt {1+c^2 x^2}+9 c x \text {arcsinh}(c x)-3 c^3 x^3 \text {arcsinh}(c x)-9 i \text {arcsinh}(c x) \log \left (1-i e^{\text {arcsinh}(c x)}\right )+9 i \text {arcsinh}(c x) \log \left (1+i e^{\text {arcsinh}(c x)}\right )+9 i \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )-9 i \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )\right )+3 b^2 \left (\frac {5}{2} \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-\frac {5}{4} c x \left (2+\text {arcsinh}(c x)^2\right )-\frac {1}{18} \text {arcsinh}(c x) \cosh (3 \text {arcsinh}(c x))+i \left (-\text {arcsinh}(c x)^2 \left (\log \left (1-i e^{-\text {arcsinh}(c x)}\right )-\log \left (1+i e^{-\text {arcsinh}(c x)}\right )\right )-2 \text {arcsinh}(c x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{-\text {arcsinh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{-\text {arcsinh}(c x)}\right )\right )+\frac {1}{108} \left (2+9 \text {arcsinh}(c x)^2\right ) \sinh (3 \text {arcsinh}(c x))\right )}{3 c^5 d} \]
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\[\int \frac {x^{4} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}{c^{2} d \,x^{2}+d}d x\]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} + d} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\frac {\int \frac {a^{2} x^{4}}{c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} + 1}\, dx}{d} \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{c^{2} d x^{2} + d} \,d x } \]
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Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{d+c^2 d x^2} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{d\,c^2\,x^2+d} \,d x \]
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