Integrand size = 20, antiderivative size = 670 \[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}-\frac {3 d^2 e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}+\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}-\frac {5 e^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^7}+\frac {3 d^2 e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}-\frac {9 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}+\frac {9 e^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {3 d e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {5 e^3 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {e^3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}+\frac {3 d^2 e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}-\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}+\frac {5 e^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^7}-\frac {3 d^2 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}+\frac {9 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {9 e^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {3 d e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}+\frac {5 e^3 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {e^3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7} \]
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Time = 0.84 (sec) , antiderivative size = 670, normalized size of antiderivative = 1.00, number of steps used = 42, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5793, 5774, 3384, 3379, 3382, 5780, 5556} \[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=-\frac {5 e^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^7}+\frac {9 e^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {5 e^3 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {e^3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {5 e^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b c^7}-\frac {9 e^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {5 e^3 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}-\frac {e^3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b c^7}+\frac {3 d e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}-\frac {9 d e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}+\frac {3 d e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {3 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 b c^5}+\frac {9 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {3 d e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 b c^5}-\frac {3 d^2 e \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}+\frac {3 d^2 e \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}+\frac {3 d^2 e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 b c^3}-\frac {3 d^2 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 b c^3}+\frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5774
Rule 5780
Rule 5793
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3}{a+b \text {arcsinh}(c x)}+\frac {3 d^2 e x^2}{a+b \text {arcsinh}(c x)}+\frac {3 d e^2 x^4}{a+b \text {arcsinh}(c x)}+\frac {e^3 x^6}{a+b \text {arcsinh}(c x)}\right ) \, dx \\ & = d^3 \int \frac {1}{a+b \text {arcsinh}(c x)} \, dx+\left (3 d^2 e\right ) \int \frac {x^2}{a+b \text {arcsinh}(c x)} \, dx+\left (3 d e^2\right ) \int \frac {x^4}{a+b \text {arcsinh}(c x)} \, dx+e^3 \int \frac {x^6}{a+b \text {arcsinh}(c x)} \, dx \\ & = \frac {d^3 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c}+\frac {\left (3 d^2 e\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3}+\frac {\left (3 d e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^5}+\frac {e^3 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^6\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^7} \\ & = \frac {\left (3 d^2 e\right ) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}-\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^3}+\frac {\left (3 d e^2\right ) \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}-\frac {3 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^5}+\frac {e^3 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 x}-\frac {5 \cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}+\frac {9 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}-\frac {5 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{64 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c^7}+\frac {\left (d^3 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c}-\frac {\left (d^3 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b c} \\ & = \frac {d^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}-\frac {d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b c}+\frac {\left (3 d^2 e\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b c^3}-\frac {\left (3 d^2 e\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b c^3}+\frac {\left (3 d e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b c^5}+\frac {\left (3 d e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b c^5}-\frac {\left (9 d e^2\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b c^5}+\frac {e^3 \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^7}-\frac {\left (5 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^7}-\frac {\left (5 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^7}+\frac {\left (9 e^3\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b c^7} \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\frac {\left (64 c^6 d^3-48 c^4 d^2 e+24 c^2 d e^2-5 e^3\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+3 e \left (16 c^4 d^2-12 c^2 d e+3 e^2\right ) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+12 c^2 d e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-5 e^3 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+e^3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-64 c^6 d^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+48 c^4 d^2 e \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-24 c^2 d e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+5 e^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-48 c^4 d^2 e \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+36 c^2 d e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-9 e^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-12 c^2 d e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 e^3 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-e^3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b c^7} \]
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Time = 1.76 (sec) , antiderivative size = 654, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {e^{3} {\mathrm e}^{\frac {7 a}{b}} \operatorname {Ei}_{1}\left (7 \,\operatorname {arcsinh}\left (c x \right )+\frac {7 a}{b}\right )}{128 c^{6} b}-\frac {e^{3} {\mathrm e}^{-\frac {7 a}{b}} \operatorname {Ei}_{1}\left (-7 \,\operatorname {arcsinh}\left (c x \right )-\frac {7 a}{b}\right )}{128 c^{6} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{2} e}{8 c^{2} b}-\frac {3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d \,e^{2}}{16 c^{4} b}+\frac {5 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e^{3}}{128 c^{6} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{2} e}{8 c^{2} b}-\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d \,e^{2}}{16 c^{4} b}+\frac {5 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e^{3}}{128 c^{6} b}-\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right ) d^{2}}{8 c^{2} b}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right ) d}{32 c^{4} b}-\frac {9 e^{3} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{128 c^{6} b}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) d^{2}}{8 c^{2} b}+\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) d}{32 c^{4} b}-\frac {9 e^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{128 c^{6} b}-\frac {3 e^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right ) d}{32 c^{4} b}+\frac {5 e^{3} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right )}{128 c^{6} b}-\frac {3 e^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right ) d}{32 c^{4} b}+\frac {5 e^{3} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right )}{128 c^{6} b}}{c}\) | \(654\) |
default | \(\frac {-\frac {e^{3} {\mathrm e}^{\frac {7 a}{b}} \operatorname {Ei}_{1}\left (7 \,\operatorname {arcsinh}\left (c x \right )+\frac {7 a}{b}\right )}{128 c^{6} b}-\frac {e^{3} {\mathrm e}^{-\frac {7 a}{b}} \operatorname {Ei}_{1}\left (-7 \,\operatorname {arcsinh}\left (c x \right )-\frac {7 a}{b}\right )}{128 c^{6} b}-\frac {{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d^{2} e}{8 c^{2} b}-\frac {3 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) d \,e^{2}}{16 c^{4} b}+\frac {5 \,{\mathrm e}^{\frac {a}{b}} \operatorname {Ei}_{1}\left (\operatorname {arcsinh}\left (c x \right )+\frac {a}{b}\right ) e^{3}}{128 c^{6} b}-\frac {{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{3}}{2 b}+\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d^{2} e}{8 c^{2} b}-\frac {3 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) d \,e^{2}}{16 c^{4} b}+\frac {5 \,{\mathrm e}^{-\frac {a}{b}} \operatorname {Ei}_{1}\left (-\operatorname {arcsinh}\left (c x \right )-\frac {a}{b}\right ) e^{3}}{128 c^{6} b}-\frac {3 e \,{\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right ) d^{2}}{8 c^{2} b}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right ) d}{32 c^{4} b}-\frac {9 e^{3} {\mathrm e}^{\frac {3 a}{b}} \operatorname {Ei}_{1}\left (3 \,\operatorname {arcsinh}\left (c x \right )+\frac {3 a}{b}\right )}{128 c^{6} b}-\frac {3 e \,{\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) d^{2}}{8 c^{2} b}+\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right ) d}{32 c^{4} b}-\frac {9 e^{3} {\mathrm e}^{-\frac {3 a}{b}} \operatorname {Ei}_{1}\left (-3 \,\operatorname {arcsinh}\left (c x \right )-\frac {3 a}{b}\right )}{128 c^{6} b}-\frac {3 e^{2} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right ) d}{32 c^{4} b}+\frac {5 e^{3} {\mathrm e}^{\frac {5 a}{b}} \operatorname {Ei}_{1}\left (5 \,\operatorname {arcsinh}\left (c x \right )+\frac {5 a}{b}\right )}{128 c^{6} b}-\frac {3 e^{2} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right ) d}{32 c^{4} b}+\frac {5 e^{3} {\mathrm e}^{-\frac {5 a}{b}} \operatorname {Ei}_{1}\left (-5 \,\operatorname {arcsinh}\left (c x \right )-\frac {5 a}{b}\right )}{128 c^{6} b}}{c}\) | \(654\) |
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\[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {\left (d + e x^{2}\right )^{3}}{a + b \operatorname {asinh}{\left (c x \right )}}\, dx \]
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\[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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\[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{b \operatorname {arsinh}\left (c x\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^2\right )^3}{a+b \text {arcsinh}(c x)} \, dx=\int \frac {{\left (e\,x^2+d\right )}^3}{a+b\,\mathrm {asinh}\left (c\,x\right )} \,d x \]
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