Integrand size = 28, antiderivative size = 400 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}} \]
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Time = 0.44 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5810, 5812, 5783, 5776, 327, 221, 5797, 3799, 2221, 2317, 2438} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{2 b c^5 d \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^5 d \sqrt {c^2 d x^2+d}}-\frac {2 b \sqrt {c^2 x^2+1} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))}{c^5 d \sqrt {c^2 d x^2+d}}+\frac {3 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {b x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^3 d \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {c^2 d x^2+d}}-\frac {b^2 \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{4 c^5 d \sqrt {c^2 d x^2+d}}+\frac {b^2 x \left (c^2 x^2+1\right )}{4 c^4 d \sqrt {c^2 d x^2+d}} \]
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Rule 221
Rule 327
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5776
Rule 5783
Rule 5797
Rule 5810
Rule 5812
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}} \\ & = \frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {3 \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{2 c^4 d}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{1+c^2 x^2} \, dx}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{c^2 d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2 x \left (1+c^2 x^2\right )}{2 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \tanh (x) \, dx,x,\text {arcsinh}(c x))}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 c^4 d \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 c^2 d \sqrt {d+c^2 d x^2}} \\ & = \frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{2 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 c^4 d \sqrt {d+c^2 d x^2}} \\ & = \frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arcsinh}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}} \\ & = \frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}} \\ & = \frac {b^2 x \left (1+c^2 x^2\right )}{4 c^4 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 1.70 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.72 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {4 a^2 c \sqrt {d} x \left (3+c^2 x^2\right )-12 a^2 \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b^2 \sqrt {d} \left (8 c x \text {arcsinh}(c x)^2+8 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+\sqrt {1+c^2 x^2} \left (-4 \text {arcsinh}(c x)^3-2 \text {arcsinh}(c x) \left (\cosh (2 \text {arcsinh}(c x))+8 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )+2 \text {arcsinh}(c x)^2 (-4+\sinh (2 \text {arcsinh}(c x)))+\sinh (2 \text {arcsinh}(c x))\right )\right )+2 a b \sqrt {d} \left (8 c x \text {arcsinh}(c x)-\sqrt {1+c^2 x^2} \left (6 \text {arcsinh}(c x)^2+\cosh (2 \text {arcsinh}(c x))+4 \log \left (1+c^2 x^2\right )-2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))\right )\right )}{8 c^5 d^{3/2} \sqrt {d+c^2 d x^2}} \]
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Time = 0.29 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-2 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+2 \operatorname {arcsinh}\left (c x \right )^{3} x^{2} c^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}+8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-6 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x +3 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 \operatorname {arcsinh}\left (c x \right )^{3}-c x \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2}+8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right )+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{4 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{5} d^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-8 \,\operatorname {arcsinh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{4 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{5} d^{2}}\) | \(579\) |
parts | \(\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-2 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+2 \operatorname {arcsinh}\left (c x \right )^{3} x^{2} c^{2}-c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}+8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-6 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x +3 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 \operatorname {arcsinh}\left (c x \right )^{3}-c x \sqrt {c^{2} x^{2}+1}-4 \operatorname {arcsinh}\left (c x \right )^{2}+8 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (c x \right )+4 \operatorname {polylog}\left (2, -\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{4 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{5} d^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-8 \,\operatorname {arcsinh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{4 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{5} d^{2}}\) | \(579\) |
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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