Integrand size = 26, antiderivative size = 206 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^5 d \sqrt {d+c^2 d x^2}} \]
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Time = 0.18 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5810, 5812, 5783, 30, 272, 45} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{4 b c^5 d \sqrt {c^2 d x^2+d}}+\frac {3 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{2 c^4 d^2}-\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c^5 d \sqrt {c^2 d x^2+d}}-\frac {b x^2 \sqrt {c^2 x^2+1}}{4 c^3 d \sqrt {c^2 d x^2+d}} \]
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Rule 30
Rule 45
Rule 272
Rule 5783
Rule 5810
Rule 5812
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx}{c^2 d}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}} \\ & = -\frac {x^3 (a+b \text {arcsinh}(c x))}{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 c^4 d^2}-\frac {3 \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx}{2 c^4 d}-\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 c^3 d \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{2 c d \sqrt {d+c^2 d x^2}} \\ & = -\frac {3 b x^2 \sqrt {1+c^2 x^2}}{4 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{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 c^4 d^2}-\frac {3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{2 c d \sqrt {d+c^2 d x^2}} \\ & = -\frac {b x^2 \sqrt {1+c^2 x^2}}{4 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))}{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 c^4 d^2}-\frac {3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^5 d \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 0.55 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.78 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {4 a c \sqrt {d} x \left (3+c^2 x^2\right )-12 a \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b \sqrt {d} \left (8 c x \text {arcsinh}(c x)-\sqrt {1+c^2 x^2} \left (6 \text {arcsinh}(c x)^2+\cosh (2 \text {arcsinh}(c x))+4 \log \left (1+c^2 x^2\right )-2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))\right )\right )}{8 c^5 d^{3/2} \sqrt {d+c^2 d x^2}} \]
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Time = 0.24 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.33
method | result | size |
default | \(\frac {a \,x^{3}}{2 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a x}{2 c^{4} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-8 \,\operatorname {arcsinh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{5} d^{2}}\) | \(274\) |
parts | \(\frac {a \,x^{3}}{2 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a x}{2 c^{4} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-12 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+3 c^{2} x^{2}+6 \operatorname {arcsinh}\left (c x \right )^{2}-8 \,\operatorname {arcsinh}\left (c x \right )+8 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+1\right )}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{5} d^{2}}\) | \(274\) |
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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