Integrand size = 28, antiderivative size = 265 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {4 a b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {14 b^2 \left (1+c^2 x^2\right )}{9 c^4 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^2}{27 c^4 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d} \]
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Time = 0.22 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5812, 5798, 5772, 267, 5776, 272, 45} \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c \sqrt {c^2 d x^2+d}}-\frac {2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^4 d}+\frac {4 a b x \sqrt {c^2 x^2+1}}{3 c^3 \sqrt {c^2 d x^2+d}}+\frac {4 b^2 x \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{3 c^3 \sqrt {c^2 d x^2+d}}+\frac {2 b^2 \left (c^2 x^2+1\right )^2}{27 c^4 \sqrt {c^2 d x^2+d}}-\frac {14 b^2 \left (c^2 x^2+1\right )}{9 c^4 \sqrt {c^2 d x^2+d}} \]
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Rule 45
Rule 267
Rule 272
Rule 5772
Rule 5776
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{3 c^2}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int x^2 (a+b \text {arcsinh}(c x)) \, dx}{3 c \sqrt {d+c^2 d x^2}} \\ & = -\frac {2 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2}} \, dx}{9 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{3 c^3 \sqrt {d+c^2 d x^2}} \\ & = \frac {4 a b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{9 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \int \text {arcsinh}(c x) \, dx}{3 c^3 \sqrt {d+c^2 d x^2}} \\ & = \frac {4 a b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2 \sqrt {1+c^2 x}}+\frac {\sqrt {1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{9 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^2 \sqrt {d+c^2 d x^2}} \\ & = \frac {4 a b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {14 b^2 \left (1+c^2 x^2\right )}{9 c^4 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^2}{27 c^4 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^2 d} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.66 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {-6 a b c x \left (-6+c^2 x^2\right ) \sqrt {1+c^2 x^2}+2 b^2 \left (-20-19 c^2 x^2+c^4 x^4\right )+9 a^2 \left (-2-c^2 x^2+c^4 x^4\right )-6 b \left (b c x \left (-6+c^2 x^2\right ) \sqrt {1+c^2 x^2}+a \left (6+3 c^2 x^2-3 c^4 x^4\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (-2-c^2 x^2+c^4 x^4\right ) \text {arcsinh}(c x)^2}{27 c^4 \sqrt {d+c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(705\) vs. \(2(231)=462\).
Time = 0.29 (sec) , antiderivative size = 706, normalized size of antiderivative = 2.66
method | result | size |
default | \(a^{2} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (9 \operatorname {arcsinh}\left (c x \right )^{2}-6 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (9 \operatorname {arcsinh}\left (c x \right )^{2}+6 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\) | \(706\) |
parts | \(a^{2} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (9 \operatorname {arcsinh}\left (c x \right )^{2}-6 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (9 \operatorname {arcsinh}\left (c x \right )^{2}+6 \,\operatorname {arcsinh}\left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\) | \(706\) |
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Time = 0.29 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {9 \, {\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b c^{4} x^{4} - 3 \, a b c^{2} x^{2} - 6 \, a b - {\left (b^{2} c^{3} x^{3} - 6 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} - {\left (9 \, a^{2} + 38 \, b^{2}\right )} c^{2} x^{2} - 18 \, a^{2} - 40 \, b^{2} - 6 \, {\left (a b c^{3} x^{3} - 6 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{27 \, {\left (c^{6} d x^{2} + c^{4} d\right )}} \]
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\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.92 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {1}{3} \, b^{2} {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{3} \, a b {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a^{2} {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} + \frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {c^{2} x^{2} + 1}}{c^{2}}}{c^{2} \sqrt {d}} - \frac {3 \, {\left (c^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (c x\right )}{c^{3} \sqrt {d}}\right )} - \frac {2 \, {\left (c^{2} x^{3} - 6 \, x\right )} a b}{9 \, c^{3} \sqrt {d}} \]
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Exception generated. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \]
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