Integrand size = 18, antiderivative size = 239 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d^2 x-\frac {4 b^2 e^2 x}{9 c^2}+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3-\frac {2 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {b d e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {2 b e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}-\frac {d^3 (a+b \text {arcsinh}(c x))^2}{3 e}+\frac {d e (a+b \text {arcsinh}(c x))^2}{2 c^2}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e} \]
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Time = 0.35 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5828, 5838, 5783, 5798, 8, 5812, 30} \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {2 b d^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}-\frac {b d e x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}+\frac {d e (a+b \text {arcsinh}(c x))^2}{2 c^2}-\frac {2 b e^2 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c}+\frac {4 b e^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {d^3 (a+b \text {arcsinh}(c x))^2}{3 e}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}-\frac {4 b^2 e^2 x}{9 c^2}+2 b^2 d^2 x+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3 \]
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Rule 8
Rule 30
Rule 5783
Rule 5798
Rule 5812
Rule 5828
Rule 5838
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}-\frac {(2 b c) \int \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{3 e} \\ & = \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}-\frac {(2 b c) \int \left (\frac {d^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {3 d^2 e x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {3 d e^2 x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {e^3 x^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}\right ) \, dx}{3 e} \\ & = \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}-\left (2 b c d^2\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {\left (2 b c d^3\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{3 e}-(2 b c d e) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{3} \left (2 b c e^2\right ) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = -\frac {2 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {b d e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {2 b e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}-\frac {d^3 (a+b \text {arcsinh}(c x))^2}{3 e}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}+\left (2 b^2 d^2\right ) \int 1 \, dx+\left (b^2 d e\right ) \int x \, dx+\frac {(b d e) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{c}+\frac {1}{9} \left (2 b^2 e^2\right ) \int x^2 \, dx+\frac {\left (4 b e^2\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{9 c} \\ & = 2 b^2 d^2 x+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3-\frac {2 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {b d e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {2 b e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}-\frac {d^3 (a+b \text {arcsinh}(c x))^2}{3 e}+\frac {d e (a+b \text {arcsinh}(c x))^2}{2 c^2}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}-\frac {\left (4 b^2 e^2\right ) \int 1 \, dx}{9 c^2} \\ & = 2 b^2 d^2 x-\frac {4 b^2 e^2 x}{9 c^2}+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3-\frac {2 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {b d e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {2 b e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}-\frac {d^3 (a+b \text {arcsinh}(c x))^2}{3 e}+\frac {d e (a+b \text {arcsinh}(c x))^2}{2 c^2}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.04 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {18 a^2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+b^2 c x \left (-24 e^2+c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )-6 b \left (-3 a \left (3 c d e+2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right )+b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right ) \text {arcsinh}(c x)+9 b^2 c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d \left (e+2 c^2 e x^2\right )\right ) \text {arcsinh}(c x)^2}{54 c^3} \]
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Time = 0.46 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.52
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {e^{2} \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {d c e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{2}+d^{2} c^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(364\) |
default | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {e^{2} \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {d c e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{2}+d^{2} c^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(364\) |
parts | \(\frac {a^{2} \left (e x +d \right )^{3}}{3 e}+\frac {b^{2} \left (18 \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3} e^{2}+54 \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{2} d e +54 \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x \,d^{2}-12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2} e^{2}-54 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x d e -108 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} d^{2}+27 \operatorname {arcsinh}\left (c x \right )^{2} c d e +4 c^{3} x^{3} e^{2}+27 c^{3} x^{2} d e +108 c^{3} x \,d^{2}+24 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, e^{2}-24 c x \,e^{2}+27 c d e \right )}{54 c^{3}}+\frac {2 a b \left (\frac {c \,e^{2} \operatorname {arcsinh}\left (c x \right ) x^{3}}{3}+c \,\operatorname {arcsinh}\left (c x \right ) d e \,x^{2}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{2}+\frac {c \,\operatorname {arcsinh}\left (c x \right ) d^{3}}{3 e}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 c^{2} e}\right )}{c}\) | \(380\) |
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Time = 0.26 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.33 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {2 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e^{2} x^{3} + 27 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} d e x^{2} + 9 \, {\left (2 \, b^{2} c^{3} e^{2} x^{3} + 6 \, b^{2} c^{3} d e x^{2} + 6 \, b^{2} c^{3} d^{2} x + 3 \, b^{2} c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (9 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d^{2} - 4 \, b^{2} c e^{2}\right )} x + 6 \, {\left (6 \, a b c^{3} e^{2} x^{3} + 18 \, a b c^{3} d e x^{2} + 18 \, a b c^{3} d^{2} x + 9 \, a b c d e - {\left (2 \, b^{2} c^{2} e^{2} x^{2} + 9 \, b^{2} c^{2} d e x + 18 \, b^{2} c^{2} d^{2} - 4 \, b^{2} e^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (2 \, a b c^{2} e^{2} x^{2} + 9 \, a b c^{2} d e x + 18 \, a b c^{2} d^{2} - 4 \, a b e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{54 \, c^{3}} \]
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Time = 0.33 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.90 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} d^{2} x + a^{2} d e x^{2} + \frac {a^{2} e^{2} x^{3}}{3} + 2 a b d^{2} x \operatorname {asinh}{\left (c x \right )} + 2 a b d e x^{2} \operatorname {asinh}{\left (c x \right )} + \frac {2 a b e^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {a b d e x \sqrt {c^{2} x^{2} + 1}}{c} - \frac {2 a b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {a b d e \operatorname {asinh}{\left (c x \right )}}{c^{2}} + \frac {4 a b e^{2} \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + b^{2} d e x^{2} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {b^{2} d e x^{2}}{2} + \frac {b^{2} e^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} e^{2} x^{3}}{27} - \frac {2 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {b^{2} d e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {2 b^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} + \frac {b^{2} d e \operatorname {asinh}^{2}{\left (c x \right )}}{2 c^{2}} - \frac {4 b^{2} e^{2} x}{9 c^{2}} + \frac {4 b^{2} e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.58 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{3} \, b^{2} e^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + b^{2} d e x^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} e^{2} x^{3} + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + a^{2} d e x^{2} + {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} a b d e + \frac {1}{2} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} d e + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e^{2} - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} e^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]
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Exception generated. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \]
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