Optimal. Leaf size=140 \[ 2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}+\frac {e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e} \]
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Rubi [A]
time = 0.22, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {5828, 5838,
5783, 5798, 8, 5812, 30} \begin {gather*} -\frac {2 b d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}+\frac {e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}+2 b^2 d x+\frac {1}{4} b^2 e x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5783
Rule 5798
Rule 5812
Rule 5828
Rule 5838
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}-\frac {(b c) \int \left (\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {2 d e x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {e^2 x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\right ) \, dx}{e}\\ &=\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}-(2 b c d) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {\left (b c d^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{e}-(b c e) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {2 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}+\left (2 b^2 d\right ) \int 1 \, dx+\frac {1}{2} \left (b^2 e\right ) \int x \, dx+\frac {(b e) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 c}\\ &=2 b^2 d x+\frac {1}{4} b^2 e x^2-\frac {2 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {b e x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 c}-\frac {d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}+\frac {e \left (a+b \sinh ^{-1}(c x)\right )^2}{4 c^2}+\frac {(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{2 e}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 142, normalized size = 1.01 \begin {gather*} \frac {c \left (2 a^2 c x (2 d+e x)+b^2 c x (8 d+e x)-2 a b (4 d+e x) \sqrt {1+c^2 x^2}\right )+2 b \left (-b c (4 d+e x) \sqrt {1+c^2 x^2}+a \left (e+4 c^2 d x+2 c^2 e x^2\right )\right ) \sinh ^{-1}(c x)+b^2 \left (e+4 c^2 d x+2 c^2 e x^2\right ) \sinh ^{-1}(c x)^2}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.86, size = 193, normalized size = 1.38
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {e \left (2 \arcsinh \left (c x \right )^{2} c^{2} x^{2}-2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x +\arcsinh \left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsinh \left (c x \right ) d \,c^{2} x +\frac {\arcsinh \left (c x \right ) e \,c^{2} x^{2}}{2}-d c \sqrt {c^{2} x^{2}+1}-\frac {e \left (\frac {\sqrt {c^{2} x^{2}+1}\, c x}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )}{2}\right )}{c}}{c}\) | \(193\) |
default | \(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b^{2} \left (d c \left (\arcsinh \left (c x \right )^{2} c x -2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {e \left (2 \arcsinh \left (c x \right )^{2} c^{2} x^{2}-2 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x +\arcsinh \left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}\right )}{c}+\frac {2 a b \left (\arcsinh \left (c x \right ) d \,c^{2} x +\frac {\arcsinh \left (c x \right ) e \,c^{2} x^{2}}{2}-d c \sqrt {c^{2} x^{2}+1}-\frac {e \left (\frac {\sqrt {c^{2} x^{2}+1}\, c x}{2}-\frac {\arcsinh \left (c x \right )}{2}\right )}{2}\right )}{c}}{c}\) | \(193\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 223, normalized size = 1.59 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} \operatorname {arsinh}\left (c x\right )^{2} e + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{2} \, a^{2} x^{2} e + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {1}{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 e + \frac {1}{4} \, {\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} e + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 257 vs.
\(2 (128) = 256\).
time = 0.38, size = 257, normalized size = 1.84 \begin {gather*} \frac {{\left (2 \, a^{2} + b^{2}\right )} c^{2} x^{2} \cosh \left (1\right ) + {\left (2 \, a^{2} + b^{2}\right )} c^{2} x^{2} \sinh \left (1\right ) + 4 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{2} d x + {\left (4 \, b^{2} c^{2} d x + {\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \cosh \left (1\right ) + {\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (4 \, a b c^{2} d x + {\left (2 \, a b c^{2} x^{2} + a b\right )} \cosh \left (1\right ) + {\left (2 \, a b c^{2} x^{2} + a b\right )} \sinh \left (1\right ) - {\left (b^{2} c x \cosh \left (1\right ) + b^{2} c x \sinh \left (1\right ) + 4 \, b^{2} c d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (a b c x \cosh \left (1\right ) + a b c x \sinh \left (1\right ) + 4 \, a b c d\right )} \sqrt {c^{2} x^{2} + 1}}{4 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 233, normalized size = 1.66 \begin {gather*} \begin {cases} a^{2} d x + \frac {a^{2} e x^{2}}{2} + 2 a b d x \operatorname {asinh}{\left (c x \right )} + a b e x^{2} \operatorname {asinh}{\left (c x \right )} - \frac {2 a b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {a b e x \sqrt {c^{2} x^{2} + 1}}{2 c} + \frac {a b e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac {b^{2} e x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {b^{2} e x^{2}}{4} - \frac {2 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {b^{2} e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c} + \frac {b^{2} e \operatorname {asinh}^{2}{\left (c x \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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