Optimal. Leaf size=153 \[ 2 b^2 d x-\frac {4 b^2 e x}{9 c^2}+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {2 b e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \]
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Rubi [A]
time = 0.20, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {5793, 5772,
5798, 8, 5776, 5812, 30} \begin {gather*} -\frac {2 b d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c}-\frac {2 b e x^2 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {4 b e \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {4 b^2 e x}{9 c^2}+2 b^2 d x+\frac {2}{27} b^2 e x^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 5772
Rule 5776
Rule 5793
Rule 5798
Rule 5812
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\int \left (d \left (a+b \sinh ^{-1}(c x)\right )^2+e x^2 \left (a+b \sinh ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+e \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx\\ &=d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2-(2 b c d) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{3} (2 b c e) \int \frac {x^3 \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 {2 b e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2+\left (2 b^2 d\right ) \int 1 \, dx+\frac {1}{9} \left (2 b^2 e\right ) \int x^2 \, dx+\frac {(4 b e) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{9 c}\\ &=2 b^2 d x+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {2 b e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (4 b^2 e\right ) \int 1 \, dx}{9 c^2}\\ &=2 b^2 d x-\frac {4 b^2 e x}{9 c^2}+\frac {2}{27} b^2 e x^3-\frac {2 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c}+\frac {4 b e \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3}-\frac {2 b e x^2 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 164, normalized size = 1.07 \begin {gather*} \frac {9 a^2 c^3 x \left (3 d+e x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )+2 b^2 c x \left (-6 e+c^2 \left (27 d+e x^2\right )\right )-6 b \left (-3 a c^3 x \left (3 d+e x^2\right )+b \sqrt {1+c^2 x^2} \left (-2 e+c^2 \left (9 d+e x^2\right )\right )\right ) \sinh ^{-1}(c x)+9 b^2 c^3 x \left (3 d+e x^2\right ) \sinh ^{-1}(c x)^2}{27 c^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.91, size = 206, normalized size = 1.35 \[\frac {\frac {a^{2} \left (\frac {1}{3} x^{3} c^{3} e +x \,c^{3} d \right )}{c^{2}}+\frac {b^{2} \left (\frac {\left (9 \arcsinh \left (c x \right )^{2} x^{2} c^{2} e +27 \arcsinh \left (c x \right )^{2} c^{2} d +2 c^{2} e \,x^{2}+54 c^{2} d -12 e \right ) x c}{27}-\frac {2 \arcsinh \left (c x \right ) \left (c^{2} e \,x^{2}+9 c^{2} d -2 e \right ) \sqrt {c^{2} x^{2}+1}}{9}\right )}{c^{2}}+\frac {2 a b \left (\frac {\arcsinh \left (c x \right ) x^{3} c^{3} e}{3}+\arcsinh \left (c x \right ) d \,c^{3} x -\frac {e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}-d \,c^{2} \sqrt {c^{2} x^{2}+1}\right )}{c^{2}}}{c}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 222, normalized size = 1.45 \begin {gather*} \frac {1}{3} \, b^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} e + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} x^{3} 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 {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 - \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 + \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. 304 vs.
\(2 (142) = 284\).
time = 0.39, size = 304, normalized size = 1.99 \begin {gather*} \frac {27 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d x + 9 \, {\left (b^{2} c^{3} x^{3} \cosh \left (1\right ) + b^{2} c^{3} x^{3} \sinh \left (1\right ) + 3 \, b^{2} c^{3} d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} x^{3} - 12 \, b^{2} c x\right )} \cosh \left (1\right ) + 6 \, {\left (3 \, a b c^{3} x^{3} \cosh \left (1\right ) + 3 \, a b c^{3} x^{3} \sinh \left (1\right ) + 9 \, a b c^{3} d x - {\left (9 \, b^{2} c^{2} d + {\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \cosh \left (1\right ) + {\left (b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} x^{3} - 12 \, b^{2} c x\right )} \sinh \left (1\right ) - 6 \, {\left (9 \, a b c^{2} d + {\left (a b c^{2} x^{2} - 2 \, a b\right )} \cosh \left (1\right ) + {\left (a b c^{2} x^{2} - 2 \, a b\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} + 1}}{27 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 279, normalized size = 1.82 \begin {gather*} \begin {cases} a^{2} d x + \frac {a^{2} e x^{3}}{3} + 2 a b d x \operatorname {asinh}{\left (c x \right )} + \frac {2 a b e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2 a b d \sqrt {c^{2} x^{2} + 1}}{c} - \frac {2 a b e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {4 a b e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d x + \frac {b^{2} e x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} e x^{3}}{27} - \frac {2 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {2 b^{2} e x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} - \frac {4 b^{2} e x}{9 c^{2}} + \frac {4 b^{2} e \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\a^{2} \left (d x + \frac {e x^{3}}{3}\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 (e\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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