Optimal. Leaf size=138 \[ -\frac {2 a b x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}+\frac {2 b^2 \left (c^2 x^2+1\right )}{c^2 \sqrt {c^2 d x^2+d}}-\frac {2 b^2 x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{c \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.12, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5717, 5653, 261} \[ -\frac {2 a b x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}+\frac {2 b^2 \left (c^2 x^2+1\right )}{c^2 \sqrt {c^2 d x^2+d}}-\frac {2 b^2 x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{c \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 261
Rule 5653
Rule 5717
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{c \sqrt {d+c^2 d x^2}}\\ &=-\frac {2 a b x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}+\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}-\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{c \sqrt {d+c^2 d x^2}}\\ &=-\frac {2 a b x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}-\frac {2 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{c \sqrt {d+c^2 d x^2}}+\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=-\frac {2 a b x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^2 \sqrt {d+c^2 d x^2}}-\frac {2 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{c \sqrt {d+c^2 d x^2}}+\frac {\sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 127, normalized size = 0.92 \[ \frac {\sqrt {c^2 d x^2+d} \left (a^2 \sqrt {c^2 x^2+1}-2 b \sinh ^{-1}(c x) \left (b c x-a \sqrt {c^2 x^2+1}\right )-2 a b c x+2 b^2 \sqrt {c^2 x^2+1}+b^2 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2\right )}{c^2 d \sqrt {c^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 179, normalized size = 1.30 \[ \frac {{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 2 \, {\left (a b c^{2} x^{2} - \sqrt {c^{2} x^{2} + 1} b^{2} c x + a b\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left ({\left (a^{2} + 2 \, b^{2}\right )} c^{2} x^{2} - 2 \, \sqrt {c^{2} x^{2} + 1} a b c x + a^{2} + 2 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d}}{c^{4} d x^{2} + c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x}{\sqrt {c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 296, normalized size = 2.14 \[ \frac {a^{2} \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\arcsinh \left (c x \right )^{2}-2 \arcsinh \left (c x \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (\arcsinh \left (c x \right )^{2}+2 \arcsinh \left (c x \right )+2\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (-1+\arcsinh \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \left (1+\arcsinh \left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}+1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 125, normalized size = 0.91 \[ -2 \, b^{2} {\left (\frac {x \operatorname {arsinh}\left (c x\right )}{c \sqrt {d}} - \frac {\sqrt {c^{2} x^{2} + 1}}{c^{2} \sqrt {d}}\right )} - \frac {2 \, a b x}{c \sqrt {d}} + \frac {\sqrt {c^{2} d x^{2} + d} b^{2} \operatorname {arsinh}\left (c x\right )^{2}}{c^{2} d} + \frac {2 \, \sqrt {c^{2} d x^{2} + d} a b \operatorname {arsinh}\left (c x\right )}{c^{2} d} + \frac {\sqrt {c^{2} d x^{2} + d} a^{2}}{c^{2} d} \]
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 \[ \int \frac {x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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