Optimal. Leaf size=29 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}},x\right )}{e} \]
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Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1}{(c e+d e x) \sqrt {a+b \sinh ^{-1}(c+d x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{(c e+d e x) \sqrt {a+b \sinh ^{-1}(c+d x)}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{e x \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c e+d e x) \sqrt {a+b \sinh ^{-1}(c+d x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d e x +c e \right ) \sqrt {a +b \arcsinh \left (d x +c \right )}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}}\, dx}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\left (c\,e+d\,e\,x\right )\,\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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