Optimal. Leaf size=82 \[ \frac {2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d e}-\frac {8 b \text {Int}\left (\frac {(e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {1+(c+d x)^2}},x\right )}{3 e} \]
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Rubi [A]
time = 0.12, 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 \sqrt {c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\text {Subst}\left (\int \sqrt {e x} \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{3 d e}-\frac {(8 b) \text {Subst}\left (\int \frac {(e x)^{3/2} \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e}\\ \end {align*}
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Mathematica [A]
time = 147.14, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {c e+d e x} \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, 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 \[\int \left (a +b \arcsinh \left (d x +c \right )\right )^{4} \sqrt {d e x +c e}\, dx\]
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 [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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{4}\, dx \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 = -0.01 \begin {gather*} \int \sqrt {c\,e+d\,e\,x}\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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