Optimal. Leaf size=87 \[ \frac {(e (c+d x))^{m+1} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e (m+1)}-\frac {4 b \text {Int}\left (\frac {(e (c+d x))^{m+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{\sqrt {(c+d x)^2+1}},x\right )}{e (m+1)} \]
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Rubi [A] time = 0.18, 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 = {} \[ \int (c e+d e x)^m \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int (c e+d e x)^m \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int (e x)^m \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e (c+d x))^{1+m} \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d e (1+m)}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {(e x)^{1+m} \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e (1+m)}\\ \end {align*}
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Mathematica [A] time = 1.92, size = 0, normalized size = 0.00 \[ \int (c e+d e x)^m \left (a+b \sinh ^{-1}(c+d x)\right )^4 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \operatorname {arsinh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arsinh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arsinh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arsinh}\left (d x + c\right ) + a^{4}\right )} {\left (d e x + c e\right )}^{m}, x\right ) \]
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 \[ \int {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} {\left (d e x + c e\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.04, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right )^{m} \left (a +b \arcsinh \left (d x +c \right )\right )^{4}\, dx \]
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 \[ \text {result too large to display} \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^m\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]
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 \[ \int \left (e \left (c + d x\right )\right )^{m} \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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