Optimal. Leaf size=88 \[ \frac{(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d e (m+1)}-\frac{4 b \text{Unintegrable}\left (\frac{(e (c+d x))^{m+1} \left (a+b \sin ^{-1}(c+d x)\right )^3}{\sqrt{1-(c+d x)^2}},x\right )}{e (m+1)} \]
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Rubi [A] time = 0.186821, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (c e+d e x)^m \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx \]
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 \sin ^{-1}(c+d x)\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^m \left (a+b \sin ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e (c+d x))^{1+m} \left (a+b \sin ^{-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 \sin ^{-1}(x)\right )^3}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d e (1+m)}\\ \end{align*}
Mathematica [A] time = 2.86894, size = 0, normalized size = 0. \[ \int (c e+d e x)^m \left (a+b \sin ^{-1}(c+d x)\right )^4 \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 1.517, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{m} \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{4} \arcsin \left (d x + c\right )^{4} + 4 \, a b^{3} \arcsin \left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \arcsin \left (d x + c\right )^{2} + 4 \, a^{3} b \arcsin \left (d x + c\right ) + a^{4}\right )}{\left (d e x + c e\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}{\left (d e x + c e\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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