Optimal. Leaf size=79 \[ \frac{2 \sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d e}-\frac{8 b \text{Unintegrable}\left (\frac{\sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )^3}{\sqrt{1-(c+d x)^2}},x\right )}{e} \]
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Rubi [A] time = 0.178986, 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 = {} \[ \int \frac{\left (a+b \sin ^{-1}(c+d x)\right )^4}{\sqrt{c e+d e x}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (a+b \sin ^{-1}(c+d x)\right )^4}{\sqrt{c e+d e x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sin ^{-1}(x)\right )^4}{\sqrt{e x}} \, dx,x,c+d x\right )}{d}\\ &=\frac{2 \sqrt{e (c+d x)} \left (a+b \sin ^{-1}(c+d x)\right )^4}{d e}-\frac{(8 b) \operatorname{Subst}\left (\int \frac{\sqrt{e x} \left (a+b \sin ^{-1}(x)\right )^3}{\sqrt{1-x^2}} \, dx,x,c+d x\right )}{d e}\\ \end{align*}
Mathematica [A] time = 14.1969, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sin ^{-1}(c+d x)\right )^4}{\sqrt{c e+d e x}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.286, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{4}{\frac{1}{\sqrt{dex+ce}}}}\, 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 (\frac{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}}{\sqrt{d e x + c e}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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 \frac{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{4}}{\sqrt{d e x + c e}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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