Optimal. Leaf size=81 \[ \frac{2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{7 d e}-\frac{6 b \text{Unintegrable}\left (\frac{(e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt{(c+d x)^2+1}},x\right )}{7 e} \]
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Rubi [A] time = 0.203318, 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 = {} \[ \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int (e x)^{5/2} \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{2 (e (c+d x))^{7/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{7 d e}-\frac{(6 b) \operatorname{Subst}\left (\int \frac{(e x)^{7/2} \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{7 d e}\\ \end{align*}
Mathematica [A] time = 99.8167, size = 0, normalized size = 0. \[ \int (c e+d e x)^{5/2} \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.255, size = 0, normalized size = 0. \begin{align*} \int \left ( dex+ce \right ) ^{{\frac{5}{2}}} \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{3}\, 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 (a^{3} d^{2} e^{2} x^{2} + 2 \, a^{3} c d e^{2} x + a^{3} c^{2} e^{2} +{\left (b^{3} d^{2} e^{2} x^{2} + 2 \, b^{3} c d e^{2} x + b^{3} c^{2} e^{2}\right )} \operatorname{arsinh}\left (d x + c\right )^{3} + 3 \,{\left (a b^{2} d^{2} e^{2} x^{2} + 2 \, a b^{2} c d e^{2} x + a b^{2} c^{2} e^{2}\right )} \operatorname{arsinh}\left (d x + c\right )^{2} + 3 \,{\left (a^{2} b d^{2} e^{2} x^{2} + 2 \, a^{2} b c d e^{2} x + a^{2} b c^{2} e^{2}\right )} \operatorname{arsinh}\left (d x + c\right )\right )} \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(-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 (d e x + c e\right )}^{\frac{5}{2}}{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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