Optimal. Leaf size=77 \[ \frac{6 b \text{Unintegrable}\left (\frac{\left (a+b \sinh ^{-1}(c+d x)\right )^2}{\sqrt{(c+d x)^2+1} \sqrt{e (c+d x)}},x\right )}{e}-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e \sqrt{e (c+d x)}} \]
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Rubi [A] time = 0.192779, 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 = {} \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^3}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d e \sqrt{e (c+d x)}}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{\left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt{e x} \sqrt{1+x^2}} \, dx,x,c+d x\right )}{d e}\\ \end{align*}
Mathematica [A] time = 18.9413, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \sinh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.251, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{3} \left ( dex+ce \right ) ^{-{\frac{3}{2}}}}\, 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{{\left (b^{3} \operatorname{arsinh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arsinh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arsinh}\left (d x + c\right ) + a^{3}\right )} \sqrt{d e x + c e}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{asinh}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac{3}{2}}}\, dx \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 \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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