Optimal. Leaf size=150 \[ \frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}-\frac{3 b \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d} \]
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Rubi [A] time = 0.253528, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5863, 5653, 5717, 5657, 3307, 2180, 2205, 2204} \[ \frac{3 \sqrt{\pi } b^{3/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{3 \sqrt{\pi } b^{3/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}-\frac{3 b \sqrt{(c+d x)^2+1} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d} \]
Antiderivative was successfully verified.
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Rule 5863
Rule 5653
Rule 5717
Rule 5657
Rule 3307
Rule 2180
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{x \sqrt{a+b \sinh ^{-1}(x)}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac{3 b \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac{3 b \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{4 d}\\ &=-\frac{3 b \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{e^{-i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{8 d}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{e^{i \left (\frac{i a}{b}-\frac{i x}{b}\right )}}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c+d x)\right )}{8 d}\\ &=-\frac{3 b \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d}+\frac{(3 b) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 d}+\frac{(3 b) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{4 d}\\ &=-\frac{3 b \sqrt{1+(c+d x)^2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{d}+\frac{3 b^{3/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}+\frac{3 b^{3/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{8 d}\\ \end{align*}
Mathematica [A] time = 0.181756, size = 272, normalized size = 1.81 \[ \frac{a e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right )}{2 d}+\frac{\sqrt{b} \left (\sqrt{\pi } (3 b-2 a) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{\pi } (2 a+3 b) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+4 \sqrt{b} \left (2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{asinh}{\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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