Optimal. Leaf size=179 \[ \frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d} \]
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Rubi [A] time = 0.388462, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5863, 5653, 5717, 5779, 3308, 2180, 2204, 2205} \[ \frac{15 \sqrt{\pi } b^{5/2} e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{15 \sqrt{\pi } b^{5/2} e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}+\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d} \]
Antiderivative was successfully verified.
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Rule 5863
Rule 5653
Rule 5717
Rule 5779
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{x \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt{1+x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac{5 b \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int \sqrt{a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{4 d}\\ &=\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1+x^2} \sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{8 d}\\ &=\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d}\\ &=\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}-\frac{\left (15 b^3\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}\\ &=\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d}-\frac{\left (15 b^2\right ) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{8 d}\\ &=\frac{15 b^2 (c+d x) \sqrt{a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac{5 b \sqrt{1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac{(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac{15 b^{5/2} e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}-\frac{15 b^{5/2} e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{16 d}\\ \end{align*}
Mathematica [B] time = 1.43188, size = 458, normalized size = 2.56 \[ \frac{8 a^2 e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt{-\frac{a+b \sinh ^{-1}(c+d x)}{b}}}-\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)}}\right )+\sqrt{b} \left (\sqrt{\pi } \left (4 a^2-12 a b+15 b^2\right ) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{\pi } \left (4 a^2+12 a b+15 b^2\right ) \left (\sinh \left (\frac{a}{b}\right )-\cosh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+4 \sqrt{b} \sqrt{a+b \sinh ^{-1}(c+d x)} \left (2 \sqrt{(c+d x)^2+1} \left (a-5 b \sinh ^{-1}(c+d x)\right )+b (c+d x) \left (4 \sinh ^{-1}(c+d x)^2+15\right )\right )\right )+4 a \sqrt{b} \left (\sqrt{\pi } (3 b-2 a) \left (\sinh \left (\frac{a}{b}\right )+\cosh \left (\frac{a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+\sqrt{\pi } (2 a+3 b) \left (\cosh \left (\frac{a}{b}\right )-\sinh \left (\frac{a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )+4 \sqrt{b} \left (2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt{(c+d x)^2+1}\right ) \sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{16 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( dx+c \right ) \right ) ^{{\frac{5}{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{5}{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(-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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsinh}\left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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