Optimal. Leaf size=209 \[ -\frac{2 \sqrt{2 \pi } e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{2 \sqrt{2 \pi } e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e \sqrt{(c+d x)^2+1} (c+d x)}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \]
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Rubi [A] time = 0.54193, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {5865, 12, 5667, 5774, 5669, 5448, 3308, 2180, 2204, 2205, 5675} \[ -\frac{2 \sqrt{2 \pi } e e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{2 \sqrt{2 \pi } e e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e \sqrt{(c+d x)^2+1} (c+d x)}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5865
Rule 12
Rule 5667
Rule 5774
Rule 5669
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rule 5675
Rubi steps
\begin{align*} \int \frac{c e+d e x}{\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{e x}{\left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac{e \operatorname{Subst}\left (\int \frac{x}{\left (a+b \sinh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac{2 e (c+d x) \sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}+\frac{(4 e) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1+x^2} \left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac{2 e (c+d x) \sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{(16 e) \operatorname{Subst}\left (\int \frac{x}{\sqrt{a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{(16 e) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{(16 e) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}+\frac{(8 e) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{(4 e) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d}+\frac{(4 e) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac{2 e (c+d x) \sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{(8 e) \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{3 b^3 d}+\frac{(8 e) \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c+d x)}\right )}{3 b^3 d}\\ &=-\frac{2 e (c+d x) \sqrt{1+(c+d x)^2}}{3 b d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}-\frac{4 e}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{8 e (c+d x)^2}{3 b^2 d \sqrt{a+b \sinh ^{-1}(c+d x)}}-\frac{2 e e^{\frac{2 a}{b}} \sqrt{2 \pi } \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}+\frac{2 e e^{-\frac{2 a}{b}} \sqrt{2 \pi } \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c+d x)}}{\sqrt{b}}\right )}{3 b^{5/2} d}\\ \end{align*}
Mathematica [A] time = 0.673967, size = 227, normalized size = 1.09 \[ \frac{e e^{-2 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )} \left (e^{\frac{2 a}{b}} \left (4 \sqrt{2} e^{2 \left (\frac{a}{b}+\sinh ^{-1}(c+d x)\right )} \sqrt{\frac{a}{b}+\sinh ^{-1}(c+d x)} \left (a+b \sinh ^{-1}(c+d x)\right ) \text{Gamma}\left (\frac{1}{2},\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )-4 a e^{4 \sinh ^{-1}(c+d x)}-4 a-b e^{4 \sinh ^{-1}(c+d x)}-4 b \left (e^{4 \sinh ^{-1}(c+d x)}+1\right ) \sinh ^{-1}(c+d x)+b\right )-4 \sqrt{2} b e^{2 \sinh ^{-1}(c+d x)} \left (-\frac{a+b \sinh ^{-1}(c+d x)}{b}\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-\frac{2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{6 b^2 d \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.089, size = 0, normalized size = 0. \begin{align*} \int{(dex+ce) \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 \frac{d e x + c e}{{\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] time = 0., size = 0, normalized size = 0. \begin{align*} e \left (\int \frac{c}{a^{2} \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}} + 2 a b \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}} \operatorname{asinh}{\left (c + d x \right )} + b^{2} \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}} \operatorname{asinh}^{2}{\left (c + d x \right )}}\, dx + \int \frac{d x}{a^{2} \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}} + 2 a b \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}} \operatorname{asinh}{\left (c + d x \right )} + b^{2} \sqrt{a + b \operatorname{asinh}{\left (c + d x \right )}} \operatorname{asinh}^{2}{\left (c + d x \right )}}\, dx\right ) \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 \frac{d e x + c e}{{\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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