Optimal. Leaf size=236 \[ \frac{\sqrt{\pi } d e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}+\frac{\sqrt{\pi } d e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}-\frac{2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}} \]
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Rubi [A] time = 0.766615, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5777, 5699, 3312, 3307, 2180, 2204, 2205, 5779, 5448} \[ \frac{\sqrt{\pi } d e^{\frac{4 a}{b}} \text{Erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{\frac{2 a}{b}} \text{Erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}+\frac{\sqrt{\pi } d e^{-\frac{4 a}{b}} \text{Erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{\sqrt{\frac{\pi }{2}} d e^{-\frac{2 a}{b}} \text{Erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}-\frac{2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 5777
Rule 5699
Rule 3312
Rule 3307
Rule 2180
Rule 2204
Rule 2205
Rule 5779
Rule 5448
Rubi steps
\begin{align*} \int \frac{x \left (d+c^2 d x^2\right )}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(2 d) \int \frac{\sqrt{1+c^2 x^2}}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}+\frac{(8 c d) \int \frac{x^2 \sqrt{1+c^2 x^2}}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(2 d) \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac{(8 d) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac{2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(2 d) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{a+b x}}+\frac{\cosh (2 x)}{2 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac{(8 d) \operatorname{Subst}\left (\int \left (-\frac{1}{8 \sqrt{a+b x}}+\frac{\cosh (4 x)}{8 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac{2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac{d \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac{2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}+\frac{d \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}+\frac{d \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}+\frac{d \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}\\ &=-\frac{2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d \operatorname{Subst}\left (\int e^{\frac{4 a}{b}-\frac{4 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}+\frac{d \operatorname{Subst}\left (\int e^{\frac{2 a}{b}-\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}+\frac{d \operatorname{Subst}\left (\int e^{-\frac{2 a}{b}+\frac{2 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}+\frac{d \operatorname{Subst}\left (\int e^{-\frac{4 a}{b}+\frac{4 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}\\ &=-\frac{2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d e^{\frac{4 a}{b}} \sqrt{\pi } \text{erf}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{d e^{\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}+\frac{d e^{-\frac{4 a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{2 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 b^{3/2} c^2}+\frac{d e^{-\frac{2 a}{b}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{\sqrt{2} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 b^{3/2} c^2}\\ \end{align*}
Mathematica [A] time = 0.44492, size = 227, normalized size = 0.96 \[ \frac{d e^{-\frac{4 a}{b}} \left (\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\sqrt{2} e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-e^{\frac{4 a}{b}} \left (\sqrt{2} e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{\frac{4 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+2 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )\right )\right )}{4 b c^2 \sqrt{a+b \sinh ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.128, size = 0, normalized size = 0. \begin{align*} \int{x \left ({c}^{2}d{x}^{2}+d \right ) \left ( a+b{\it Arcsinh} \left ( cx \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 \frac{{\left (c^{2} d x^{2} + d\right )} x}{{\left (b \operatorname{arsinh}\left (c x\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*} d \left (\int \frac{x}{a \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} \operatorname{asinh}{\left (c x \right )}}\, dx + \int \frac{c^{2} x^{3}}{a \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} + b \sqrt{a + b \operatorname{asinh}{\left (c x \right )}} \operatorname{asinh}{\left (c 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{{\left (c^{2} d x^{2} + d\right )} x}{{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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