Optimal. Leaf size=335 \[ \frac{\sqrt{\pi } d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}-\frac{\sqrt{3 \pi } d e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{5 \pi } d e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{\sqrt{5 \pi } d e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{2 d x^2 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}} \]
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Rubi [A] time = 1.37859, antiderivative size = 335, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {5777, 5779, 5448, 3308, 2180, 2204, 2205} \[ \frac{\sqrt{\pi } d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}-\frac{\sqrt{3 \pi } d e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{5 \pi } d e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{\sqrt{\pi } d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{\sqrt{3 \pi } d e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{\sqrt{5 \pi } d e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{2 d x^2 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 5777
Rule 5779
Rule 5448
Rule 3308
Rule 2180
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \frac{x^2 \left (d+c^2 d x^2\right )}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac{2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(4 d) \int \frac{x \sqrt{1+c^2 x^2}}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b c}+\frac{(10 c d) \int \frac{x^3 \sqrt{1+c^2 x^2}}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac{2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac{(10 d) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^3(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{(4 d) \operatorname{Subst}\left (\int \left (\frac{\sinh (x)}{4 \sqrt{a+b x}}+\frac{\sinh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac{(10 d) \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{8 \sqrt{a+b x}}-\frac{\sinh (3 x)}{16 \sqrt{a+b x}}+\frac{\sinh (5 x)}{16 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}\\ &=-\frac{2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}+\frac{d \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac{d \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{\sinh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{4 b c^3}\\ &=-\frac{2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{-5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{5 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^3}-\frac{d \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}-\frac{d \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac{d \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac{d \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}-\frac{(5 d) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac{2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}-\frac{(5 d) \operatorname{Subst}\left (\int e^{\frac{5 a}{b}-\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c^3}+\frac{(5 d) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c^3}-\frac{(5 d) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c^3}+\frac{(5 d) \operatorname{Subst}\left (\int e^{-\frac{5 a}{b}+\frac{5 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{8 b^2 c^3}-\frac{d \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2 c^3}-\frac{d \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2 c^3}+\frac{d \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2 c^3}+\frac{d \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{b^2 c^3}+\frac{(5 d) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{4 b^2 c^3}-\frac{(5 d) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{4 b^2 c^3}\\ &=-\frac{2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt{a+b \sinh ^{-1}(c x)}}+\frac{d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}-\frac{d e^{\frac{3 a}{b}} \sqrt{3 \pi } \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{d e^{\frac{5 a}{b}} \sqrt{5 \pi } \text{erf}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}-\frac{d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 b^{3/2} c^3}+\frac{d e^{-\frac{3 a}{b}} \sqrt{3 \pi } \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}+\frac{d e^{-\frac{5 a}{b}} \sqrt{5 \pi } \text{erfi}\left (\frac{\sqrt{5} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 b^{3/2} c^3}\\ \end{align*}
Mathematica [A] time = 0.695819, size = 436, normalized size = 1.3 \[ \frac{d e^{-5 \left (\frac{a}{b}+\sinh ^{-1}(c x)\right )} \left (-2 e^{\frac{6 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )+\sqrt{5} e^{5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\sqrt{3} e^{\frac{2 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-2 e^{\frac{4 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )+\sqrt{3} e^{\frac{8 a}{b}+5 \sinh ^{-1}(c x)} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\sqrt{5} e^{5 \left (\frac{2 a}{b}+\sinh ^{-1}(c x)\right )} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-e^{\frac{5 a}{b}+2 \sinh ^{-1}(c x)}+2 e^{\frac{5 a}{b}+4 \sinh ^{-1}(c x)}+2 e^{\frac{5 a}{b}+6 \sinh ^{-1}(c x)}-e^{\frac{5 a}{b}+8 \sinh ^{-1}(c x)}-e^{\frac{5 a}{b}+10 \sinh ^{-1}(c x)}-e^{\frac{5 a}{b}}\right )}{16 b c^3 \sqrt{a+b \sinh ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.226, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \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^{2}}{{\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^{2}}{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^{4}}{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^{2}}{{\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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