Optimal. Leaf size=427 \[ -\frac{3 \sqrt{\pi } b^{3/2} e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}-\frac{3 \sqrt{\pi } b^{3/2} e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}+\frac{3 \sqrt{\pi } b^{3/2} d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}+\frac{3 \sqrt{\pi } b^{3/2} d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{3 b d \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}-\frac{b e x^2 \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+\frac{b e \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]
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Rubi [A] time = 1.2637, antiderivative size = 427, normalized size of antiderivative = 1., number of steps used = 32, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5706, 5653, 5717, 5657, 3307, 2180, 2205, 2204, 5663, 5758, 5669, 5448} \[ -\frac{3 \sqrt{\pi } b^{3/2} e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}-\frac{3 \sqrt{\pi } b^{3/2} e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{\sqrt{\frac{\pi }{3}} b^{3/2} e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}+\frac{3 \sqrt{\pi } b^{3/2} d e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}+\frac{3 \sqrt{\pi } b^{3/2} d e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{3 b d \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}-\frac{b e x^2 \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+\frac{b e \sqrt{c^2 x^2+1} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \]
Antiderivative was successfully verified.
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Rule 5706
Rule 5653
Rule 5717
Rule 5657
Rule 3307
Rule 2180
Rule 2205
Rule 2204
Rule 5663
Rule 5758
Rule 5669
Rule 5448
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \, dx &=\int \left (d \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+e x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}\right ) \, dx\\ &=d \int \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \, dx+e \int x^2 \left (a+b \sinh ^{-1}(c x)\right )^{3/2} \, dx\\ &=d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}-\frac{1}{2} (3 b c d) \int \frac{x \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{1+c^2 x^2}} \, dx-\frac{1}{2} (b c e) \int \frac{x^3 \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{1+c^2 x^2}} \, dx\\ &=-\frac{3 b d \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}-\frac{b e x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{4} \left (3 b^2 d\right ) \int \frac{1}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx+\frac{1}{12} \left (b^2 e\right ) \int \frac{x^2}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx+\frac{(b e) \int \frac{x \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{1+c^2 x^2}} \, dx}{3 c}\\ &=-\frac{3 b d \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac{b e x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{(3 b d) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{4 c}+\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^2(x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{12 c^3}-\frac{\left (b^2 e\right ) \int \frac{1}{\sqrt{a+b \sinh ^{-1}(c x)}} \, dx}{6 c^2}\\ &=-\frac{3 b d \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac{b e x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{(3 b d) \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 x)\right )}{8 c}+\frac{(3 b d) \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 x)\right )}{8 c}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}-\frac{x}{b}\right )}{\sqrt{x}} \, dx,x,a+b \sinh ^{-1}(c x)\right )}{6 c^3}+\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \left (-\frac{\cosh (x)}{4 \sqrt{a+b x}}+\frac{\cosh (3 x)}{4 \sqrt{a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{12 c^3}\\ &=-\frac{3 b d \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac{b e x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{(3 b d) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{4 c}+\frac{(3 b d) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{4 c}-\frac{(b e) \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 x)\right )}{12 c^3}-\frac{(b e) \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 x)\right )}{12 c^3}-\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cosh (x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{48 c^3}+\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{\cosh (3 x)}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{48 c^3}\\ &=-\frac{3 b d \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac{b e x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{3 b^{3/2} d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}+\frac{3 b^{3/2} d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{(b e) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{6 c^3}-\frac{(b e) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{6 c^3}+\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{e^{-3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}-\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{e^{-x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}-\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}+\frac{\left (b^2 e\right ) \operatorname{Subst}\left (\int \frac{e^{3 x}}{\sqrt{a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{96 c^3}\\ &=-\frac{3 b d \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac{b e x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{3 b^{3/2} d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{b^{3/2} e e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}+\frac{3 b^{3/2} d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{b^{3/2} e e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{12 c^3}+\frac{(b e) \operatorname{Subst}\left (\int e^{\frac{3 a}{b}-\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{48 c^3}-\frac{(b e) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{48 c^3}-\frac{(b e) \operatorname{Subst}\left (\int e^{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{48 c^3}+\frac{(b e) \operatorname{Subst}\left (\int e^{-\frac{3 a}{b}+\frac{3 x^2}{b}} \, dx,x,\sqrt{a+b \sinh ^{-1}(c x)}\right )}{48 c^3}\\ &=-\frac{3 b d \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{2 c}+\frac{b e \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{3 c^3}-\frac{b e x^2 \sqrt{1+c^2 x^2} \sqrt{a+b \sinh ^{-1}(c x)}}{6 c}+d x \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{1}{3} e x^3 \left (a+b \sinh ^{-1}(c x)\right )^{3/2}+\frac{3 b^{3/2} d e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{3 b^{3/2} e e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{b^{3/2} e e^{\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}+\frac{3 b^{3/2} d e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{8 c}-\frac{3 b^{3/2} e e^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\left (\frac{\sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 c^3}+\frac{b^{3/2} e e^{-\frac{3 a}{b}} \sqrt{\frac{\pi }{3}} \text{erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )}{96 c^3}\\ \end{align*}
Mathematica [A] time = 4.19707, size = 770, normalized size = 1.8 \[ \frac{a e e^{-\frac{3 a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left (9 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )+\sqrt{3} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-9 e^{\frac{2 a}{b}} \sqrt{\frac{a}{b}+\sinh ^{-1}(c x)} \text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )-\sqrt{3} e^{\frac{6 a}{b}} \sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{3}{2},\frac{3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )}{72 c^3 \sqrt{-\frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}}}+\frac{a d e^{-\frac{a}{b}} \sqrt{a+b \sinh ^{-1}(c x)} \left (\frac{\text{Gamma}\left (\frac{3}{2},-\frac{a+b \sinh ^{-1}(c x)}{b}\right )}{\sqrt{-\frac{a+b \sinh ^{-1}(c x)}{b}}}-\frac{e^{\frac{2 a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{a}{b}+\sinh ^{-1}(c x)\right )}{\sqrt{\frac{a}{b}+\sinh ^{-1}(c x)}}\right )}{2 c}+\frac{\sqrt{b} d \left (4 \sqrt{b} \left (2 c x \sinh ^{-1}(c x)-3 \sqrt{c^2 x^2+1}\right ) \sqrt{a+b \sinh ^{-1}(c x)}+\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 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 x)}}{\sqrt{b}}\right )\right )}{8 c}+\frac{\sqrt{b} e \left (-9 \left (4 \sqrt{b} \left (2 c x \sinh ^{-1}(c x)-3 \sqrt{c^2 x^2+1}\right ) \sqrt{a+b \sinh ^{-1}(c x)}+\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 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 x)}}{\sqrt{b}}\right )\right )+\sqrt{3 \pi } (b-2 a) \left (\sinh \left (\frac{3 a}{b}\right )+\cosh \left (\frac{3 a}{b}\right )\right ) \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )+\sqrt{3 \pi } (2 a+b) \left (\cosh \left (\frac{3 a}{b}\right )-\sinh \left (\frac{3 a}{b}\right )\right ) \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \sinh ^{-1}(c x)}}{\sqrt{b}}\right )+12 \sqrt{b} \left (2 \sinh ^{-1}(c x) \sinh \left (3 \sinh ^{-1}(c x)\right )-\cosh \left (3 \sinh ^{-1}(c x)\right )\right ) \sqrt{a+b \sinh ^{-1}(c x)}\right )}{288 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int \left ( e{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{\left (e x^{2} + d\right )}{\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*} \int \left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{\frac{3}{2}} \left (d + e x^{2}\right )\, 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 (e x^{2} + d\right )}{\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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