Optimal. Leaf size=319 \[ \frac {3}{8} c x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.28, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5786, 5785,
5783, 5780, 5556, 12, 3389, 2211, 2235, 2236, 5819} \begin {gather*} \frac {\sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}-\frac {\sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {a^2 x^2+1}}-\frac {\sqrt {\frac {\pi }{2}} c \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {a^2 x^2+1}}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {c \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt {a^2 x^2+1}}+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5780
Rule 5783
Rule 5785
Rule 5786
Rule 5819
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)} \, dx &=\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)} \, dx-\frac {\left (a c \sqrt {c+a^2 c x^2}\right ) \int \frac {x \left (1+a^2 x^2\right )}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{8 \sqrt {1+a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{8 a \sqrt {1+a^2 x^2}}-\frac {\left (3 a c \sqrt {c+a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{16 \sqrt {1+a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {\sinh (2 x)}{4 \sqrt {x}}+\frac {\sinh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{8 a \sqrt {1+a^2 x^2}}-\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt {1+a^2 x^2}}-\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{16 a \sqrt {1+a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt {1+a^2 x^2}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt {1+a^2 x^2}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}-\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{32 a \sqrt {1+a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt {1+a^2 x^2}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}-\frac {\left (c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}-\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}-\frac {\left (3 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{32 a \sqrt {1+a^2 x^2}}\\ &=\frac {3}{8} c x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \sqrt {\sinh ^{-1}(a x)}+\frac {c \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{4 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}+\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}-\frac {c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}-\frac {c \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{16 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 142, normalized size = 0.45 \begin {gather*} \frac {c \sqrt {c+a^2 c x^2} \left (-\sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-4 \sinh ^{-1}(a x)\right )-8 \sqrt {2} \sqrt {-\sinh ^{-1}(a x)} \Gamma \left (\frac {3}{2},-2 \sinh ^{-1}(a x)\right )+\sqrt {\sinh ^{-1}(a x)} \left (32 \sinh ^{-1}(a x)^{3/2}-8 \sqrt {2} \Gamma \left (\frac {3}{2},2 \sinh ^{-1}(a x)\right )-\Gamma \left (\frac {3}{2},4 \sinh ^{-1}(a x)\right )\right )\right )}{128 a \sqrt {1+a^2 x^2} \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \sqrt {\arcsinh \left (a x \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \sqrt {\operatorname {asinh}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {\mathrm {asinh}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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