Optimal. Leaf size=298 \[ \frac {15 \sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {a^2 x^2+1}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}-\frac {5 a x^2 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {a^2 x^2+1}}-\frac {5 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt {a^2 x^2+1}}+\frac {15}{32} x \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)} \]
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Rubi [A] time = 0.30, antiderivative size = 298, normalized size of antiderivative = 1.00, 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 = {5682, 5675, 5663, 5758, 5669, 5448, 12, 3308, 2180, 2204, 2205} \[ \frac {15 \sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {a^2 x^2+1}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {a^2 x^2+1}}+\frac {\sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {a^2 x^2+1}}+\frac {1}{2} x \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{5/2}-\frac {5 a x^2 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {a^2 x^2+1}}-\frac {5 \sqrt {a^2 c x^2+c} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt {a^2 x^2+1}}+\frac {15}{32} x \sqrt {a^2 c x^2+c} \sqrt {\sinh ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5663
Rule 5669
Rule 5675
Rule 5682
Rule 5758
Rubi steps
\begin {align*} \int \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2} \, dx &=\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \int \frac {\sinh ^{-1}(a x)^{5/2}}{\sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {1+a^2 x^2}}-\frac {\left (5 a \sqrt {c+a^2 c x^2}\right ) \int x \sinh ^{-1}(a x)^{3/2} \, dx}{4 \sqrt {1+a^2 x^2}}\\ &=-\frac {5 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {1+a^2 x^2}}+\frac {\left (15 a^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2 \sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {1+a^2 x^2}}\\ &=\frac {15}{32} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}-\frac {5 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {1+a^2 x^2}}-\frac {\left (15 \sqrt {c+a^2 c x^2}\right ) \int \frac {\sqrt {\sinh ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{32 \sqrt {1+a^2 x^2}}-\frac {\left (15 a \sqrt {c+a^2 c x^2}\right ) \int \frac {x}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{64 \sqrt {1+a^2 x^2}}\\ &=\frac {15}{32} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt {1+a^2 x^2}}-\frac {5 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {1+a^2 x^2}}-\frac {\left (15 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}\\ &=\frac {15}{32} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt {1+a^2 x^2}}-\frac {5 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {1+a^2 x^2}}-\frac {\left (15 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a \sqrt {1+a^2 x^2}}\\ &=\frac {15}{32} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt {1+a^2 x^2}}-\frac {5 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {1+a^2 x^2}}-\frac {\left (15 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{128 a \sqrt {1+a^2 x^2}}\\ &=\frac {15}{32} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt {1+a^2 x^2}}-\frac {5 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {1+a^2 x^2}}+\frac {\left (15 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}-\frac {\left (15 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{256 a \sqrt {1+a^2 x^2}}\\ &=\frac {15}{32} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt {1+a^2 x^2}}-\frac {5 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {1+a^2 x^2}}+\frac {\left (15 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}-\frac {\left (15 \sqrt {c+a^2 c x^2}\right ) \operatorname {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{128 a \sqrt {1+a^2 x^2}}\\ &=\frac {15}{32} x \sqrt {c+a^2 c x^2} \sqrt {\sinh ^{-1}(a x)}-\frac {5 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{16 a \sqrt {1+a^2 x^2}}-\frac {5 a x^2 \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{3/2}}{8 \sqrt {1+a^2 x^2}}+\frac {1}{2} x \sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{5/2}+\frac {\sqrt {c+a^2 c x^2} \sinh ^{-1}(a x)^{7/2}}{7 a \sqrt {1+a^2 x^2}}+\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}-\frac {15 \sqrt {\frac {\pi }{2}} \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{256 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 135, normalized size = 0.45 \[ \frac {\sqrt {c \left (a^2 x^2+1\right )} \left (105 \sqrt {2 \pi } \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )-105 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )+8 \sqrt {\sinh ^{-1}(a x)} \left (64 \sinh ^{-1}(a x)^3+7 \left (16 \sinh ^{-1}(a x)^2+15\right ) \sinh \left (2 \sinh ^{-1}(a x)\right )-140 \sinh ^{-1}(a x) \cosh \left (2 \sinh ^{-1}(a x)\right )\right )\right )}{3584 a \sqrt {a^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.34, size = 0, normalized size = 0.00 \[ \int \arcsinh \left (a x \right )^{\frac {5}{2}} \sqrt {a^{2} c \,x^{2}+c}\, dx \]
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 \[ \int \sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\mathrm {asinh}\left (a\,x\right )}^{5/2}\,\sqrt {c\,a^2\,x^2+c} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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