Optimal. Leaf size=259 \[ -\frac {\sqrt {\pi } \sqrt {b} c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}+\frac {\sqrt {\pi } \sqrt {b} c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^2}+\frac {\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d^2} \]
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Rubi [A] time = 0.70, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5865, 5805, 6741, 6742, 5325, 5298, 2205, 2204, 5324, 5299} \[ -\frac {\sqrt {\pi } \sqrt {b} c e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}+\frac {\sqrt {\pi } \sqrt {b} c e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^2}+\frac {\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d^2} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5298
Rule 5299
Rule 5324
Rule 5325
Rule 5805
Rule 5865
Rule 6741
Rule 6742
Rubi steps
\begin {align*} \int x \sqrt {a+b \sinh ^{-1}(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right ) \sqrt {a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \sqrt {a+b x} \cosh (x) \left (-\frac {c}{d}+\frac {\sinh (x)}{d}\right ) \, dx,x,\sinh ^{-1}(c+d x)\right )}{d}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^2 \cosh \left (\frac {a-x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \left (c+\sinh \left (\frac {a-x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac {2 \operatorname {Subst}\left (\int \left (c x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+\frac {1}{2} x^2 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right )\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac {\operatorname {Subst}\left (\int x^2 \sinh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}-\frac {(2 c) \operatorname {Subst}\left (\int x^2 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b d^2}\\ &=-\frac {c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^2}+\frac {\sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac {\operatorname {Subst}\left (\int \cosh \left (\frac {2 a}{b}-\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{4 d^2}-\frac {c \operatorname {Subst}\left (\int \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{d^2}\\ &=-\frac {c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^2}+\frac {\sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac {\operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{8 d^2}-\frac {\operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{8 d^2}-\frac {c \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2 d^2}+\frac {c \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{2 d^2}\\ &=-\frac {c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{d^2}+\frac {\sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac {\sqrt {b} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {b} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}+\frac {\sqrt {b} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4 d^2}-\frac {\sqrt {b} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}\\ \end {align*}
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Mathematica [A] time = 1.82, size = 251, normalized size = 0.97 \[ \frac {-\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \sqrt {b} \left (\sinh \left (\frac {2 a}{b}\right )-\cosh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}-16 c e^{-\frac {a}{b}} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}}}-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)}}\right )}{32 d^2} \]
Warning: Unable to verify antiderivative.
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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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.52, size = 0, normalized size = 0.00 \[ \int x \sqrt {a +b \arcsinh \left (d x +c \right )}\, dx \]
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 \[ \int \sqrt {b \operatorname {arsinh}\left (d x + c\right ) + a} x\,{d x} \]
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 \[ \int x\,\sqrt {a+b\,\mathrm {asinh}\left (c+d\,x\right )} \,d x \]
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 \[ \int x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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