Optimal. Leaf size=205 \[ -\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {3 b^{3/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {3 b^{3/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d} \]
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Rubi [A]
time = 0.33, antiderivative size = 205, 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 = {5859, 12,
5777, 5812, 5783, 5780, 5556, 3389, 2211, 2236, 2235} \begin {gather*} -\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {3 b e \sqrt {(c+d x)^2+1} (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5777
Rule 5780
Rule 5783
Rule 5812
Rule 5859
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int e x \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int x \left (a+b \sinh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {(3 b e) \text {Subst}\left (\int \frac {x^2 \sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(3 b e) \text {Subst}\left (\int \frac {\sqrt {a+b \sinh ^{-1}(x)}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}\\ &=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{32 d}\\ &=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64 d}+\frac {\left (3 b^2 e\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{64 d}\\ &=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {(3 b e) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{32 d}+\frac {(3 b e) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{32 d}\\ &=-\frac {3 b e (c+d x) \sqrt {1+(c+d x)^2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{8 d}+\frac {e \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}-\frac {3 b^{3/2} e 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 )}{64 d}+\frac {3 b^{3/2} e 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 )}{64 d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 142, normalized size = 0.69 \begin {gather*} \frac {b e e^{-\frac {2 a}{b}} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (-\sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {5}{2},\frac {2 \left (a+b \sinh ^{-1}(c+d x)\right )}{b}\right )\right )}{16 \sqrt {2} d \sqrt {-\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^2}{b^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right ) \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {3}{2}}\, dx\]
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*} e \left (\int a c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int a d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b c \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int b d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx\right ) \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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