Optimal. Leaf size=179 \[ \frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {15 b^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d} \]
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Rubi [A] time = 0.39, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5863, 5653, 5717, 5779, 3308, 2180, 2204, 2205} \[ \frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {15 b^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5653
Rule 5717
Rule 5779
Rule 5863
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^{3/2}}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {5 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int \sqrt {a+b \sinh ^{-1}(x)} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{8 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{8 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{8 d}-\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{8 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}\\ \end {align*}
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Mathematica [B] time = 2.45, size = 458, normalized size = 2.56 \[ \frac {\sqrt {b} \left (\sqrt {\pi } \left (4 a^2-12 a b+15 b^2\right ) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {\pi } \left (4 a^2+12 a b+15 b^2\right ) \left (\sinh \left (\frac {a}{b}\right )-\cosh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+4 \sqrt {b} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (2 \sqrt {(c+d x)^2+1} \left (a-5 b \sinh ^{-1}(c+d x)\right )+b (c+d x) \left (4 \sinh ^{-1}(c+d x)^2+15\right )\right )\right )+8 a^2 e^{-\frac {a}{b}} \sqrt {a+b \sinh ^{-1}(c+d x)} \left (\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}}}-\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)}}\right )+4 a \sqrt {b} \left (\sqrt {\pi } (3 b-2 a) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {\pi } (2 a+3 b) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )+4 \sqrt {b} \left (2 (c+d x) \sinh ^{-1}(c+d x)-3 \sqrt {(c+d x)^2+1}\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{16 d} \]
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 \[ \int {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
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 \[ \int \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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