Optimal. Leaf size=186 \[ -\frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {15 b^2 (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{d} \]
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Rubi [A] time = 0.61, antiderivative size = 186, 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 = {5864, 5654, 5718, 5781, 3307, 2180, 2204, 2205} \[ -\frac {15 \sqrt {\pi } b^{5/2} e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {15 \sqrt {\pi } b^{5/2} e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {15 b^2 (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{d} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5654
Rule 5718
Rule 5781
Rule 5864
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac {(5 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d}\\ &=-\frac {5 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{d}+\frac {\left (15 b^2\right ) \operatorname {Subst}\left (\int \sqrt {a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{4 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{8 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{d}-\frac {\left (15 b^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-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,\cosh ^{-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,\cosh ^{-1}(c+d x)\right )}{16 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-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 \cosh ^{-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 \cosh ^{-1}(c+d x)}\right )}{8 d}\\ &=\frac {15 b^2 (c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}}{4 d}-\frac {5 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{2 d}+\frac {(c+d x) \left (a+b \cosh ^{-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 \cosh ^{-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 \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}\\ \end {align*}
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Mathematica [B] time = 3.56, size = 494, normalized size = 2.66 \[ \frac {-\sqrt {\pi } \sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-\sqrt {\pi } \sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+8 a^2 e^{-\frac {a}{b}} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (\frac {e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{\sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)}}+\frac {\Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{\sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}}}\right )+4 a b \left (\frac {\sqrt {\pi } (2 a-3 b) \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {\sqrt {\pi } (2 a+3 b) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{\sqrt {b}}-12 \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \sqrt {a+b \cosh ^{-1}(c+d x)}+8 (c+d x) \cosh ^{-1}(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}\right )+4 b \sqrt {a+b \cosh ^{-1}(c+d x)} \left (2 \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1) \left (a-5 b \cosh ^{-1}(c+d x)\right )+b (c+d x) \left (4 \cosh ^{-1}(c+d x)^2+15\right )\right )}{16 d} \]
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 \[ \int {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
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 \[ \int \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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