Optimal. Leaf size=389 \[ -\frac {15 \sqrt {\pi } b^{5/2} c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}+\frac {15 \sqrt {\pi } b^{5/2} c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}-\frac {15 b^2 c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac {15 b^2 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}}{64 d^2}+\frac {5 b c \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac {5 b \sinh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{16 d^2}-\frac {c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac {\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d^2} \]
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Rubi [A] time = 1.13, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 18, 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, 5324, 5298, 2205, 2204, 5299} \[ -\frac {15 \sqrt {\pi } b^{5/2} c e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}+\frac {15 \sqrt {\pi } b^{5/2} c e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{256 d^2}-\frac {15 b^2 c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac {15 b^2 \cosh \left (2 \sinh ^{-1}(c+d x)\right ) \sqrt {a+b \sinh ^{-1}(c+d x)}}{64 d^2}+\frac {5 b c \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac {5 b \sinh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{16 d^2}-\frac {c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac {\cosh \left (2 \sinh ^{-1}(c+d x)\right ) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{4 d^2} \]
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 \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {c}{d}+\frac {x}{d}\right ) \left (a+b \sinh ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int (a+b x)^{5/2} \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^6 \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^6 \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^6 \cosh \left (\frac {a}{b}-\frac {x^2}{b}\right )+\frac {1}{2} x^6 \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^6 \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^6 \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) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac {5 \operatorname {Subst}\left (\int x^4 \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 {(5 c) \operatorname {Subst}\left (\int x^4 \sinh \left (\frac {a}{b}-\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{d^2}\\ &=\frac {5 b c \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac {c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac {5 b \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}-\frac {(15 b) \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 )}{16 d^2}-\frac {(15 b 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 )}{2 d^2}\\ &=-\frac {15 b^2 c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac {5 b c \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac {c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{64 d^2}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac {5 b \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}-\frac {\left (15 b^2\right ) \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 )}{64 d^2}-\frac {\left (15 b^2 c\right ) \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 )}{4 d^2}\\ &=-\frac {15 b^2 c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac {5 b c \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac {c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{64 d^2}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac {5 b \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}-\frac {\left (15 b^2\right ) \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 )}{128 d^2}-\frac {\left (15 b^2\right ) \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 )}{128 d^2}-\frac {\left (15 b^2 c\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^2}+\frac {\left (15 b^2 c\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^2}\\ &=-\frac {15 b^2 c (c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)}}{4 d^2}+\frac {5 b c \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}}{2 d^2}-\frac {c (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^{5/2}}{d^2}+\frac {15 b^2 \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{64 d^2}+\frac {\left (a+b \sinh ^{-1}(c+d x)\right )^{5/2} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{4 d^2}-\frac {15 b^{5/2} c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 b^{5/2} 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 )}{256 d^2}+\frac {15 b^{5/2} c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d^2}-\frac {15 b^{5/2} 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 )}{256 d^2}-\frac {5 b \left (a+b \sinh ^{-1}(c+d x)\right )^{3/2} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{16 d^2}\\ \end {align*}
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Mathematica [B] time = 10.19, size = 939, normalized size = 2.41 \[ \frac {480 c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) b^{5/2}-15 \sqrt {2 \pi } \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) b^{5/2}-480 c \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right ) b^{5/2}+15 \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right ) b^{5/2}-15 \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right ) b^{5/2}+128 \sinh ^{-1}(c+d x)^2 \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right ) b^2+120 \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right ) b^2-160 \sinh ^{-1}(c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)} \sinh \left (2 \sinh ^{-1}(c+d x)\right ) b^2-1920 c^2 \sqrt {a+b \sinh ^{-1}(c+d x)} b^2-512 c^2 \sinh ^{-1}(c+d x)^2 \sqrt {a+b \sinh ^{-1}(c+d x)} b^2-512 c d x \sinh ^{-1}(c+d x)^2 \sqrt {a+b \sinh ^{-1}(c+d x)} b^2-1920 c d x \sqrt {a+b \sinh ^{-1}(c+d x)} b^2+1280 c \sqrt {c^2+2 d x c+d^2 x^2+1} \sinh ^{-1}(c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)} b^2+256 a \sinh ^{-1}(c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right ) b+\frac {256 a^2 c e^{a/b} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {3}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right ) b}{\sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {256 a^2 c e^{-\frac {a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {3}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right ) b}{\sqrt {a+b \sinh ^{-1}(c+d x)}}-160 a \sqrt {a+b \sinh ^{-1}(c+d x)} \sinh \left (2 \sinh ^{-1}(c+d x)\right ) b-1024 a c^2 \sinh ^{-1}(c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)} b-1024 a c d x \sinh ^{-1}(c+d x) \sqrt {a+b \sinh ^{-1}(c+d x)} b+1280 a c \sqrt {c^2+2 d x c+d^2 x^2+1} \sqrt {a+b \sinh ^{-1}(c+d x)} b-128 a^2 c \sqrt {\pi } \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sqrt {b}+128 a^2 c \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right ) \sqrt {b}+32 \left (4 a^2-15 b^2\right ) c \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right ) \sqrt {b}+128 a^2 \sqrt {a+b \sinh ^{-1}(c+d x)} \cosh \left (2 \sinh ^{-1}(c+d x)\right )}{512 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 \[ \int {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int x \left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {5}{2}}\, dx \]
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 \[ \int {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} x\,{d x} \]
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 \[ \int x\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \]
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 \[ \int x \left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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