Optimal. Leaf size=269 \[ \frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {c e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}-\frac {c e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2} \]
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Rubi [A]
time = 0.38, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5859, 5829,
5773, 5819, 3389, 2211, 2236, 2235, 5778, 3388} \begin {gather*} \frac {\sqrt {\pi } c e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}-\frac {\sqrt {\pi } c e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {\sqrt {\frac {\pi }{2}} e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {2 c \sqrt {(c+d x)^2+1}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {(c+d x)^2+1}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3389
Rule 5773
Rule 5778
Rule 5819
Rule 5829
Rule 5859
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \sinh ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {-\frac {c}{d}+\frac {x}{d}}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {c}{d \left (a+b \sinh ^{-1}(x)\right )^{3/2}}+\frac {x}{d \left (a+b \sinh ^{-1}(x)\right )^{3/2}}\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {x}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \text {Subst}\left (\int \frac {1}{\left (a+b \sinh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {2 \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}-\frac {(2 c) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sqrt {a+b \sinh ^{-1}(x)}} \, dx,x,c+d x\right )}{b d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {\text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}+\frac {\text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}-\frac {(2 c) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {2 \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 )}{b^2 d^2}+\frac {2 \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 )}{b^2 d^2}+\frac {c \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}-\frac {c \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c+d x)\right )}{b d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {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 )}{b^{3/2} d^2}+\frac {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 )}{b^{3/2} d^2}+\frac {(2 c) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b^2 d^2}-\frac {(2 c) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c+d x)}\right )}{b^2 d^2}\\ &=\frac {2 c \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {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 )}{b^{3/2} d^2}-\frac {c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \sinh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d^2}+\frac {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 )}{b^{3/2} d^2}\\ \end {align*}
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Mathematica [A]
time = 1.77, size = 301, normalized size = 1.12 \begin {gather*} -\frac {c e^{-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \left (-e^{a/b} \left (1+e^{2 \sinh ^{-1}(c+d x)}\right )+e^{\frac {2 a}{b}+\sinh ^{-1}(c+d x)} \sqrt {\frac {a}{b}+\sinh ^{-1}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\sinh ^{-1}(c+d x)\right )+e^{\sinh ^{-1}(c+d x)} \sqrt {-\frac {a+b \sinh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \sinh ^{-1}(c+d x)}{b}\right )\right )}{b d^2 \sqrt {a+b \sinh ^{-1}(c+d x)}}+\frac {\sqrt {2 \pi } \text {Erfi}\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 )+\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 )-\frac {2 \sqrt {b} \sinh \left (2 \sinh ^{-1}(c+d x)\right )}{\sqrt {a+b \sinh ^{-1}(c+d x)}}}{2 b^{3/2} d^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x}{\left (a +b \arcsinh \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
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*} \int \frac {x}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \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 \frac {x}{{\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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