Optimal. Leaf size=355 \[ \frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{a/b} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-1-n} e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5819, 5556,
3389, 2212} \begin {gather*} \frac {3^{-n-1} e^{-\frac {3 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt {c^2 x^2+1}}+\frac {e^{-\frac {a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt {c^2 x^2+1}}+\frac {e^{a/b} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt {c^2 x^2+1}}+\frac {3^{-n-1} e^{\frac {3 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3389
Rule 5556
Rule 5819
Rubi steps
\begin {align*} \int x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx &=\frac {\sqrt {d+c^2 d x^2} \int x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int (a+b x)^n \cosh ^2(x) \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int \left (\frac {1}{4} (a+b x)^n \sinh (x)+\frac {1}{4} (a+b x)^n \sinh (3 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int (a+b x)^n \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^2 \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int (a+b x)^n \sinh (3 x) \, dx,x,\sinh ^{-1}(c x)\right )}{4 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt {1+c^2 x^2}}-\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^x (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {\sqrt {d+c^2 d x^2} \text {Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {3^{-1-n} e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {e^{a/b} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-1-n} e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{8 c^2 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.69, size = 229, normalized size = 0.65 \begin {gather*} \frac {d e^{-\frac {3 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (3 e^{\frac {4 a}{b}} \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^{-n} \Gamma \left (1+n,\frac {a}{b}+\sinh ^{-1}(c x)\right )+\left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \left (3^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+3 e^{\frac {2 a}{b}} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )+3^{-n} e^{\frac {6 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )}{24 c^2 \sqrt {d \left (1+c^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x \left (a +b \arcsinh \left (c x \right )\right )^{n} \sqrt {c^{2} d \,x^{2}+d}\, dx\]
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]
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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \sqrt {d \left (c^{2} x^{2} + 1\right )} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{n}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,\sqrt {d\,c^2\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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