Optimal. Leaf size=542 \[ \frac {5^{-1-n} d e^{-\frac {5 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-n} d e^{-\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {d e^{-\frac {a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {d e^{a/b} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-n} d e^{\frac {3 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {5^{-1-n} d e^{\frac {5 a}{b}} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}} \]
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Rubi [A]
time = 0.35, antiderivative size = 542, normalized size of antiderivative = 1.00, number of steps
used = 12, 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 {d 5^{-n-1} e^{-\frac {5 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 {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d 3^{-n} 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 )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d 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 )}{16 c^2 \sqrt {c^2 x^2+1}}+\frac {d 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 )}{16 c^2 \sqrt {c^2 x^2+1}}+\frac {d 3^{-n} 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 )}{32 c^2 \sqrt {c^2 x^2+1}}+\frac {d 5^{-n-1} e^{\frac {5 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 {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 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 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh ^4(x) \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{8} (a+b x)^n \sinh (x)+\frac {3}{16} (a+b x)^n \sinh (3 x)+\frac {1}{16} (a+b x)^n \sinh (5 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \sinh (5 x) \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \sinh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{8 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (3 d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \sinh (3 x) \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2 \sqrt {1+c^2 x^2}}\\ &=-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-5 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{5 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^x (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (3 d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {\left (3 d \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^2 \sqrt {1+c^2 x^2}}\\ &=\frac {5^{-1-n} d e^{-\frac {5 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 {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-n} d 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 )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {d 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 )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {d 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 )}{16 c^2 \sqrt {1+c^2 x^2}}+\frac {3^{-n} d 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 )}{32 c^2 \sqrt {1+c^2 x^2}}+\frac {5^{-1-n} d e^{\frac {5 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 {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 c^2 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.24, size = 390, normalized size = 0.72 \begin {gather*} \frac {15^{-1-n} d^2 e^{-\frac {5 a}{b}} \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^n \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^{-2 n} \left (2\ 15^{1+n} e^{\frac {6 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (1+n,\frac {a}{b}+\sinh ^{-1}(c x)\right )+3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \left (3^n \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+5^{1+n} e^{\frac {2 a}{b}} \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 )+2\ 3^n 5^{1+n} e^{\frac {4 a}{b}} \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (1+n,-\frac {a+b \sinh ^{-1}(c x)}{b}\right )+5^{1+n} e^{\frac {8 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+3^n e^{\frac {10 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{2 n} \Gamma \left (1+n,\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )}{32 c^2 \sqrt {d+c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{n}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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