Optimal. Leaf size=816 \[ -\frac {5 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-11-3 n} d^2 e^{-\frac {8 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 {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 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 {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-2 (4+n)} d^2 e^{-\frac {4 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 {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} d^2 e^{-\frac {2 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 {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d^2 e^{\frac {2 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 {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-2 (4+n)} d^2 e^{\frac {4 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 {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 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 {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-11-3 n} d^2 e^{\frac {8 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 {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}} \]
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Rubi [A]
time = 0.55, antiderivative size = 816, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5819, 5556,
3388, 2212} \begin {gather*} \frac {2^{-3 n-11} d^2 e^{-\frac {8 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}+\frac {2^{-n-7} 3^{-n-1} d^2 e^{-\frac {6 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}+\frac {2^{-2 (n+4)} d^2 e^{-\frac {4 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-n-7} d^2 e^{-\frac {2 a}{b}} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c^2 x^2+1}}-\frac {5 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^{n+1}}{128 b c^3 (n+1) \sqrt {c^2 x^2+1}}+\frac {2^{-n-7} d^2 e^{\frac {2 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 {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-2 (n+4)} d^2 e^{\frac {4 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 {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-n-7} 3^{-n-1} d^2 e^{\frac {6 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 {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}}-\frac {2^{-3 n-11} d^2 e^{\frac {8 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 {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c^2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2212
Rule 3388
Rule 5556
Rule 5819
Rubi steps
\begin {align*} \int x^2 \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^n \, dx}{\sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh ^6(x) \sinh ^2(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt {1+c^2 x^2}}\\ &=\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {5}{128} (a+b x)^n-\frac {1}{32} (a+b x)^n \cosh (2 x)+\frac {1}{32} (a+b x)^n \cosh (4 x)+\frac {1}{32} (a+b x)^n \cosh (6 x)+\frac {1}{128} (a+b x)^n \cosh (8 x)\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {5 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (8 x) \, dx,x,\sinh ^{-1}(c x)\right )}{128 c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (4 x) \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (6 x) \, dx,x,\sinh ^{-1}(c x)\right )}{32 c^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {5 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-8 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{256 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{8 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{256 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-6 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-4 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}-\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{4 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int e^{6 x} (a+b x)^n \, dx,x,\sinh ^{-1}(c x)\right )}{64 c^3 \sqrt {1+c^2 x^2}}\\ &=-\frac {5 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {1+c^2 x^2}}+\frac {2^{-11-3 n} d^2 e^{-\frac {8 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 {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 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 {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {4^{-4-n} d^2 e^{-\frac {4 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 {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} d^2 e^{-\frac {2 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 {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}+\frac {2^{-7-n} d^2 e^{\frac {2 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 {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {4^{-4-n} d^2 e^{\frac {4 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 {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 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 {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}-\frac {2^{-11-3 n} d^2 e^{\frac {8 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 {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 5.19, size = 667, normalized size = 0.82 \begin {gather*} -\frac {2^{-11-3 n} 3^{-1-n} d^3 e^{-\frac {8 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 )^{-n} \left (-3^{1+n} b (1+n) \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (1+n,-\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-4^{2+n} b e^{\frac {2 a}{b}} (1+n) \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (1+n,-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-2^{3+n} 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (1+n,-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+3^{1+n} 4^{2+n} b e^{\frac {6 a}{b}} (1+n) \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )^n \Gamma \left (1+n,-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{\frac {8 a}{b}} \left (5\ 2^{4+3 n} 3^{1+n} a \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n+5\ 2^{4+3 n} 3^{1+n} b \sinh ^{-1}(c x) \left (-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{b^2}\right )^n-3^{1+n} 4^{2+n} b e^{\frac {2 a}{b}} (1+n) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+2^{3+n} 3^{1+n} b e^{\frac {4 a}{b}} (1+n) \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+4^{2+n} b e^{\frac {6 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+4^{2+n} b e^{\frac {6 a}{b}} n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+3^{1+n} b e^{\frac {8 a}{b}} \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+3^{1+n} b e^{\frac {8 a}{b}} n \left (-\frac {a+b \sinh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {8 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )\right )\right )}{b c^3 (1+n) \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 \[\int x^{2} \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \arcsinh \left (c x \right )\right )^{n}\, dx\]
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]
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 x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^n\,{\left (d\,c^2\,x^2+d\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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