Optimal. Leaf size=870 \[ \frac {2^{-3 n-11} d^2 e^{-\frac {8 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2^{-n-7} 3^{-n-1} d^2 e^{-\frac {6 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2^{-2 (n+4)} d^2 e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2^{-n-7} d^2 e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{128 b c^3 (n+1) \sqrt {c x-1} \sqrt {c x+1}}-\frac {2^{-n-7} d^2 e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2^{-2 (n+4)} d^2 e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2^{-n-7} 3^{-n-1} d^2 e^{\frac {6 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2^{-3 n-11} d^2 e^{\frac {8 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 1.24, antiderivative size = 870, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {5798, 5781, 5448, 3307, 2181} \[ \frac {2^{-3 n-11} d^2 e^{-\frac {8 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2^{-n-7} 3^{-n-1} d^2 e^{-\frac {6 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2^{-2 (n+4)} d^2 e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2^{-n-7} d^2 e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \text {Gamma}\left (n+1,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right ) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n}}{c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{128 b c^3 (n+1) \sqrt {c x-1} \sqrt {c x+1}}-\frac {2^{-n-7} d^2 e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2^{-2 (n+4)} d^2 e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {2^{-n-7} 3^{-n-1} d^2 e^{\frac {6 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2^{-3 n-11} d^2 e^{\frac {8 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 2181
Rule 3307
Rule 5448
Rule 5781
Rule 5798
Rubi steps
\begin {align*} \int x^2 \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int x^2 (-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh ^2(x) \sinh ^6(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {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,\cosh ^{-1}(c x)\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (8 x) \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (4 x) \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (6 x) \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-8 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{8 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-6 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{6 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{128 b c^3 (1+n) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-11-3 n} d^2 e^{-\frac {8 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-7-n} 3^{-1-n} d^2 e^{-\frac {6 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4^{-4-n} d^2 e^{-\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-7-n} d^2 e^{-\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-7-n} d^2 e^{\frac {2 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {4^{-4-n} d^2 e^{\frac {4 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-7-n} 3^{-1-n} d^2 e^{\frac {6 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-11-3 n} d^2 e^{\frac {8 a}{b}} \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 7.41, size = 677, normalized size = 0.78 \[ \frac {d^3 2^{-3 n-11} 3^{-n-1} e^{-\frac {8 a}{b}} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (e^{\frac {8 a}{b}} \left (5 a 2^{3 n+4} 3^{n+1} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n+5 b 2^{3 n+4} 3^{n+1} \cosh ^{-1}(c x) \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n+b 3^{n+1} 4^{n+2} (n+1) e^{\frac {2 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 2^{n+3} 3^{n+1} (n+1) e^{\frac {4 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b 4^{n+2} e^{\frac {6 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b 4^{n+2} n e^{\frac {6 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} e^{\frac {8 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 3^{n+1} n e^{\frac {8 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )+b \left (-3^{n+1}\right ) (n+1) \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+b 4^{n+2} (n+1) e^{\frac {2 a}{b}} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b 2^{n+3} 3^{n+1} (n+1) e^{\frac {4 a}{b}} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-b 3^{n+1} 4^{n+2} (n+1) e^{\frac {6 a}{b}} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{b c^3 (n+1) \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}\right )} \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}\, dx \]
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 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x^{2}\,{d x} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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