Optimal. Leaf size=793 \[ \frac {d^2 7^{-n-1} e^{-\frac {7 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 {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 5^{-n} e^{-\frac {5 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 {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 3^{1-n} e^{-\frac {3 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 {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 e^{-\frac {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 {a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d^2 e^{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 {a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 3^{1-n} e^{\frac {3 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 {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 5^{-n} e^{\frac {5 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 {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 7^{-n-1} e^{\frac {7 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 {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 1.05, antiderivative size = 793, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {5798, 5781, 5448, 3307, 2181} \[ \frac {d^2 7^{-n-1} e^{-\frac {7 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 {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 5^{-n} e^{-\frac {5 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 {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 3^{1-n} e^{-\frac {3 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 {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {5 d^2 e^{-\frac {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 {a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 d^2 e^{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 {a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 3^{1-n} e^{\frac {3 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 {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {d^2 5^{-n} e^{\frac {5 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 {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 7^{-n-1} e^{\frac {7 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 {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \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 \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 (-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 (x) \sinh ^6(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \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}{64} (a+b x)^n \cosh (x)+\frac {9}{64} (a+b x)^n \cosh (3 x)-\frac {5}{64} (a+b x)^n \cosh (5 x)+\frac {1}{64} (a+b x)^n \cosh (7 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \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 (7 x) \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (5 x) \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (9 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (3 x) \, dx,x,\cosh ^{-1}(c x)\right )}{64 c^2 \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^{-7 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2 \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^{7 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-5 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{5 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (9 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (9 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {7^{-1-n} d^2 e^{-\frac {7 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 {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5^{-n} d^2 e^{-\frac {5 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 {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3^{1-n} d^2 e^{-\frac {3 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 {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d^2 e^{-\frac {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 {a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 d^2 e^{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 {a+b \cosh ^{-1}(c x)}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3^{1-n} d^2 e^{\frac {3 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 {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5^{-n} d^2 e^{\frac {5 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 {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {7^{-1-n} d^2 e^{\frac {7 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 {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{128 c^2 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 3.98, size = 633, normalized size = 0.80 \[ \frac {d^3 5^{-n} 21^{-n-1} e^{-\frac {7 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 )^{-3 n} \left (\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \left (e^{\frac {2 a}{b}} \left (21^{n+1} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,-\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-9\ 5^n 7^{n+1} e^{\frac {2 a}{b}} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+105^{n+1} e^{\frac {4 a}{b}} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )+16\ 5^n 7^{n+1} e^{\frac {8 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{2 n} \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n \Gamma \left (n+1,\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-5^n 7^{n+2} e^{\frac {8 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )-21^{n+1} e^{\frac {10 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,\frac {5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3^{n+1} 5^n e^{\frac {12 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{3 n} \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )-3^{n+1} 5^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,-\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )-105^{n+1} e^{\frac {8 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{2 n} \Gamma \left (n+1,\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )}{128 c^2 \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}\, dx \]
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 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n} x\,{d x} \]
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 \[ \int x\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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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