Optimal. Leaf size=314 \[ -\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 d x^9}-\frac {2 c^2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{63 d x^7}-\frac {b c d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c^9 d^2 \log (x) \sqrt {d-c^2 d x^2}}{63 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{21 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^5 d^2 \sqrt {d-c^2 d x^2}}{42 x^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^3 d^2 \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.53, antiderivative size = 448, normalized size of antiderivative = 1.43, number of steps used = 7, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5798, 97, 12, 103, 95, 5733, 446, 78, 43} \[ \frac {2 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac {5 c^2 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac {d^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{21 x^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^5 d^2 \sqrt {d-c^2 d x^2}}{42 x^4 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^3 d^2 \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b c^9 d^2 \log (x) \sqrt {d-c^2 d x^2}}{63 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 78
Rule 95
Rule 97
Rule 103
Rule 446
Rule 5733
Rule 5798
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{10}} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{x^{10}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac {2 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-7-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{63 x^9} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac {2 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-7-2 c^2 x^2\right ) \left (1-c^2 x^2\right )^3}{x^9} \, dx}{63 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac {2 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (-7-2 c^2 x\right ) \left (1-c^2 x\right )^3}{x^5} \, dx,x,x^2\right )}{126 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac {2 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {\left (1-c^2 x\right )^3}{x^4} \, dx,x,x^2\right )}{63 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac {2 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}+\frac {\left (b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^4}-\frac {3 c^2}{x^3}+\frac {3 c^4}{x^2}-\frac {c^6}{x}\right ) \, dx,x,x^2\right )}{63 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c^3 d^2 \sqrt {d-c^2 d x^2}}{189 x^6 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 \sqrt {d-c^2 d x^2}}{42 x^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^7 d^2 \sqrt {d-c^2 d x^2}}{21 x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \left (1-c^2 x^2\right )^4 \sqrt {d-c^2 d x^2}}{72 x^8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {c^4 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{21 x^5}+\frac {c^6 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^3}+\frac {2 c^8 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x}+\frac {5 c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 x^7}-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 x^9}-\frac {2 b c^9 d^2 \sqrt {d-c^2 d x^2} \log (x)}{63 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 147, normalized size = 0.47 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (48 c^2 x^2 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )+168 (c x-1)^{7/2} (c x+1)^{7/2} \left (a+b \cosh ^{-1}(c x)\right )-b c x \left (48 c^8 x^8 \log (x)-12 c^6 x^6+90 c^4 x^4-76 c^2 x^2+21\right )\right )}{1512 x^9 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 795, normalized size = 2.53 \[ \left [\frac {24 \, {\left (2 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 16 \, b c^{6} d^{2} x^{6} + 34 \, b c^{4} d^{2} x^{4} - 26 \, b c^{2} d^{2} x^{2} + 7 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 24 \, {\left (b c^{11} d^{2} x^{11} - b c^{9} d^{2} x^{9}\right )} \sqrt {-d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{4} - 1\right )} \sqrt {-d} - d}{c^{2} x^{4} - x^{2}}\right ) + {\left (12 \, b c^{7} d^{2} x^{7} - 90 \, b c^{5} d^{2} x^{5} - {\left (12 \, b c^{7} - 90 \, b c^{5} + 76 \, b c^{3} - 21 \, b c\right )} d^{2} x^{9} + 76 \, b c^{3} d^{2} x^{3} - 21 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 24 \, {\left (2 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 16 \, a c^{6} d^{2} x^{6} + 34 \, a c^{4} d^{2} x^{4} - 26 \, a c^{2} d^{2} x^{2} + 7 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{1512 \, {\left (c^{2} x^{11} - x^{9}\right )}}, -\frac {48 \, {\left (b c^{11} d^{2} x^{11} - b c^{9} d^{2} x^{9}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} {\left (x^{2} + 1\right )} \sqrt {d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - 24 \, {\left (2 \, b c^{10} d^{2} x^{10} - b c^{8} d^{2} x^{8} - 16 \, b c^{6} d^{2} x^{6} + 34 \, b c^{4} d^{2} x^{4} - 26 \, b c^{2} d^{2} x^{2} + 7 \, b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (12 \, b c^{7} d^{2} x^{7} - 90 \, b c^{5} d^{2} x^{5} - {\left (12 \, b c^{7} - 90 \, b c^{5} + 76 \, b c^{3} - 21 \, b c\right )} d^{2} x^{9} + 76 \, b c^{3} d^{2} x^{3} - 21 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} - 24 \, {\left (2 \, a c^{10} d^{2} x^{10} - a c^{8} d^{2} x^{8} - 16 \, a c^{6} d^{2} x^{6} + 34 \, a c^{4} d^{2} x^{4} - 26 \, a c^{2} d^{2} x^{2} + 7 \, a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{1512 \, {\left (c^{2} x^{11} - x^{9}\right )}}\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 [B] time = 1.09, size = 5006, normalized size = 15.94 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 187, normalized size = 0.60 \[ -\frac {1}{1512} \, {\left (48 \, c^{8} \sqrt {-d} d^{2} \log \relax (x) - \frac {12 \, c^{6} \sqrt {-d} d^{2} x^{6} - 90 \, c^{4} \sqrt {-d} d^{2} x^{4} + 76 \, c^{2} \sqrt {-d} d^{2} x^{2} - 21 \, \sqrt {-d} d^{2}}{x^{8}}\right )} b c - \frac {1}{63} \, b {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{7}} + \frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{9}}\right )} \operatorname {arcosh}\left (c x\right ) - \frac {1}{63} \, a {\left (\frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} c^{2}}{d x^{7}} + \frac {7 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{9}}\right )} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^{10}} \,d x \]
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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