Optimal. Leaf size=218 \[ -\frac {\left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2 d}+\frac {b d^2 x \sqrt {d-c^2 d x^2}}{7 c \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c^5 d^2 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.28, antiderivative size = 233, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5798, 5718, 194} \[ -\frac {d^2 (1-c x)^3 (c x+1)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {b c^5 d^2 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d^2 x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d^2 x \sqrt {d-c^2 d x^2}}{7 c \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 194
Rule 5718
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 ) \, 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 ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^3 \, dx}{7 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}-\frac {\left (b d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+3 c^2 x^2-3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d^2 x \sqrt {d-c^2 d x^2}}{7 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 x^3 \sqrt {d-c^2 d x^2}}{7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b c^3 d^2 x^5 \sqrt {d-c^2 d x^2}}{35 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^5 d^2 x^7 \sqrt {d-c^2 d x^2}}{49 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d^2 (1-c x)^3 (1+c x)^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^2}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 117, normalized size = 0.54 \[ \frac {d^2 \sqrt {d-c^2 d x^2} \left (35 a \left (c^2 x^2-1\right )^4+35 b \left (c^2 x^2-1\right )^4 \cosh ^{-1}(c x)+b c x \sqrt {c x-1} \sqrt {c x+1} \left (-5 c^6 x^6+21 c^4 x^4-35 c^2 x^2+35\right )\right )}{245 c^2 \left (c^2 x^2-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 241, normalized size = 1.11 \[ \frac {35 \, {\left (b c^{8} d^{2} x^{8} - 4 \, b c^{6} d^{2} x^{6} + 6 \, b c^{4} d^{2} x^{4} - 4 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (5 \, b c^{7} d^{2} x^{7} - 21 \, b c^{5} d^{2} x^{5} + 35 \, b c^{3} d^{2} x^{3} - 35 \, b c d^{2} x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 35 \, {\left (a c^{8} d^{2} x^{8} - 4 \, a c^{6} d^{2} x^{6} + 6 \, a c^{4} d^{2} x^{4} - 4 \, a c^{2} d^{2} x^{2} + a d^{2}\right )} \sqrt {-c^{2} d x^{2} + d}}{245 \, {\left (c^{4} x^{2} - c^{2}\right )}} \]
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 = 0.35, size = 956, normalized size = 4.39 \[ -\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{7 c^{2} d}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (64 c^{8} x^{8}-144 c^{6} x^{6}+64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+104 c^{4} x^{4}-112 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}-25 c^{2} x^{2}+56 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-7 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) \left (-1+7 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{6272 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (16 c^{6} x^{6}-28 c^{4} x^{4}+16 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}+13 c^{2} x^{2}-20 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+5 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -1\right ) \left (-1+5 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{640 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (4 c^{4} x^{4}-5 c^{2} x^{2}+4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +1\right ) \left (-1+3 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {5 \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \left (1+\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-4 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+4 c^{4} x^{4}+3 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -5 c^{2} x^{2}+1\right ) \left (1+3 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-16 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}+16 c^{6} x^{6}+20 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}-28 c^{4} x^{4}-5 \sqrt {c x +1}\, \sqrt {c x -1}\, x c +13 c^{2} x^{2}-1\right ) \left (1+5 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{640 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-64 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{7} c^{7}+64 c^{8} x^{8}+112 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{5} c^{5}-144 c^{6} x^{6}-56 \sqrt {c x +1}\, \sqrt {c x -1}\, x^{3} c^{3}+104 c^{4} x^{4}+7 \sqrt {c x +1}\, \sqrt {c x -1}\, x c -25 c^{2} x^{2}+1\right ) \left (1+7 \,\mathrm {arccosh}\left (c x \right )\right ) d^{2}}{6272 \left (c x +1\right ) c^{2} \left (c x -1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 118, normalized size = 0.54 \[ -\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} b \operatorname {arcosh}\left (c x\right )}{7 \, c^{2} d} - \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}} a}{7 \, c^{2} d} - \frac {{\left (5 \, c^{6} \sqrt {-d} d^{3} x^{7} - 21 \, c^{4} \sqrt {-d} d^{3} x^{5} + 35 \, c^{2} \sqrt {-d} d^{3} x^{3} - 35 \, \sqrt {-d} d^{3} x\right )} b}{245 \, c d} \]
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 \[ \int x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]
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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