Optimal. Leaf size=321 \[ -\frac {\left (d-c^2 d x^2\right )^{9/2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^6 d^3}+\frac {2 \left (d-c^2 d x^2\right )^{7/2} \left (a+b \cosh ^{-1}(c x)\right )}{7 c^6 d^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{5 c^6 d}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 0.44, antiderivative size = 366, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5798, 100, 12, 74, 5733, 1153} \[ -\frac {d x^4 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}-\frac {4 d x^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {8 d (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {c x-1} \sqrt {c x+1}}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 100
Rule 1153
Rule 5733
Rule 5798
Rubi steps
\begin {align*} \int x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int x^5 (-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {8 d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}-\frac {4 d x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d x^4 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right )^2 \left (8+20 c^2 x^2+35 c^4 x^4\right )}{315 c^6} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {8 d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}-\frac {4 d x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d x^4 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right )^2 \left (8+20 c^2 x^2+35 c^4 x^4\right ) \, dx}{315 c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {8 d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}-\frac {4 d x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d x^4 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int \left (8+4 c^2 x^2+3 c^4 x^4-50 c^6 x^6+35 c^8 x^8\right ) \, dx}{315 c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {8 b d x \sqrt {d-c^2 d x^2}}{315 c^5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b d x^3 \sqrt {d-c^2 d x^2}}{945 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^5 \sqrt {d-c^2 d x^2}}{525 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {10 b c d x^7 \sqrt {d-c^2 d x^2}}{441 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^9 \sqrt {d-c^2 d x^2}}{81 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {8 d (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{315 c^6}-\frac {4 d x^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{63 c^4}-\frac {d x^4 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^2}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 164, normalized size = 0.51 \[ -\frac {d \sqrt {d-c^2 d x^2} \left (35 c^3 x^4 (c x-1)^{5/2} (c x+1)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {4 (c x-1)^{5/2} (c x+1)^{5/2} \left (5 c^2 x^2+2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c}-b \left (\frac {35 c^8 x^9}{9}-\frac {50 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}+\frac {4 c^2 x^3}{3}+8 x\right )\right )}{315 c^5 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 245, normalized size = 0.76 \[ -\frac {315 \, {\left (35 \, b c^{10} d x^{10} - 85 \, b c^{8} d x^{8} + 53 \, b c^{6} d x^{6} + b c^{4} d x^{4} + 4 \, b c^{2} d x^{2} - 8 \, b d\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (1225 \, b c^{9} d x^{9} - 2250 \, b c^{7} d x^{7} + 189 \, b c^{5} d x^{5} + 420 \, b c^{3} d x^{3} + 2520 \, b c d x\right )} \sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} + 315 \, {\left (35 \, a c^{10} d x^{10} - 85 \, a c^{8} d x^{8} + 53 \, a c^{6} d x^{6} + a c^{4} d x^{4} + 4 \, a c^{2} d x^{2} - 8 \, a d\right )} \sqrt {-c^{2} d x^{2} + d}}{99225 \, {\left (c^{8} x^{2} - c^{6}\right )}} \]
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 = 0.65, size = 1376, normalized size = 4.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 223, normalized size = 0.69 \[ -\frac {1}{315} \, {\left (\frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{2} d} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{6} d}\right )} b \operatorname {arcosh}\left (c x\right ) - \frac {1}{315} \, {\left (\frac {35 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}{c^{2} d} + \frac {20 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{4} d} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{6} d}\right )} a + \frac {{\left (1225 \, c^{8} \sqrt {-d} d x^{9} - 2250 \, c^{6} \sqrt {-d} d x^{7} + 189 \, c^{4} \sqrt {-d} d x^{5} + 420 \, c^{2} \sqrt {-d} d x^{3} + 2520 \, \sqrt {-d} d x\right )} b}{99225 \, c^{5}} \]
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 \[ \int x^5\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
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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