Optimal. Leaf size=360 \[ \frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^5 \sqrt {c x-1} \sqrt {c x+1}}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^4}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}} \]
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Rubi [A] time = 1.06, antiderivative size = 372, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5798, 5745, 5743, 5759, 5676, 30, 14} \[ \frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^4}-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {c x-1} \sqrt {c x+1}} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 5676
Rule 5743
Rule 5745
Rule 5759
Rule 5798
Rubi steps
\begin {align*} \int x^4 \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^4 (-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 {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int x^4 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^5 \left (-1+c^2 x^2\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int x^5 \, dx}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \left (-x^5+c^2 x^7\right ) \, dx}{8 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{64 c^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b d \sqrt {d-c^2 d x^2}\right ) \int x^3 \, dx}{64 c \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{128 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b d \sqrt {d-c^2 d x^2}\right ) \int x \, dx}{128 c^3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 b d x^2 \sqrt {d-c^2 d x^2}}{256 c^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b d x^4 \sqrt {d-c^2 d x^2}}{256 c \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d x^6 \sqrt {d-c^2 d x^2}}{32 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^3 d x^8 \sqrt {d-c^2 d x^2}}{64 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 d x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{128 c^4}-\frac {d x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{64 c^2}+\frac {1}{16} d x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{8} d x^5 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 d \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{256 b c^5 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A] time = 4.82, size = 337, normalized size = 0.94 \[ \frac {d \left (-1728 a \sqrt {d} \tan ^{-1}\left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (c^2 x^2-1\right )}\right )-576 a c x \left (16 c^6 x^6-24 c^4 x^4+2 c^2 x^2+3\right ) \sqrt {d-c^2 d x^2}+\frac {32 b \sqrt {d-c^2 d x^2} \left (-72 \cosh ^{-1}(c x)^2+18 \cosh \left (2 \cosh ^{-1}(c x)\right )-9 \cosh \left (4 \cosh ^{-1}(c x)\right )-2 \cosh \left (6 \cosh ^{-1}(c x)\right )+12 \cosh ^{-1}(c x) \left (-3 \sinh \left (2 \cosh ^{-1}(c x)\right )+3 \sinh \left (4 \cosh ^{-1}(c x)\right )+\sinh \left (6 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}+\frac {b \sqrt {d-c^2 d x^2} \left (1440 \cosh ^{-1}(c x)^2-576 \cosh \left (2 \cosh ^{-1}(c x)\right )+144 \cosh \left (4 \cosh ^{-1}(c x)\right )+64 \cosh \left (6 \cosh ^{-1}(c x)\right )+9 \cosh \left (8 \cosh ^{-1}(c x)\right )-24 \cosh ^{-1}(c x) \left (-48 \sinh \left (2 \cosh ^{-1}(c x)\right )+24 \sinh \left (4 \cosh ^{-1}(c x)\right )+16 \sinh \left (6 \cosh ^{-1}(c x)\right )+3 \sinh \left (8 \cosh ^{-1}(c x)\right )\right )\right )}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}\right )}{73728 c^5} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c^{2} d x^{6} - a d x^{4} + {\left (b c^{2} d x^{6} - b d x^{4}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 561, normalized size = 1.56 \[ -\frac {a \,x^{3} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8 c^{2} d}-\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{16 c^{4} d}+\frac {a x \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{64 c^{4}}+\frac {3 a d x \sqrt {-c^{2} d \,x^{2}+d}}{128 c^{4}}+\frac {3 a \,d^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{128 c^{4} \sqrt {c^{2} d}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{3} x^{8}}{64 \sqrt {c x +1}\, \sqrt {c x -1}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d c \,x^{6}}{32 \sqrt {c x +1}\, \sqrt {c x -1}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{4}}{256 \sqrt {c x +1}\, c \sqrt {c x -1}}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,x^{2}}{256 \sqrt {c x +1}\, c^{3} \sqrt {c x -1}}-\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (c x \right )^{2} d}{256 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{5}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{4} \mathrm {arccosh}\left (c x \right ) x^{9}}{8 \left (c x +1\right ) \left (c x -1\right )}+\frac {5 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,c^{2} \mathrm {arccosh}\left (c x \right ) x^{7}}{16 \left (c x +1\right ) \left (c x -1\right )}-\frac {13 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) x^{5}}{64 \left (c x +1\right ) \left (c x -1\right )}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) x^{3}}{128 \left (c x +1\right ) c^{2} \left (c x -1\right )}+\frac {3 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d \,\mathrm {arccosh}\left (c x \right ) x}{128 \left (c x +1\right ) c^{4} \left (c x -1\right )}-\frac {15 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d}{8192 \sqrt {c x +1}\, c^{5} \sqrt {c x -1}} \]
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 \[ -\frac {1}{128} \, {\left (\frac {16 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{3}}{c^{2} d} - \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x}{c^{4}} + \frac {8 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x}{c^{4} d} - \frac {3 \, \sqrt {-c^{2} d x^{2} + d} d x}{c^{4}} - \frac {3 \, d^{\frac {3}{2}} \arcsin \left (c x\right )}{c^{5}}\right )} a + b \int {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]
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 \[ \int x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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