Optimal. Leaf size=138 \[ \frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (25 c^2 d+12 e\right )}{225 c^5}-\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (25 c^2 d+12 e\right )}{225 c^3}-\frac {b e x^4 \sqrt {c x-1} \sqrt {c x+1}}{25 c} \]
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Rubi [A] time = 0.12, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5786, 460, 100, 12, 74} \[ \frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1} \left (25 c^2 d+12 e\right )}{225 c^3}-\frac {2 b \sqrt {c x-1} \sqrt {c x+1} \left (25 c^2 d+12 e\right )}{225 c^5}-\frac {b e x^4 \sqrt {c x-1} \sqrt {c x+1}}{25 c} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 100
Rule 460
Rule 5786
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{15} (b c) \int \frac {x^3 \left (5 d+3 e x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{75} \left (b c \left (25 d+\frac {12 e}{c^2}\right )\right ) \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b \left (25 c^2 d+12 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c^3}-\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b \left (25 c^2 d+12 e\right )\right ) \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{225 c^3}\\ &=-\frac {b \left (25 c^2 d+12 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c^3}-\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (2 b \left (25 c^2 d+12 e\right )\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{225 c^3}\\ &=-\frac {2 b \left (25 c^2 d+12 e\right ) \sqrt {-1+c x} \sqrt {1+c x}}{225 c^5}-\frac {b \left (25 c^2 d+12 e\right ) x^2 \sqrt {-1+c x} \sqrt {1+c x}}{225 c^3}-\frac {b e x^4 \sqrt {-1+c x} \sqrt {1+c x}}{25 c}+\frac {1}{3} d x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 101, normalized size = 0.73 \[ \frac {1}{225} \left (15 a x^3 \left (5 d+3 e x^2\right )-\frac {b \sqrt {c x-1} \sqrt {c x+1} \left (c^4 \left (25 d x^2+9 e x^4\right )+2 c^2 \left (25 d+6 e x^2\right )+24 e\right )}{c^5}+15 b x^3 \cosh ^{-1}(c x) \left (5 d+3 e x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 119, normalized size = 0.86 \[ \frac {45 \, a c^{5} e x^{5} + 75 \, a c^{5} d x^{3} + 15 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (9 \, b c^{4} e x^{4} + 50 \, b c^{2} d + {\left (25 \, b c^{4} d + 12 \, b c^{2} e\right )} x^{2} + 24 \, b e\right )} \sqrt {c^{2} x^{2} - 1}}{225 \, c^{5}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 115, normalized size = 0.83 \[ \frac {\frac {a \left (\frac {1}{5} c^{5} x^{5} e +\frac {1}{3} c^{5} x^{3} d \right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) c^{5} x^{5} e}{5}+\frac {\mathrm {arccosh}\left (c x \right ) c^{5} x^{3} d}{3}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (9 c^{4} e \,x^{4}+25 c^{4} d \,x^{2}+12 c^{2} x^{2} e +50 c^{2} d +24 e \right )}{225}\right )}{c^{2}}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 139, normalized size = 1.01 \[ \frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,\left (e\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.15, size = 178, normalized size = 1.29 \[ \begin {cases} \frac {a d x^{3}}{3} + \frac {a e x^{5}}{5} + \frac {b d x^{3} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {b e x^{5} \operatorname {acosh}{\left (c x \right )}}{5} - \frac {b d x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {b e x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {2 b d \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} - \frac {4 b e x^{2} \sqrt {c^{2} x^{2} - 1}}{75 c^{3}} - \frac {8 b e \sqrt {c^{2} x^{2} - 1}}{75 c^{5}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d x^{3}}{3} + \frac {e x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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