Optimal. Leaf size=85 \[ \frac {6 b \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{\sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}},x\right )}{e}-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e \sqrt {e (c+d x)}} \]
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Rubi [A] time = 0.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d e \sqrt {e (c+d x)}}+\frac {(6 b) \operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^2}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e}\\ \end {align*}
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Mathematica [A] time = 32.21, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{(c e+d e x)^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname {arcosh}\left (d x + c\right ) + a^{3}\right )} \sqrt {d e x + c e}}{d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}{\left (d e x +c e \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, b^{3} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3}}{\sqrt {d x + c} d e^{\frac {3}{2}}} - \frac {2 \, a^{3}}{\sqrt {d e x + c e} d e} + \int \frac {3 \, {\left ({\left ({\left (c^{3} - c\right )} a b^{2} + 2 \, {\left (c^{3} - c\right )} b^{3} + {\left (a b^{2} d^{3} + 2 \, b^{3} d^{3}\right )} x^{3} + 3 \, {\left (a b^{2} c d^{2} + 2 \, b^{3} c d^{2}\right )} x^{2} + {\left (2 \, b^{3} c^{2} + {\left (c^{2} - 1\right )} a b^{2} + {\left (a b^{2} d^{2} + 2 \, b^{3} d^{2}\right )} x^{2} + 2 \, {\left (a b^{2} c d + 2 \, b^{3} c d\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left ({\left (3 \, c^{2} d - d\right )} a b^{2} + 2 \, {\left (3 \, c^{2} d - d\right )} b^{3}\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2} + {\left (a^{2} b d^{3} x^{3} + 3 \, a^{2} b c d^{2} x^{2} + {\left (3 \, c^{2} d - d\right )} a^{2} b x + {\left (c^{3} - c\right )} a^{2} b + {\left (a^{2} b d^{2} x^{2} + 2 \, a^{2} b c d x + {\left (c^{2} - 1\right )} a^{2} b\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )\right )}}{{\left (d^{3} e^{\frac {3}{2}} x^{3} + 3 \, c d^{2} e^{\frac {3}{2}} x^{2} + c^{3} e^{\frac {3}{2}} - c e^{\frac {3}{2}} + {\left (3 \, c^{2} d e^{\frac {3}{2}} - d e^{\frac {3}{2}}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} + {\left (d^{4} e^{\frac {3}{2}} x^{4} + 4 \, c d^{3} e^{\frac {3}{2}} x^{3} + c^{4} e^{\frac {3}{2}} - c^{2} e^{\frac {3}{2}} + {\left (6 \, c^{2} d^{2} e^{\frac {3}{2}} - d^{2} e^{\frac {3}{2}}\right )} x^{2} + 2 \, {\left (2 \, c^{3} d e^{\frac {3}{2}} - c d e^{\frac {3}{2}}\right )} x\right )} \sqrt {d x + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{3}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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