Optimal. Leaf size=89 \[ \frac {8 b \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^3}{\sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{3/2}},x\right )}{3 e}-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e (e (c+d x))^{3/2}} \]
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Rubi [A] time = 0.31, 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 = {} \[ \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{(c e+d e x)^{5/2}} \, dx \]
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 )^4}{(c e+d e x)^{5/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^4}{(e x)^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )^4}{3 d e (e (c+d x))^{3/2}}+\frac {(8 b) \operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt {-1+x} (e x)^{3/2} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e}\\ \end {align*}
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Mathematica [A] time = 39.12, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \cosh ^{-1}(c+d x)\right )^4}{(c e+d e x)^{5/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{4} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arcosh}\left (d x + c\right ) + a^{4}\right )} \sqrt {d e x + c e}}{d^{3} e^{3} x^{3} + 3 \, c d^{2} e^{3} x^{2} + 3 \, c^{2} d e^{3} x + c^{3} e^{3}}, x\right ) \]
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 [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{4}}{\left (d e x +c e \right )^{\frac {5}{2}}}\, dx \]
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 \, \sqrt {d x + c} b^{4} \sqrt {e} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{4}}{3 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} - \frac {2 \, a^{4}}{3 \, {\left (d e x + c e\right )}^{\frac {3}{2}} d e} + \int \frac {2 \, {\left (2 \, {\left ({\left (2 \, b^{4} c^{2} \sqrt {e} + 3 \, {\left (c^{2} \sqrt {e} - \sqrt {e}\right )} a b^{3} + {\left (3 \, a b^{3} d^{2} \sqrt {e} + 2 \, b^{4} d^{2} \sqrt {e}\right )} x^{2} + 2 \, {\left (3 \, a b^{3} c d \sqrt {e} + 2 \, b^{4} c d \sqrt {e}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} + {\left (3 \, {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} a b^{3} + 2 \, {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} b^{4} + {\left (3 \, a b^{3} d^{3} \sqrt {e} + 2 \, b^{4} d^{3} \sqrt {e}\right )} x^{3} + 3 \, {\left (3 \, a b^{3} c d^{2} \sqrt {e} + 2 \, b^{4} c d^{2} \sqrt {e}\right )} x^{2} + {\left (3 \, {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} a b^{3} + 2 \, {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} b^{4}\right )} x\right )} \sqrt {d x + c}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3} + 9 \, {\left ({\left (a^{2} b^{2} d^{2} \sqrt {e} x^{2} + 2 \, a^{2} b^{2} c d \sqrt {e} x + {\left (c^{2} \sqrt {e} - \sqrt {e}\right )} a^{2} b^{2}\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} + {\left (a^{2} b^{2} d^{3} \sqrt {e} x^{3} + 3 \, a^{2} b^{2} c d^{2} \sqrt {e} x^{2} + {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} a^{2} b^{2} x + {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} a^{2} b^{2}\right )} \sqrt {d x + c}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2} + 6 \, {\left ({\left (a^{3} b d^{2} \sqrt {e} x^{2} + 2 \, a^{3} b c d \sqrt {e} x + {\left (c^{2} \sqrt {e} - \sqrt {e}\right )} a^{3} b\right )} \sqrt {d x + c + 1} \sqrt {d x + c} \sqrt {d x + c - 1} + {\left (a^{3} b d^{3} \sqrt {e} x^{3} + 3 \, a^{3} b c d^{2} \sqrt {e} x^{2} + {\left (3 \, c^{2} d \sqrt {e} - d \sqrt {e}\right )} a^{3} b x + {\left (c^{3} \sqrt {e} - c \sqrt {e}\right )} a^{3} b\right )} \sqrt {d x + c}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )\right )}}{3 \, {\left (d^{6} e^{3} x^{6} + 6 \, c d^{5} e^{3} x^{5} + c^{6} e^{3} - c^{4} e^{3} + {\left (15 \, c^{2} d^{4} e^{3} - d^{4} e^{3}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{3} - c d^{3} e^{3}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{2} e^{3} - 2 \, c^{2} d^{2} e^{3}\right )} x^{2} + {\left (d^{5} e^{3} x^{5} + 5 \, c d^{4} e^{3} x^{4} + c^{5} e^{3} - c^{3} e^{3} + {\left (10 \, c^{2} d^{3} e^{3} - d^{3} e^{3}\right )} x^{3} + {\left (10 \, c^{3} d^{2} e^{3} - 3 \, c d^{2} e^{3}\right )} x^{2} + {\left (5 \, c^{4} d e^{3} - 3 \, c^{2} d e^{3}\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + 2 \, {\left (3 \, c^{5} d e^{3} - 2 \, c^{3} d e^{3}\right )} x\right )}}\,{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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4}{{\left (c\,e+d\,e\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )^{4}}{\left (e \left (c + d x\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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