Optimal. Leaf size=127 \[ \frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}}{9 d}-\frac {4 b \sqrt {e} \sqrt {-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d \sqrt {c+d x-1}} \]
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Rubi [A] time = 0.10, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5866, 5662, 102, 12, 117, 116} \[ \frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}}{9 d}-\frac {4 b \sqrt {e} \sqrt {-c-d x+1} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d \sqrt {c+d x-1}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 102
Rule 116
Rule 117
Rule 5662
Rule 5866
Rubi steps
\begin {align*} \int \sqrt {c e+d e x} \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {e x} \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^2}{2 \sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d e}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {(2 b e) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {\left (2 b e \sqrt {1-c-d x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d \sqrt {-1+c+d x}}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d e}-\frac {4 b \sqrt {e} \sqrt {1-c-d x} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right )\right |-1\right )}{9 d \sqrt {-1+c+d x}}\\ \end {align*}
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Mathematica [C] time = 0.46, size = 131, normalized size = 1.03 \[ \frac {\sqrt {e (c+d x)} \left (\frac {2}{3} (c+d x)^{3/2} \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {4 b \left (c^2+\sqrt {1-(c+d x)^2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};(c+d x)^2\right )+2 c d x+d^2 x^2-1\right )}{9 \sqrt {\frac {c+d x-1}{c+d x}} \sqrt {c+d x+1}}\right )}{d \sqrt {c+d x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {d e x + c e} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}, x\right ) \]
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 \[ \int \sqrt {d e x + c e} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 194, normalized size = 1.53 \[ \frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \mathrm {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \EllipticF \left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, \sqrt {d e x +c e}\, e^{2}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e} \]
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 \[ \frac {1}{9} \, {\left (\frac {6 \, {\left (d \sqrt {e} x + c \sqrt {e}\right )} \sqrt {d x + c} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{d} - \frac {4 \, {\left (d x + c\right )}^{\frac {3}{2}} \sqrt {e} - 3 i \, \sqrt {e} {\left (\log \left (i \, \sqrt {d x + c} + 1\right ) - \log \left (-i \, \sqrt {d x + c} + 1\right )\right )} - 3 \, \sqrt {e} \log \left (\sqrt {d x + c} + 1\right ) + 3 \, \sqrt {e} \log \left (\sqrt {d x + c} - 1\right )}{d} + 9 \, \int \frac {2 \, {\left (d \sqrt {e} x + c \sqrt {e}\right )} \sqrt {d x + c}}{3 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c\right )}}\,{d x}\right )} b + \frac {2 \, {\left (d e x + c e\right )}^{\frac {3}{2}} a}{3 \, d e} \]
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 \[ \int \sqrt {c\,e+d\,e\,x}\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]
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 \[ \int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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