Integrand size = 23, antiderivative size = 169 \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {20 b e^{5/2} \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{147 d \sqrt {-1+c+d x}} \]
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Time = 0.11 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5996, 5883, 104, 12, 118, 117} \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {20 b e^{5/2} \sqrt {-c-d x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{147 d \sqrt {c+d x-1}}-\frac {20 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}}{147 d}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} (e (c+d x))^{5/2}}{49 d} \]
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Rule 12
Rule 104
Rule 117
Rule 118
Rule 5883
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (e x)^{5/2} (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{7/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{7 d e} \\ & = -\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {(4 b) \text {Subst}\left (\int \frac {5 e^2 (e x)^{3/2}}{2 \sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{49 d e} \\ & = -\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {(10 b e) \text {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{49 d} \\ & = -\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {(20 b e) \text {Subst}\left (\int \frac {e^2}{2 \sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d} \\ & = -\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {\left (10 b e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d} \\ & = -\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {\left (10 b e^3 \sqrt {1-c-d x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{147 d \sqrt {-1+c+d x}} \\ & = -\frac {20 b e^2 \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{147 d}-\frac {4 b \sqrt {-1+c+d x} (e (c+d x))^{5/2} \sqrt {1+c+d x}}{49 d}+\frac {2 (e (c+d x))^{7/2} (a+b \text {arccosh}(c+d x))}{7 d e}-\frac {20 b e^{5/2} \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{147 d \sqrt {-1+c+d x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.88 \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{5/2} \left ((c+d x)^{7/2} (a+b \text {arccosh}(c+d x))+\frac {2 b \left (5 \left (1-(c+d x)^2\right )+3 (c+d x)^2 \left (1-(c+d x)^2\right )-5 \sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{21 \sqrt {\frac {-1+c+d x}{c+d x}} \sqrt {1+c+d x}}\right )}{7 d (c+d x)^{5/2}} \]
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Time = 3.84 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {5}{2}}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, e^{4} \sqrt {d e x +c e}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(218\) |
default | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {7}{2}} a}{7}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {5}{2}}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, e^{4} \sqrt {d e x +c e}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(218\) |
parts | \(\frac {2 a \left (d e x +c e \right )^{\frac {7}{2}}}{7 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {7}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{7}-\frac {2 \left (3 \sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {9}{2}}+2 \sqrt {-\frac {1}{e}}\, e^{2} \left (d e x +c e \right )^{\frac {5}{2}}+5 e^{4} \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-5 \sqrt {-\frac {1}{e}}\, e^{4} \sqrt {d e x +c e}\right )}{147 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\) | \(223\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.50 \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {2 \, {\left (10 \, \sqrt {d^{3} e} b e^{2} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) - 21 \, {\left (b d^{5} e^{2} x^{3} + 3 \, b c d^{4} e^{2} x^{2} + 3 \, b c^{2} d^{3} e^{2} x + b c^{3} d^{2} e^{2}\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + 2 \, {\left (3 \, b d^{4} e^{2} x^{2} + 6 \, b c d^{3} e^{2} x + {\left (3 \, b c^{2} + 5 \, b\right )} d^{2} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} \sqrt {d e x + c e} - 21 \, {\left (a d^{5} e^{2} x^{3} + 3 \, a c d^{4} e^{2} x^{2} + 3 \, a c^{2} d^{3} e^{2} x + a c^{3} d^{2} e^{2}\right )} \sqrt {d e x + c e}\right )}}{147 \, d^{3}} \]
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Timed out. \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=\text {Timed out} \]
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Exception generated. \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=\text {Exception raised: ValueError} \]
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\[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=\int { {\left (d e x + c e\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (c e+d e x)^{5/2} (a+b \text {arccosh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^{5/2}\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]
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