Optimal. Leaf size=135 \[ \frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}-\frac {b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{25 d}-\frac {4 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{75 d}-\frac {8 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1}}{75 d} \]
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Rubi [A] time = 0.09, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5866, 12, 5662, 100, 74} \[ \frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}-\frac {b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{25 d}-\frac {4 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{75 d}-\frac {8 b e^4 \sqrt {c+d x-1} \sqrt {c+d x+1}}{75 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 100
Rule 5662
Rule 5866
Rubi steps
\begin {align*} \int (c e+d e x)^4 \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int e^4 x^4 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \operatorname {Subst}\left (\int x^4 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}-\frac {\left (b e^4\right ) \operatorname {Subst}\left (\int \frac {x^5}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d}\\ &=-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}-\frac {\left (b e^4\right ) \operatorname {Subst}\left (\int \frac {4 x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}-\frac {\left (4 b e^4\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{25 d}\\ &=-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}-\frac {\left (4 b e^4\right ) \operatorname {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{75 d}\\ &=-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}-\frac {\left (8 b e^4\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{75 d}\\ &=-\frac {8 b e^4 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{75 d}-\frac {4 b e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{75 d}-\frac {b e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{25 d}+\frac {e^4 (c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 103, normalized size = 0.76 \[ \frac {e^4 \left ((c+d x)^5 \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {4}{15} b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (c^2+2 c d x+d^2 x^2+2\right )-\frac {1}{5} b \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4\right )}{5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 279, normalized size = 2.07 \[ \frac {15 \, a d^{5} e^{4} x^{5} + 75 \, a c d^{4} e^{4} x^{4} + 150 \, a c^{2} d^{3} e^{4} x^{3} + 150 \, a c^{3} d^{2} e^{4} x^{2} + 75 \, a c^{4} d e^{4} x + 15 \, {\left (b d^{5} e^{4} x^{5} + 5 \, b c d^{4} e^{4} x^{4} + 10 \, b c^{2} d^{3} e^{4} x^{3} + 10 \, b c^{3} d^{2} e^{4} x^{2} + 5 \, b c^{4} d e^{4} x + b c^{5} e^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (3 \, b d^{4} e^{4} x^{4} + 12 \, b c d^{3} e^{4} x^{3} + 2 \, {\left (9 \, b c^{2} + 2 \, b\right )} d^{2} e^{4} x^{2} + 4 \, {\left (3 \, b c^{3} + 2 \, b c\right )} d e^{4} x + {\left (3 \, b c^{4} + 4 \, b c^{2} + 8 \, b\right )} e^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{75 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 5.61, size = 822, normalized size = 6.09 \[ \frac {1}{600} \, {\left (120 \, a d^{4} x^{5} + 600 \, a c d^{3} x^{4} + 1200 \, a c^{2} d^{2} x^{3} + 1200 \, a c^{3} d x^{2} - 600 \, {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c^{4} + 600 \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b c^{3} d + 200 \, {\left (6 \, x^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (x {\left (\frac {2 \, x}{d^{2}} - \frac {5 \, c}{d^{3}}\right )} + \frac {11 \, c^{2} d + 4 \, d}{d^{5}}\right )} + \frac {3 \, {\left (2 \, c^{3} + 3 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{3} {\left | d \right |}}\right )} d\right )} b c^{2} d^{2} + 25 \, {\left (24 \, x^{4} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (2 \, x {\left (\frac {3 \, x}{d^{2}} - \frac {7 \, c}{d^{3}}\right )} + \frac {26 \, c^{2} d^{3} + 9 \, d^{3}}{d^{7}}\right )} x - \frac {5 \, {\left (10 \, c^{3} d^{2} + 11 \, c d^{2}\right )}}{d^{7}}\right )} - \frac {3 \, {\left (8 \, c^{4} + 24 \, c^{2} + 3\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{4} {\left | d \right |}}\right )} d\right )} b c d^{3} + {\left (120 \, x^{5} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (2 \, {\left (3 \, x {\left (\frac {4 \, x}{d^{2}} - \frac {9 \, c}{d^{3}}\right )} + \frac {47 \, c^{2} d^{5} + 16 \, d^{5}}{d^{9}}\right )} x - \frac {7 \, {\left (22 \, c^{3} d^{4} + 23 \, c d^{4}\right )}}{d^{9}}\right )} x + \frac {274 \, c^{4} d^{3} + 607 \, c^{2} d^{3} + 64 \, d^{3}}{d^{9}}\right )} + \frac {15 \, {\left (8 \, c^{5} + 40 \, c^{3} + 15 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{5} {\left | d \right |}}\right )} d\right )} b d^{4} + 600 \, a c^{4} x\right )} e^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 78, normalized size = 0.58 \[ \frac {\frac {\left (d x +c \right )^{5} e^{4} a}{5}+e^{4} b \left (\frac {\left (d x +c \right )^{5} \mathrm {arccosh}\left (d x +c \right )}{5}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (3 \left (d x +c \right )^{4}+4 \left (d x +c \right )^{2}+8\right )}{75}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.39, size = 1241, normalized size = 9.19 \[ \text {result too large to display} \]
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 \[ \int {\left (c\,e+d\,e\,x\right )}^4\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.25, size = 527, normalized size = 3.90 \[ \begin {cases} a c^{4} e^{4} x + 2 a c^{3} d e^{4} x^{2} + 2 a c^{2} d^{2} e^{4} x^{3} + a c d^{3} e^{4} x^{4} + \frac {a d^{4} e^{4} x^{5}}{5} + \frac {b c^{5} e^{4} \operatorname {acosh}{\left (c + d x \right )}}{5 d} + b c^{4} e^{4} x \operatorname {acosh}{\left (c + d x \right )} - \frac {b c^{4} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25 d} + 2 b c^{3} d e^{4} x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {4 b c^{3} e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} + 2 b c^{2} d^{2} e^{4} x^{3} \operatorname {acosh}{\left (c + d x \right )} - \frac {6 b c^{2} d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {4 b c^{2} e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75 d} + b c d^{3} e^{4} x^{4} \operatorname {acosh}{\left (c + d x \right )} - \frac {4 b c d^{2} e^{4} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {8 b c e^{4} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75} + \frac {b d^{4} e^{4} x^{5} \operatorname {acosh}{\left (c + d x \right )}}{5} - \frac {b d^{3} e^{4} x^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{25} - \frac {4 b d e^{4} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75} - \frac {8 b e^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{75 d} & \text {for}\: d \neq 0 \\c^{4} e^{4} x \left (a + b \operatorname {acosh}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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